Martin R. Bridson
Andre Haefliger
Metric Spaces
of
Non-Positive Curvature
With 84 Figures
Шй Springer

Martin R. Bridson
University of Oxfod
Mathematical Institute
24-29 St. Giles'
Oxford OXi 3LB
Great Britain
email: bridson@maths.ox.ac.uk
Andre Haefliger
Universite de Geneve
Section de Mathematiques
2-4 Rue du Lievre, CP-240
CH-1211 Geneve 24
Switzerland
email: Andre.Haefliger@math.unige.ch
Library of Congress Cataloging-in-Publication Data
Bridson, Martin R., 1964-
Metric spaces of non-positive curvature / Martin R. Bridson, Andre Haefliger.
p. cm. - (Grundlehren der mathematischen Wissenschaften, ISSN 0072-7830; 319)
includes bibliographical references and index.
ISBN 3-540-64324-9 (hardcover: alk. paper)
1. Metric spaces. 2. Geometry, Differential. I. Haefliger, Andre. II. Title. III. Series.
QA611.28.B75 1999 5i4'.32-dc2i 99-38163 CIP
Mathematics Subject Classification (1991): 57023, 20F32, 57MXX, 53C70
ISSN 0072-7830
ISBN 3-540-64324-9 Springer-Verlag Berlin Heidelberg New York
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For Julie and Minouche

Introduction
The purpose of this book is to describe the global properties of complete simply-
connected spaces that are non-positively curved in the sense of A. D. Alexandrov and
to examine the structure of groups that act properly on such spaces by isometries.
Thus the central objects of study are metric spaces in which every pair of points can
be joined by an arc isometric to a compact interval of the real line and in which every
triangle satisfies the CAT(O) inequality. This inequality encapsulates the concept
of non-positive curvature in Riemannian geometry and allows one to reflect the
same concept faithfully in a much wider setting — that of geodesic metric spaces.
Because the CAT(O) condition captures the essence of non-positive curvature so well,
spaces that satisfy this condition display many of the elegant features inherent in the
geometry of non-positively curved manifolds. There is therefore a great deal to be
said about the global structure of CAT(O) spaces, and also about the structure of
groups that act on them by isometries — such is the theme of this book.
The origins of our study lie in the fundamental work of A. D. Alexandrov1.
He gave several equivalent definitions of what it means for a metric space to have
curvature bounded above by a real number к. Let us begin by explaining one of
Alexandrov's definitions; this formulation has been given prominence by M. Gromov,
who termed it the CAT(/c) inequality. (The initial A is in honour of Alexandrov, and
the initials С and Τ are in honour of E. Cartan and A. Toponogov, each of whom
made an important contribution to the understanding of curvature via inequalities for
the distance function.)
Given a real number к, let M2 denote the following space: if к < 0 then M2 is real
hyperbolic space H2 with the distance function scaled by a factor of \/у/—к; if к = 0
then M2 is the Euclidean plane; if к > 0 then M2 is the 2-sphere S2 with the metric
scaled by a factor 1/-s/k. Alexandrov pointed out that one could define curvature
bounds on a space by comparing triangles in that space to triangles in M2. A natural
class of spaces in which to study triangles is the following. A metric space X is called
a geodesic space if every pair of points x,y e X can be joined by a continuous path
of length d(x, y); the image of such a path is called a geodesic segment. In general
there may be many geodesic segments joining χ to y, but nevertheless it is convenient
to use the notation [x, y] for a choice of such a segment. A geodesic triangle Δ in X
consists of three points x, y, ζ € X and three geodesic segments [x, у], [y, z], [z, x]. A
comparison triangle for Δ in M2 is a geodesic triangle Δ in M2 with vertices x, y, ζ
' In 1957 Alexandrov wrote an article summarizing his ideas [Ale57]

VIII Introduction
such that d(x, y) = d(x, y), d(y, z) = d(y, z) and d{z, x) = d(z, x). (If к < 0 then such
a Δ always exists; if к > 0 then it exists provided the perimeter of Δ is less than
2π/*/κ; in both cases it is unique up to an isometry of M2.) The point ρ e \x,y] is
called a comparison point in Δ for ρ e [x, y] ifd(x, p) = d(x, p). Comparison points
on \y, z] and [z, x] are defined similarly. A geodesic space X is said to satisfy the
CAT(/c) inequality (more briefly, X is а С AT (к) space) if, for all geodesic triangles
ΔίηΧ,
d(p, q) < d(p, q)
for all comparison points p, q e Δ с Μ2.
Alexandrov defines a metric space to be of curvature < к if each point of the
space has a neighbourhood which, equipped with the induced metric, is a CAT(/c)
space. He and the Russian school which he founded have made an extensive study of
the local properties of such spaces. A complete Riemannian manifold has curvature
< к in the above sense if and only if all of its sectional curvatures are < к. The
main point of making the above definition, though, is that there are many examples
of spaces other than Riemannian manifolds whose curvature is bounded above. An
interesting class of non-positively curved polyhedral complexes is provided by the
buildings of Euclidean (or affine) type which arose in the work of Bruhat and Tits on
algebraic groups. Many other examples will be described in the course of this book.
In recent years, CAT(—1) and CAT(O) spaces have come to play an important
role both in the study of groups from a geometrical viewpoint and in the proofs of
certain rigidity theorems in geometry. This is due in large part to the influence of
Mikhael Gromov. Of particular importance are the lectures which Gromov gave in
February 1981 at College de France in Paris. In these lectures (an excellent account
of which was written by Viktor Schroeder [BGS95]) Gromov explained the main
features of the global geometry of manifolds of non-positive curvature, essentially
by basing his account on the CAT(O) inequality. In the present book we shall pursue
this approach further in order to describe the global properties of С AT(0) spaces and
the structure of groups which act on them by isometries. Two particular features of
our treatment are that we give very detailed proofs of the basic theorems, and we
describe many examples.
We have divided our book into three parts: Part I is an introduction to the geometry
of geodesic spaces, in Part II the basic theory of spaces with upper curvature bounds
is developed, and more specialized topics are covered in Part III. We shall now
outline the contents of each part. Before doing so, we should emphasize that many
of the chapters can be read independently, and we therefore suggest that if you are
particularly interested in the material from a certain chapter, then you should turn
directly to that chapter. (References are given in the text whenever material from
earlier chapters2 is needed.)
2 We shall write (7.11) to direct readers to item 7.11 in the part of the book that they are
reading, and (1.7.11) to direct readers to item 7.11 in Part I. The chapters in Part III are
labelled by letters and subdivided into smaller sections, giving rise to references of the form
(Ш.Г.7.11).

Introduction IX
In Part I we examine such basic concepts as distance (metric spaces), geodesies,
the length of a curve, length (inner) metrics, and the notion of the (upper) angle
between two geodesies issuing from the same point in a metric space. (This concept
of angle, which is due to Alexandrov, plays an essential role throughout the book.)
Part I also contains various examples of geodesic spaces. Of these, the most important
are the model spaces M", which we introduce in Chapter 1.2 and study further in 1.6.
One can describe M" as the complete, simply connected, Riemannian и-manifold of
constant sectional curvature к. However, in keeping with the spirit of this book, we
shall define M" directly as a metric space and deduce the desired properties of the
space and its group of isometries directly from this definition.
We shall augment the supply of basic examples in Part I by describing several
methods for constructing new examples of geodesic metric spaces out of more
familiar ones: products, gluing, cones, spherical joins, quotients, induced path metrics and
limits. Most of these constructions are due to Alexandrov and the Russian school.
In Chapter 1.7 we shall describe the general properties of geometric complexes, as
established by Bridson in his thesis. And in the final chapter of Part I we shall turn
our attention to groups: after gathering some basic facts about group actions, we
shall describe some of the basic ideas in geometric group theory.
In Part II we set about our main task — exploring the geometry of CAT(/c)
spaces. We shall give several different formulations of the CAT(/c) condition, all due
to Alexandrov, and prove that they are equivalent. One quickly sees that CAT(/c)
spaces enjoy significant properties. For example, one can see almost immediately
that in a complete С АТ(0) space angles exist in a strong sense, the distance function is
convex, every bounded set has a unique circumcentre, one has orthogonal projections
onto closed convex subsets, etc. Early in Part II we shall also examine how CAT(/c)
spaces behave with regard to the basic constructions introduced in Chapter 1.5.
Following these basic considerations, we turn our attention to a richer circle of
ideas based on a key observation of Alexandrov: when considering a triangle Δ
in a complete CAT(O) space X, if one gets any non-trivial equality in the CAT(O)
condition, then Δ spans an isometrically embedded Euclidean triangle in X. This
observation leads quickly to results concerning the existence of flat polygons and
fiat strips, and thence a product decomposition theorem.
Much of the force and elegance of the theory of non-positively curved spaces
rests on the fact that there is a local-to-global theorem which allows one to use
local information about the space to make deductions about the global geometry
of its universal cover and about the structure of groups which act by isometries
on the universal cover. More precisely, we have the following generalization of
the Cartan-Hadamard theorem: for к < 0, a complete simply-connected geodesic
space satisfies the CAT(/c) inequality locally if and only if it satisfies the CAT(/c)
inequality globally. (In Chapter II.4, following a proof of Alexander and Bishop,
we shall actually prove a more general statement concerning metric spaces whose
metrics are locally convex.)

X Introduction
A more concise account of much of the material presented in Chapters Π. 1-ΙΙ.4
and Π.8-Π.9 of the present book can be found in the first two chapters of Ballmann's
lecture notes [Ba95].
Even if one were ultimately interested only in CAT(O) spaces, there are aspects of
the subject that force one to consider geodesic metric spaces satisfying the CAT(/c)
inequality for arbitrary к. An important link between CAT(O) spaces and CAT(l)
spaces is provided by a theorem of Berestovskii, which shows that the Euclidean
cone CqY over a geodesic space Υ is a CAT(O) space if and only if Υ is a CAT(l)
space. (A similar statement holds with regard to the к -cone CK Y, where к is arbitrary.)
This theorem is used in Chapter II.5 to establish the link condition, a necessary and
sufficient condition (highlighted by Gromov) which translates questions concerning
the existence of CAT(O) metrics on polyhedral complexes into questions concerning
the structure of links of vertices. The importance of the link condition is that in many
circumstances (particularly in dimension two) it provides a practical method for
deciding if a given complex supports a metric of non-positive curvature. Thus we are
able to construct interesting examples. Two-dimensional complexes are a particularly
rich source of examples, partly because the link condition is easier to check than
in higher dimensions, but also because the connections between group theory and
geometry are closest in dimension two, and in dimension two any subcomplex of a
non-positively curved complex is itself non-positively curved.
In Chapter II.6 we begin our study of groups that act by isometries on CAT(O)
spaces. First we establish basic properties of individual isometries and groups of
isometries. Individual isometries are divided into three classes according to the
behaviour of their displacement functions. If the displacement function is constant then
the isometry is called a Clifford translation. The Clifford translations of a CAT(O)
space X form a pre-Hilbert space H, which is a generalization of the Euclidean de
Rham factor in Riemannian geometry: if X is complete then there is an isometric
splitting X = Χ' χ Η. We also show that the group of isometries of a compact
non-positively curved space is a topological group with finitely many connected
components, the component of the identity being a torus.
In the early nineteen seventies, Gromoll-Wolf and Lawson-Yau proved several
striking theorems concerning the structure of those groups that are the
fundamental groups of compact non-positively curved Riemannian manifolds, including the
Flat Torus Theorem, the Solvable Subgroup Theorem and the Splitting Theorem.
In Chapters II.6 and II.7 we generalize these results to the case of groups that act
properly and cocompactly by isometries on CAT(O) spaces. These generalizations
have a variety of applications to group theory and topology.
In Chapters II. 8 and II.9 we explore the geometry at infinity in CAT(0) spaces.
Associated to any complete CAT(O) space one has a boundary at infinity ЭХ, which
can be constructed as the set of equivalence classes of geodesic rays in X, two rays
being considered equivalent if their images are a bounded distance apart. There is
a natural topology on X = X U 3X called the cone topology. If X is complete and
locally compact, X is compact. If X is a Riemannian manifold, X is homeomorphic
to a closed ball, but for more general CAT(O) spaces the topology of ЭХ can be rather

Introduction XI
complicated. An alternative construction of X is obtained by taking the closure of X
in the Banach space C*X of continuous functions on X modulo additive constants,
where X is embedded in C*X by the map that assigns to χ e X the class of the function
у ι-* d(x, у). In this description of X the points of dX emerge as classes of Busemann
functions, and we are led to examine the geometry of horoballs in CAT(O) spaces.
There is a natural metric Ζ on the set dX: given ξ, μ e ЭХ, one takes the
supremum over all points ρ e X of the angle between the geodesies issuing from ρ
in the classes ξ and μ. The topology on dX associated to this metric is in general
weaker than the cone topology. (For instance if X is a CAT(—1) space, one gets the
discrete topology.) We shall explain two significant facts concerning Z: first, if X is a
complete CAT(O) space then (ЭХ, Ζ) is a CAT(l) space; secondly, the length metric
associated to Z, called the Tits metric, encodes the geometry of flat subspaces in X,
in particular it determines how X can split as a product.
In Chapter ΠΙ.Η we shall revisit the study of boundaries in the context of Gro-
mov's 5-hyperbolic spaces. In the context of CAT(O) spaces, the 5-hyperbolic
condition is closely related to the idea of a visibility space, which was introduced in the
context of smooth manifolds by Eberlein and O'Neill. Intuitively speaking, visibility
spaces are "negatively curved on the large scale". In Chapter II.9 we shall see that
if a proper CAT(0) space X admits a cocompact group of isometries, then X is a
visibility space if and only if it does not contain an isometrically embedded copy of
the Euclidean plane.
The main purpose of the remaining three chapters in Part II is to provide examples
of CAT(O) spaces: in Chapter II. 11 we describe various gluing techniques that allow
one to build new examples out of more classical ones; in Chapter 11.10 we describe
elements of the geometry of symmetric spaces of non-compact type in terms of the
metric approach to curvature developed in earlier chapters; and in Chapter 11.12
we introduce simple complexes of groups as a forerunner to the general theory of
complexes of groups developed in Chapter III.C.
Complexes of groups were introduced by Haefliger to describe group actions
on simply-connected polyhedral complexes in terms of suitable local data on the
quotient. They are a natural generalization of the concept of a graph of groups, which
is due to Bass and Serre. In order to work effectively with polyhedral complexes in
this context, one needs a combinatorial description of them; the appropriate object
to focus on is the partially ordered set of cells in the first barycentric subdivision of
the complex, which provides the motivating example for objects that we call scwols
(small categories without loops).
Associated to any action of a group on a scwol there is a complex of groups
over the quotient scwol. If a complex of groups arises from such an action, it is
said to be developable. In contrast to the one-dimensional case (graphs of groups),
complexes of groups are not developable in general. However, if a complex of groups
is non-positively curved, in a suitable sense, then it is developable.
The foundations of the theory of complexes of groups are laid out in Chapter
III.C. The developability theorem for non-positively curved complexes of groups is

XII Introduction
proved in Chapter IH.Q, where it is treated in the more general context of groupoids
of local isometries.
There are two other chapters in Part III. In the first, Chapter ΠΙ.Η, we describe
elements of Gromov's theory of 5-hyperbolic metric spaces and discuss the
relationship between non-positive curvature and isoperimetric equalities. In the second,
Chapter ΙΠ.Γ, we shall delve more deeply into the nature of groups that act properly
and cocompactly by isometries on CAT(O) spaces. In particular, we shall analyse
the algorithmic properties of such groups and explore the diverse nature of their
subgroups. We shall also show that many theorems concerning groups of isometries
of CAT(O) spaces can be extended to larger classes of groups — hyperbolic and
semihyperbolic groups. The result is a substantial (but not comprehensive) account
of the role which non-positive curvature plays in geometric group theory.
Having talked at some length about what this book contains, we should say a few
words about what it does not contain. First we should point out that besides defining
what it means for a metric space to have curvature bounded above, Alexandrov also
defined what it means for a metric space to have curvature bounded below by a real
number к. (Не did so essentially by imposing the reverse of the CAT(/c) inequality.)
The theory of spaces with lower curvature bounds, particularly their local properties,
has been developed extensively by Alexandrov and the Russian school, and such
spaces play an important role in the study of collapsing for Riemannian manifolds.
We shall not consider the theory of such spaces at all in this book, instead we refer the
reader to the excellent survey article of Burago, Gromov and Perel'man [BGP92].
We should also make it clear that our treatment of the theory of non-positively
curved spaces is by no means exhaustive; the study of such spaces continues to be
a highly active field of research, encompassing many topics that we do not cover in
this book. In particular, we do not discuss the conformal structure on the boundary
of a CAT(— 1) space, nor do we discuss the construction of Patterson measures at
infinity, the geodesic flow in singular spaces of non-positive curvature, the theory of
harmonic maps into CAT(O) spaces, rigidity theorems etc.
It is our intention that the present book should be able to serve as an introductory
text. Although we shall arrive at non-trivial results, our lines of reasoning will be
elementary, and we have written with the intention of making the material accessible
to students whose background encompasses little more than a reasonable course in
topology and an acquaintance with the basic concepts of group theory. Thus, for
example, we expect the reader to understand what a manifold is and to be familiar
with the definition of the fundamental group of a space, but a nodding acquaintance
with the notion of a Riemannian metric will be quite sufficient for a complete
understanding of this book. In any case, all such knowledge will be much less important
than an enthusiasm for direct geometric arguments.
Acknowledgements: We thank the many colleagues whose comments helped to
improve the content and exposition of the material presented in this book. In particular
we thank Dick Bishop, Marc Burger, Mike Davis, Thomas Delzant, David Epstein,

Introduction XIII
Pierre de la Harpe, Panos Papasoglu, Frederic Paulin, John Roe, Ralph Strebel and
Dani Wise.
We offer our heartfelt gratitude to Felice Ronga for his invaluable assistance in
preparing this book for publication. We are particularly grateful for the long days that
he spent transforming our rough sketches into the many figures that accompany the
text and for his help in solving the many word processing problems we encountered.
We thank the Swiss National Science Foundation for its financial support and we
thank the mathematics department at the University of Geneva for providing us with
the facilities and equipment needed to prepare this book.
The first author thanks the Engineering and Physical Sciences Research Council
of Great Britain for the Advanced Fellowship which currently supports his research.
The National Science Foundation of America, the Alfred P. Sloane Foundation, and
the Frank Buckley Foundation have also provided financial support for Bridson's
work during the course of this project, and it is with pleasure that he takes this
opportunity to thank each of them. Above all, he thanks his wife Julie Lynch Bridson
for her constant love and support, and he thanks Minouche and Andre Haefliger for
welcoming him so warmly into their home.
The second author expresses his deep gratitude to his wife Minouche for her
unconditional support during his career, and for her dedication and interest during
the preparation of this book.
Geneva, March 1999
MRB, AH

Table of Contents
Introduction VII
Part I. Geodesic Metric Spaces 1
1. Basic Concepts 2
Metric Spaces 2
Geodesies 4
Angles 8
The Length of a Curve 12
2. The Model Spaces MnK 15
Euclidean и-Space E" 15
The и-Sphere S" 16
Hyperbolic и-Space W 18
The Model Spaces M'' 23
Alexandrov's Lemma 24
The Isometry Groups Isom(M") 26
Approximate Midpoints 30
3. Length Spaces 32
Length Metrics 32
The Hopf-Rinow Theorem 35
Riemannian Manifolds as Metric Spaces 39
Length Metrics on Covering Spaces 42
Manifolds of Constant Curvature 45
4. Normed Spaces 47
Hubert Spaces 47
Isometries of Normed Spaces 51
V Spaces 53
5. Some Basic Constructions 56
Products 56
к -Cones 59

XVI Table of Contents
Spherical Joins 63
Quotient Metrics and Gluing 64
Limits of Metric Spaces 70
Ultralimits and Asymptotic Cones 77
6. More on the Geometry of M^ 81
The Klein Model for Ш" 81
The Mobius Group 84
The Poincare Ball Model for W 86
The Poincare Half-Space Model for W 90
Isometries of H2 91
M" as a Riemannian Manifold 92
7. MK-Polyhedral Complexes 97
Metric Simplicial Complexes 97
Geometric Links and Cone Neighbourhoods 102
The Existence of Geodesies 105
The Main Argument 108
Cubical Complexes Ill
MK -Polyhedral Complexes 112
Barycentric Subdivision 115
More on the Geometry of Geodesies 118
Alternative Hypotheses 122
Appendix: Metrizing Abstract Simplicial Complexes 123
8. Group Actions and Quasi-Isometries 131
Group Actions on Metric Spaces 131
Presenting Groups of Homeomorphisms 134
Quasi-Isometries 138
Some Invariants of Quasi-Isometry 142
The Ends of a Space 144
Growth and Rigidity 148
Quasi-Isometries of the Model Spaces 150
Approximation by Metric Graphs 152
Appendix: Combinatorial 2-Complexes 153
Part П. САТ(к) Spaces 157
1. Definitions and Characterizations of САТ(к) Spaces 158
The CAT(/c) Inequality 158
Characterizations of CAT(/c) Spaces 161
CAT(/c) Implies CAT(V) if к < к' 165
Simple Examples of CAT(/c) Spaces 167

Table of Contents XVII
Historical Remarks 168
Appendix: The Curvature of Riemannian Manifolds 169
2. Convexity and Its Consequences 175
Convexity of the Metric 175
Convex Subspaces and Projection 176
The Centre of a Bounded Set 178
Flat Subspaces 180
3. Angles, Limits, Cones and Joins 184
Angles in CAT(/f) Spaces 184
4-Point Limits of CAT(/c) Spaces 186
Cones and Spherical Joins 188
The Space of Directions 190
4. The Cartan-Hadamard Theorem 193
Local-to-Global 193
An Exponential Map 196
Alexandrov's Patchwork 199
Local Isometries and 7Ti-Injectivity 200
Injectivity Radius and Systole 202
5. MK-Polyhedral Complexes of Bounded Curvature 205
Characterizations of Curvature < к 206
Extending Geodesies 207
Flag Complexes 210
Constructions with Cubical Complexes 212
Two-Dimensional Complexes 215
Subcomplexes and Subgroups in Dimension 2 216
Knot and Link Groups 220
From Group Presentations to Negatively Curved 2-Complexes 224
6. Isometries of CAT(0) Spaces 228
Individual Isometries 228
On the General Structure of Groups of Isometries 233
Clifford Translations and the Euclidean de Rham Factor 235
The Group of Isometries of a Compact Metric Space
of Non-Positive Curvature 237
A Splitting Theorem 239
7. The Flat Torus Theorem 244
The Flat Torus Theorem 244
Cocompact Actions and the Solvable Subgroup Theorem 247
Proper Actions That Are Not Cocompact 250
Actions That Are Not Proper 254
Some Applications to Topology 254

XVIII Table of Contents
8. The Boundary at Infinity of a CAT(O) Space 260
Asymptotic Rays and the Boundary ЭХ 260
The Cone Topology on X = X U ЭХ 263
Horofunctions and Busemann Functions 267
Characterizations of Horofunctions 271
Parabolic Isometries 274
9. The Tits Metric and Visibility Spaces 277
Angles in X 278
The Angular Metric 279
The Boundary (ЭХ, Ζ) is a CAT(l) Space 285
The Tits Metric 289
How the Tits Metric Determines Splittings 291
Visibility Spaces 294
10. Symmetric Spaces 299
Real, Complex and Quaternionic Hyperbolic и-Spaces 300
The Curvature of KH" 304
The Curvature of Distinguished Subspaces of KH" 306
The Group of Isometries of KH" 307
The Boundary at Infinity and Horospheres in KH" 309
Horocyclic Coordinates and Parabolic Subgroups for KH" 311
The Symmetric Space P(n, K) 314
P(n, K) as a Riemannian Manifold 314
The Exponential Map exp: M(n, K) -> GL(n, K) 316
P(n, Д) is a CAT(0) Space 318
Flats, Regular Geodesies and Weyl Chambers 320
The Iwasawa Decomposition of GL(n, K) 323
The Irreducible Symmetric space P(n, K)i 324
Reductive Subgroups of GL(n, K) 327
Semi-Simple Isometries 331
Parabolic Subgroups and Horospherical Decompositions of P(n, M.) ... 332
The Tits Boundary of P(n, K)i is a Spherical Building 337
djP{n, Ж) in the Language of Flags and Frames 340
Appendix: Spherical and Euclidean Buildings 342
11. Gluing Constructions 347
Gluing CAT(/c) Spaces Along Convex Subspaces 347
Gluing Using Local Isometries 350
Equivariant Gluing 355
Gluing Along Subspaces that are not Locally Convex 359
Truncated Hyperbolic Spaces 362
12. Simple Complexes of Groups 367
Stratified Spaces 368

Table of Contents XIX
Group Actions with a Strict Fundamental Domain 372
Simple Complexes of Groups: Definition and Examples 375
The Basic Construction 381
Local Development and Curvature 387
Constructions Using Coxeter Groups 391
Part III. Aspects of the Geometry of Group Actions 397
H. (5-Hyperbolic Spaces 398
1. Hyperbolic Metric Spaces 399
The Slim Triangles Condition 399
Quasi-Geodesics in Hyperbolic Spaces 400
fc-Local Geodesies 405
Reformulations of the Hyperbolicity Condition 407
2. Area and Isoperimetric Inequalities 414
A Coarse Notion of Area 414
The Linear Isoperimetric Inequality and Hyperbolicity 417
Sub-Quadratic Implies Linear 422
More Refined Notions of Area 425
3. The Gromov Boundary of a S-Hyperbolic Space 427
The Boundary ЭХ as a Set of Rays 427
The Topology on X U ЭХ 429
Metrizing ЭХ 432
Г. Non-Positive Curvature and Group Theory 438
1. Isometries of CAT(O) Spaces 439
A Summary of What We Already Know 439
Decision Problems for Groups of Isometries 440
The Word Problem 442
The Conjugacy Problem 445
2. Hyperbolic Groups and Their Algorithmic Properties 448
Hyperbolic Groups 448
Dehn's Algorithm 449
The Conjugacy Problem 451
Cone Types and Growth 455
3. Further Properties of Hyperbolic Groups 459
Finite Subgroups 459
Quasiconvexity and Centralizers 460
Translation Lengths 464
Free Subgroups 467
The Rips Complex 468

XX Table of Contents
4. Semihyperbolic Groups 471
Definitions 471
Basic Properties of Semihyperbolic Groups 473
Subgroups of Semihyperbolic Groups 475
5. Subgroups of Cocompact Groups of Isometries 481
Finiteness Properties 481
The Word, Conjugacy and Membership Problems 487
Isomorphism Problems 491
Distinguishing Among Non-Positively Curved
Manifolds 494
6. Amalgamating Groups of Isometries 496
Amalgamated Free Products and HNN Extensions 497
Amalgamating Along Abelian Subgroups 500
Amalgamating Along Free Subgroups 503
Subgroup Distortion and the Dehn Functions
of Doubles 506
7. Finite-Sheeted Coverings and Residual Finiteness 511
Residual Finiteness 511
Groups Without Finite Quotients 514
С Complexes of Groups 519
1. Small Categories Without Loops (Scwols) 520
Scwols and Their Geometric Realizations 521
The Fundamental Group and Coverings 526
Group Actions on Scwols 528
The Local Structure of Scwols 531
2. Complexes of Groups 534
Basic Definitions 535
Developability 538
The Basic Construction 542
3. The Fundamental Group of a Complex of Groups 546
The Universal Group FGQ?) 546
The Fundamental Group π\ (G(y), σ0) 548
A Presentation of n](G(y), σ0) 549
The Universal Covering of a Developable Complex of Groups 553
4. Local Developments of a Complex of Groups 555
The Local Structure of the Geometric Realization 555
The Geometric Realization of the Local Development 557
Local Development and Curvature 562
The Local Development as a Scwol 564
5. Coverings of Complexes of Groups 566
Definitions 566

Table of Contents XXI
The Fibres of a Covering 568
The Monodromy 572
A Appendix: Fundamental Groups and Coverings
of Small Categories 573
Basic Definitions 574
The Fundamental Group 576
Covering of a Category 579
The Relationship with Coverings of Complexes of Groups 583
Q. Groupoids of local Isometries 584
1. Orbifolds 585
Basic Definitions 585
Coverings of Orbifolds 589
Orbifolds with Geometric Structures 591
2. Etale Groupoids, Homomorphisms and Equivalences 594
Etale Groupoids 594
Equivalences and Developability 597
Groupoids of Local Isometries 601
Statement of the Main Theorem 603
3. The Fundamental Group and Coverings of Etale Groupoids .... 604
Equivalence and Homotopy of £-Paths 604
The Fundamental Group tt\({G, X), *o) 607
Coverings 609
4. Proof of the Main Theorem 613
Outline of the Proof 613
^-Geodesies 614
The Space X of ^-Geodesies Issuing from a Base Point 616
The Space X = X/Q 617
The Covering;? : X -* X 618
References 620
Index 637

Part I. Geodesic Metric Spaces
This part of the book is an introduction to the geometry of geodesic spaces. The
ideas that you will find here are elementary, and we have written with the intention
of making all of the material accessible to first year graduate students.
Our treatment begins with such basic concepts as distance, length, geodesic,
and angle. In Chapter 1.2 we introduce the model spaces M" (as metric spaces) and
establish basic facts about their geometry and their isometry groups; further aspects
of their geometry are explained in Chapter 1.6. In the intervening chapters we present
various examples of geodesic spaces, establish basic facts about length metrics, and
describe methods for manufacturing new spaces out of more familiar ones. Chapter
1.7 is an introduction to the geometry of metric polyhedral complexes, and Chapter
1.8 is an introduction to some of the basic ideas in geometric group theory.
Each chapter can be read independently (modulo references to earlier definitions).

Chapter 1.1 Basic Concepts
The fundamental concept with which we shall be concerned throughout this book is
that of distance. We shall explore the geometry of spaces whose distance functions
possess various properties that compare favourably to those of the distance function
in Euclidean space. If such properties are identified and articulated clearly, then by
proceeding in simple steps from these defining properties one can recover much of
the elegant structure of classical geometry, now adapted to a wider context.
Metric Spaces
We begin by recalling the basic properties normally required of distance functions.
1.1 Definitions. Let X be a set. A pseudometric on X is a real-valued function
d:XxX->K satisfying the following properties, for all x, y, ζ € X:
Positivity: d(x, y) > 0 and d(x, x) = 0.
Symmetry: d(x, y) = d(y, x).
Triangle Inequality: d(x, y) < d(x, z) + d(z, y).
A pseudometric is called a metric if it is positive definite, i.e.
d(x, y) > Oif χ φ у.
We shall often refer to d(x, y) as the distance between the points χ and y. A metric
space is a pair (X, d), where X is a set and d is a metric on X. A metric space is
said to be complete if every Cauchy sequence in it converges. If Υ is a subset of X,
then the restriction of d to Υ χ Υ is called the induced metric on Y. Given χ e X
and r > 0, the open ball of radius r about χ (i.e. the set {y | d(x, y) < r}) shall be
denoted B(x, r), and the closed ball {y | d(x, y) < r} shall be denoted B(x, r). (Note
that B(x, r) may be strictly larger than the closure of B(x, r).) Associated to the metric
d one has the topology with basis the set of open balls B(x, r). The metric space is
said to be proper if, in this topology, for every χ e X and every r > 0, the closed ball
B(x, r) is compact.
An isometry from one metric space (X, d) to another (X', d') is a bijection/ :
X -> X' such that d'(f(x),f(y)) = d(x, y) for all x, у е X. If such a map exists then

Metric Spaces 3
(X, d) and (X', d') are said to be isometric. The group of all isometries from a metric
space (X, d) to itself will be denoted Isom(X, d) or, more briefly, Isom(X).
1.2 First Examples. Perhaps the easiest example of a metric space is obtained by
taking the intuitive notion of distance d(x, y) = \x—y\ on the real line. More generally,
we can consider the usual Euclidean metric on W, the set of и-tuples of real numbers
χ = (jci , ..., xn). The distance d(x, y) between two points χ = (x\,..., xn) and
У = (Уи ■ ■ ■, Уп) is given by
d(x,y):= (Х>'"-У'12) ·
Throughout this book we shall use the symbol E" to denote the metric space (K", d).
The Euclidean scalar product of two vectors ijel" is the number
η
and the Euclidean norm of χ is ||jc|| := (χ \ x)l/2. A useful way to view the metric on
E" is to note that d(x, y) = \\x — y\\. The triangle inequality for E" then follows from
the Cauchy-Schwarz inequality, \(x | y)\ < \\x\\.\\y\\ (see4.1).
E" is the most familiar of the model spaces M", whose central role in this
book was described in the introduction. Another familiar model space is S", the
и-dimensional sphere, which may be described as follows. Let S" = {x e E"+1 |
||jc|| = 1}. This set inherits two obvious metrics from E"+1. The first, and least
useful, is the induced metric. The second, and most natural, is obtained by defining the
distance d between two points χ and у on the sphere to be the angle at 0 between the
line segments joining 0 to χ and 0 to у respectively. We shall denote the metric space
(S", d) by S". It requires some thought to verify the triangle inequality for S". We
shall return to this point in Chapter 2, where we consider the basic properties of the
model spaces M".
Of course, not all metric spaces enjoy as much structure as the preceding classical
examples. Indeed, given any set one can define a metric on it by defining the distance
between every pair of distinct points to be 1. Such pathological examples render the
theory of general metric spaces rather dull, and it is clear that we must restrict our
notion of distance further if we wish to obtain a class of spaces whose geometry is
in any way comparable to the preceding classical examples. For the most part, we
shall consider only spaces in which every pair of points can be joined by a geodesic,
which is defined as follows.

4 Chapter 1.1 Basic Concepts
Geodesies
1.3 Definitions. Let (X, d) be a metric space. A geodesic path joining jc e X to у е X
(or, more briefly, a geodesic from jc to y) is a map с from a closed interval [0, /] С 1
to X such that c(0) = x, c(Z) = у and d(c(t), c(O) = |i - f\ for all i, f e [0, /] (in
particular, / = d(x, y)). If c(0) = jc, then с is said to г'иие from x. The image α of с
is called a geodesic segment with endpoints jc and y. (There is a 1-1 correspondence
between geodesic paths in X and pairs (a, jc), where α is a geodesic segment in X
and χ is an endpoint of a.)
Let / с 1 be an interval. A map с : / —> X is said to be a linearly reparame-
terized geodesic or a constant speed geodesic, if there exists a constant λ such that
d(c(t), c(f)) = λ \t — f\ for all t, f el. Under the same hypotheses, we say that с
parameterizes its image proportional to arc length.
A geodesic ray in X is a map с : [0, oo) -> X such that u?(c(i), c(O) = I?- f\ for
all i, f > 0. A geodesic line in X is a map с : К -> X such that u?(c(i), с(О) = I? — f\
for all i/el (according to the context, we may also refer to the image of с as a
geodesic line).
A local geodesic in X is a map с from an interval / С R to X with the property
that for every t e I there exists ε > 0 such that d(c(l!), c(f")) = К - t"\ for all
f, r" e / with |i - г7! + |ί - f"| < ε.
(Χ, ύ?) is said to be a geodesic metric space (or, more briefly, a geodesic space)
if every two points in X are joined by a geodesic. We say that (X, d) is uniquely
geodesic if there is exactly one geodesic joining χ to y, for all x, у е X.
Given r > 0, a metric space (X, d) is said to be r-geodesic if for every pair of
points jc, у e X with d(x, y) < r there is a geodesic joining jc to y. And X is said to
be r-uniquely geodesic if there is a unique geodesic segment joining each such pair
of points jc and y.
A subset С of a metric space (X, d) is said to be convex if every pair of points
jc, у е С can be joined by a geodesic in X and the image of every such geodesic is
contained in C. If this condition holds for all points jc, у е С with d(x, y) < r, then
С is said to be r-convex.
Notation. Henceforth, when referring to a generic metric space, we shall follow the
common practice of writing X instead of (X, d), except in cases where ambiguity
may arise.
1.4 Remarks
(1) If for every pair of points jc and у in a complete metric space X there exists a
point m e X such that d(x, m) = d(y, m) = ^d{x, y), then X is a geodesic space. If
such a midpoint exists for all points jc, у e X with d{x, y) < r then X is r-geodesic.
(2) We emphasize that the paths which are commonly called geodesies in
differential geometry need not be geodesies in the above sense; in general they will only
be local geodesies. Thus, for example, a unit speed parameterization of an arc of a
great circle on S2 is a geodesic (in the sense of this book) only if it has length at most
π (cf. 2.3).

Geodesies 5
(3) According to the definition of convexity that we have adopted, a geodesic
segment in a space that is not uniquely geodesic need not be convex.
1.5 Example. The most familiar example of a uniquely geodesic metric space is E":
the unique geodesic segment joining two points χ and у is the line segment between
them, namely the set of points {(1 — t)x + ty | 0 < t < 1}. A subset С is convex in
the sense of the above definition if and only if it is convex in the linear sense, i.e.
if the linear segment joining each pair of points of С is entirely contained in С. А
subset X с Е", equipped with the induced metric, is a geodesic space if and only
if it is convex. More generally, when endowed with the induced metric, a subset of
a uniquely geodesic metric space will be geodesic if and only if it is convex. For
instance a circle in E2 with the induced metric is not a geodesic space, but a round
disc is.
From the point of view of this book, the most fundamental examples of geodesic
spaces are the model spaces of constant curvature M"; these are the subject of Chapter
2. Later in Part I we shall present three other major classes of geodesic metric spaces:
normed vector spaces, complete Riemannian manifolds, and polyhedral complexes
of piecewise constant curvature. In the last two cases the existence of geodesic paths
is not so obvious; determining when such spaces are uniquely geodesic is also a
non-trivial matter. The case of normed vector spaces is much easier.
Let У be a real vector space. Recall that a norm, denoted υ ι-> || υ ||, is a map
V -> К such that d(v, w) := \\υ - u»|| defines a metric and ||λυ|| = |λ|.||υ|| for all
υ e V and λ e №.. The triangle inequality for the metric associated to the norm is
equivalent to the statement that ||i> + w\\ < \\v\\ + \\w\\ for all v, w e V. For the
purposes of this book, we define a normed vector space to be a real vector space V
together with a choice of norm. V is called a Banach space if the metric associated
to this norm is complete.
We claim that every normed space V, equipped with the metric d(v, w) = \\v —
w ||, is a geodesic metric space. Indeed, it is easy to see that t м> (1 — f)v + tw defines
a path [0, 1] —> X which is a linearly reparameterized geodesic from Dtoiu. We
shall denote the image of this path [v, w].
1.6 Proposition. Every normed vector space V is a geodesic space. It is uniquely
geodesic if and only if the unit ball in V is strictly convex (in the sense that if и ι and
U2 are distinct vectors of norm 1, then ||(1 — t)u\ + Шг\\ < Iforallte (0, I)).
Proof. In the light of our previous observation that [υ, w] is a geodesic segment, we
see that V is uniquely geodesic if and only if, given any υ,υ',υ" e V, the equality
d(v, v') + d(v', v") = d(v, υ") implies that υ' e [υ, υ"]. Thus, writing vi in place of
v' — υ and i>2 in place of v" — υ', we see that V is uniquely geodesic if and only if
ll^i -(- U2|j < ll^i || + ||υ2|| whenever vi and i>2 are linearly independent.

6 Chapter 1.1 Basic Concepts
If we write υ, = α,-и,, where a, = |
vi +v2 = (at + a2)
υ,·||, and let t = a\/{a\ + a2), then
Й1 a2
-Щ +
-u2
ax +a2 ax + a2
= (ai +a2)(tul +(1 -f)u2).
Hence Ци! +υ2|| < ll^ill + l|i>2ll if and only if Hi^ +(1 - t)u2\\ < 1.
1.7 Example (The £' and l°° norms on K").
D
The Ix norm of a vector jc = (x\, ... , xn) e M" is defined to be J] |jc,- |, and the l°°
norm of jc is defined to be max, |jc,|. It is obvious that the maps W —> Ш thus defined
are indeed norms. We denote the associated metrics by d\ and dOo respectively. If
η > 2 then neither the unit ball in (Μ", άχ) nor that in (K", dOo) is strictly convex, so
by (1.6) these spaces are not uniquely geodesic.
ж
ж
У
Fig. 1.1 The unit ball in R2 with the norms
:i,*2,«oo
In Chapter 4 we shall use (1.6) to show that the metric space associated to the lp
norm on W is uniquely geodesic if 1 < ρ < oo.
1.8 Exercise. Give a complete description of the geodesies in (K2, d\) and (M2, dOo).
Perhaps the easiest examples of geodesic metric spaces which are not manifolds
are yielded by the following construction.
1.9 Metric Graphs. Intuitively speaking, metric graphs are the spaces that one
obtains by taking a connected graph (i.e., a connected 1-dimensional CW-complex),
metrizing the individual edges of the graph as bounded intervals of the real line,
and then defining the distance between two points to be the infimum of the lengths
of paths joining them, where "length" is measured using the chosen metrics on the
edges. It does not take long to realize that if one is not careful about the way in
which the metrics on the edges are chosen, then various unpleasant pathologies can
arise. Before considering these, we give a more precise formulation of the above
construction.

Geodesies 7
A combinatorial graph Q consists of two (possibly infinite) sets V (the vertices)
and £ (the edges) together with two maps 3o : £ —> V and 3i : £ —> V (the endpoint
maps). We assume that V is the union of the images of 3o and 3i. (A combinatorial
graph in our sense is not a graph in the sense of Serre [Ser77].)
One associates to Q the set Xg (more briefly, X) that is obtained by taking the
quotient of £ χ [0, 1] by the equivalence relation generated by (e, i) ~ (e', /') if
3,(e) = dr(e'), where e, e1 e £ and i, Ϊ e {0, 1}. Let ρ : £ χ [0, 1] -> X be the
quotient map. We identify V with the image in X of £ χ (0, 1}. For each e e £, let
fe : [0, 1] —> X denote the map that sends ί e [0, 1] top(e, t). Note that/e is injective
on (0, 1). If/e(0) = /e(l), the edge e is a called a loop.
To define a metric on X, one first specifies a map
λ : £ -* (0, oo)
associating a length X(e) to each edge e. A piecewise linear path is a map с : [0, 1] —>
X for which there is a partition 0 = ίο < h < ·■· < t„ = 1 such that each c|[f|..(i.+1]
is of the form/e, о с,-, where e, e £ and c, is an affine map from [i,, ii+i] into [0, 1].
We say that с joins χ to у if c(0) = jc and c(l) = y. The length of с is defined to
be /(c) = Σ"~0 /(с,·), where 1(a) = X{e{) |c,-(i,) - ci+i(i;+i)|. We assume that X is
connected, i.e. any two points are joined by such a path.
We define a pseudometric d : Χ χ X ^> [0, oo] by setting d(x, y) equal to the
infimum of the length of piecewise linear paths joining χ to y. The space X with
its pseudometric d is called a metric graph. For any edge e, the distance between
p(e, l/2)and3,(e)isX(e)/2.
If the graph has only one edge e and this is a loop, then the corresponding metric
graph is isometric to a circle of length X(e). Let us consider some other simple
examples.
(i) Suppose that the set of vertices V = {vn}n is indexed by the non-negative integers,
and that the set of edges {e„}„ is indexed by the positive integers. Suppose 3o(e„) =
υ„_ι and 3i(e„) = vn. In this case, if X(e„) = 1 for each n, then X is isometric
to [0, oo); if X(en) = 1/2", then X is isometric to the interval [0, 2) (which is not
complete); in general X is isometric to [0, b) where b = Σ X(e„).
(ii) Suppose that V has two elements uq and vu that the set of edges £ = {en}n is
indexed by the positive integers, and that 3,(e„) = υ,. If X(en) = l/n, then d is not a
metric, because d(vo, i>i) = 0. If X(en) = 1 + l/n, then d is a metric, d(vo, V[) = 1,
the metric space X is complete, but there is no geodesic joining u0 to i>i.
Coming back to the general situation, we leave the reader to check the following
facts.
(1) The pseudometric d is actually a metric if for each vertex ν the set {k{e) \ e e
£, and 3o(e) = υ or 3i(e) = v} is bounded away from 0.
(2) If the set of edge lengths X(e) is finite, then X with the metric d is a complete
geodesic space.
A combinatorial graph Q is called a tree if the corresponding metric graph X
where all the edges have length one is connected and simply connected.

8 Chapter 1.1 Basic Concepts
(3) If the graph Q is a tree, then the pseudometric on Xg associated to any length
function λ : £ —> (0, oo) makes Xg a geodesic metric space.
Cayley Graphs. Many interesting examples of metric graphs are given by the
following construction.
The Cayley graph 3 Сд(Г) of a group Г with respect to a generating set A is the
metric graph whose vertices are in 1 -1 correspondence with the elements of Г and
which has an edge (labelled a) of length one joining у to γα for each у e Γ and
a e A. In the notation of (1.9), V = Γ, £ = {(γ, α) | γ e Г, а е A}, 30(y, a) =
y, 3t(y, a) — γα, and λ : £ —» [0, oo) is the constant function 1.
Cayley graphs will play a significant role in Chapter 8. A simple example is the
Cayley graph of the free abelian group Z2 with basis {x, y}: this can be identified
with the integer lattice in the Euclidean plane.
Angles
We now turn to Alexandrov's definition [Ale51 and Ale57a] of the angle between
geodesies issuing from a common point in an arbitrary metric space. For this we shall
need the following tool for comparing the geometry of an arbitrary metric space to
that of the Euclidean plane. (This method of comparison was used extensively by
A.D. Alexandrov and it plays a central role in Part II of this book. For historical
references see [Rin61].)
1.10 Definition. Let X be a metric space. A comparison triangle in E2 for a triple of
points (p, q, r) in X is a triangle in the Euclidean plane with vertices p, q, 7 such that
d(p, q) = d(p, <?), d(q, r) = d(q, ?) and d(p, r) = d(p, 7). Such a triangle is unique
up to isometry, and shall be denoted A(p, q, r). The interior angle of A(p, q, r) at ρ
is called the comparison angle between q and r at ρ and is denoted Z.p{q, r). (The
comparison angle is well-defined provided q and r are both distinct from p.)
1.11 The Law of Cosines in E". We shall make frequent use of the following
standard fact from Euclidean geometry; this is commonly called the cosine rule or
the law of cosines.
In a Euclidean triangle with distinct vertices А, В, С, with sides of length a =
d(B, C), b = d(A, С), с = d(A, B), and with interior angle γ at the vertex С (opposite
the side of length c), the following identity holds:
c2 = a2 + b2 - lab cos y.
In particular, for fixed a and b, the length с is a strictly increasing function of y.
The following concept is what Alexandrov calls the upper angle between
geodesies in a metric space.
3 Cayley graphs were introduced by Arthur Cayley in 1878 [Cayl878].

Angles 9
1.12 Definition of Angle. Let X be a metric space and let с : [0, a] —> X and d :
[0, a'] -> X be two geodesic paths with c(0) = c'(0). Given t e (0, a] and f e (0, a'],
we consider the comparison triangle Д(с(0), c(t), d(t)), and the comparison angle
ZC(0)(c(i), с'(О)· The (Alexandrov) angle or the идаег angle between the geodesic
paths с and d is the number ZCiC' e [0, я] defined by:
Z(c, c') := limsup Zc(0)(c(i), c'(i')) = Hm sup Zc(0)(c(i), c'(i')).
i,i'-»0 ε-4>00<ί.ί'<ε
If the limit lim,/-^ ^с(0)(с(0> c'CO) exists, then we say the angle exists in the
strict sense.
One can express Z(c, d) purely in terms of the distance function by noting that
cos(Zc(0)(c(i), с'(/))) = ^(ί2 + f2 - d(c(t\ c'(i'))2)·
The angle between two geodesic segments which have a common endpoint is
defined to be the angle between the unique geodesies which issue from this point and
whose images are the given segments. IfX is uniquely geodesic, ρ φ хапар φ y,then
the angle between the geodesic segments [p, x] and [ρ, y] may be denoted Zp(x, y).
Note that in E", the Alexandrov angle is equal to the usual Euclidean angle.
1.13 Remarks and Examples.
(1) The angle between с and d depends only on the germs of these paths at
0: if c" : [0, /] —> X is any geodesic path for which there exists ε > 0 such that
c"|[ο ε] = c'|[o ε], then the angle between с and c" is the same as that between с and
d.
(2) The angle between the incoming and outgoing germs of a geodesic at any
interior point along its image is я. In other words, if с : [a, b] —> X is a geodesic
path with a < 0 < b, and if we define d : [0, -a] -> X and c" : [0, b] -> X by
c'(t) = c(-i) and c"(0 = c(t), then Z(d, с") = я.
(3) In a metric tree, the angle between two geodesic segments which have a
common endpoint is either 0 or π.
(4) In the above definition of angle, it is important that one takes a limsup; in
general the limit lim^c^o ^c(0)(c(0, c'CO) does not exist. For instance, we shall see
in Chapter 4 that such limits exist in a normed vector space if and only if the norm
derives from a scalar product. In contrast, in the metric spaces which are of primary
concern in this book, namely those with curvature bounded above, this limit always
exists (II.3.1).
(5) Consider (M2, dOo), where doo((x, y), (x1', /)) := max{|jc - xf\, \y — y1]}. For
every integer η > 1, the map t м> (t, [i(l — i)]") defines a geodesic path [0, l/и] —>
(Ж2 ,doo). These geodesies all issue from a common point and their germs are pairwise
disjoint, but the angle between any two of them is zero.
This last example illustrates the fact that the angle between distinct geodesies
issuing from the same point may be 0, even if their germs are distinct. Thus, in
general, (с, с') м> Z(c, d) does not define a metric on the set of (germs of) geodesies

10 Chapter 1.1 Basic Concepts
issuing from a point. However, this function clearly satisfies the axioms of symmetry
and positivity, so the following proposition, due to Alexandrov [Ale51], shows that
it is a pseudometric. If one were to take Km inf instead of lim sup in the definition of
Alexandrov angle, then this property would fail.
1.14 Proposition. Let X be a metric space and let с, с' and c" be three geodesic
paths in X issuing from the same point p. Then,
Z(c', c") < Z(c, c') + Z(c, c").
Proof. We argue by contradiction. If this inequality were not true, then there would
exist 5 > 0 such that Z(c', c") > Z(c, c') + Z(c, c") + 35. Using the definition of
the lim sup, we could then find an ε > 0 such that:
(i) 4>(c(i), c'(i')) < Ac, c) + 5 for all t, t' < ε,
(ii) Zp(c(t), c"(t")) < Z(c, c") + 5 for all t, t" < ε,
(iii) ^P(c'(f), c"(f")) > Z(c', c") - 5 for some i, t" < ε.
Let f and t" be as in (iii). Consider a triangle in E2 with vertices 0, xf, x" such
that d(0, x!) = f, d(0, x") = t", and such that the angle a at the vertex 0 satisfies:
7p(c'(i'), c"(i")) > a > Z(c', c") - 5.
In particular a < я, so the triangle is non-degenerate. Notice that the left-most
inequality implies that d(xf', x") < d(c'(f), c"(t")). The right-most inequality implies
that a > Z(c, c') + Z(c, c") + 25, and hence enables us to choose a point χ e [xf', x"\
such that the angle a' (resp. a") between the Euclidean segments [0, xf] and [0, x]
(resp. [0, xf] and [0, x]) is bigger than Z(c, c') + 5 (resp. Z(c, c") + 5).
Let t = d(0, x). Because t < maxff', f"} < ε, condition (i) implies that
Zp(c(i), c'(i')) < Ac с') + 5 < a',
and hence d(c(t), с'(О) < d{x, x1). Similarly, d(c(t), c"(f)) < d(x, χ"). Thus
d(c'(t'\ c"(f")) > d(xf, x") = d(x, xf) + d(x, x") > d(c(t), c'(f)) + d(c(t), c"(t")),
which contradicts the triangle inequality in X. D
1.15 Exercise. Generalize definition 1.12 to give a notion of (upper) angle between
arbitrary continuous paths that issue from the same point in an arbitrary metric space
and do not return to that point. Then adapt the proof of the preceding proposition so
that it applies to this more general situation.

Angles 11
The Strong Upper Angle
Following Alexandra ν ([Ale51] and [Ale57a]), we give an alternative description of
the angle between geodesies issuing from a common point. Let X be a metric space
and let с : [0, a] —> X and d : [0, a'] —> X be two geodesic paths with c(0) =
c'(0) = p. The strong upper angle γ between с and d is the number y(c, d) € [0, π]
defined by:
y(c, c') := limsup Zp(c(s), c'{t)) = lim sup Zp(c(i), c'(0)·
5^0;ί€(0,α'] ε^°ί€(0,ε];ί€(0,α']
Alexandrov proved that this notion of strong upper angle is equivalent to the
notion of angle given in (1.12).
1.16 Proposition. Let X be a metric space. For all geodesies с : [0, a] —> X and
d : [0, a'] -> X with c(0) = c'(0) one has Z(c, d) = y(c, d).
Proof. Let ρ = c(0) = c'(0). It is clear from the definition that Z(c, d) < y(c, d).
In order to establish the reverse inequality it suffices to show that for each fixed
t e (0, a'] we have lim supJ^0 Zp(c(s), d(t)) < Z(c, d). For this we need a technical
lemma:
1.17 Lemma. Let uSJ = d(c(s), c'(t)) and let ySJ = Zp(c(s), c'(t)). Then
Τ !*j ι ι Ms ι S
— < cos ys,t < + —.
s s 2t
Proof of the lemma. From the definition of ysj we have
2±f2_J , _ „ „2 _ /„ _ rt2
s2 + f - ιξ4 t - uSJ s1 - (им - tj
cos ys,t = L- = — +
2st s 1st
And by the triangle inequality \uSi, — t\ < s, whence 0 < —-jft—- < fr Π
From the lemma we have lim inf^o cos y^, = liminf,^o '—y^ for each fixed
t e (0, a']. If f < t then by the triangle inequality t — f > uSJ — uSJ,, and hence
f7 — us,f < t — usd. Therefore
lim inf cos ys t· < liminf cos ys,.
The left hand side of this inequality is equal to cos Z(c, d), so since cos is
decreasing on [0, я] we have limsupJ^0 Zp(c(s), c'(t)) < Z(c, d) for each t e (0, a'], as
required. Π

12 Chapter 1.1 Basic Concepts
The Length of a Curve
The last of the fundamental concepts which we introduce in this first chapter is the
notion of length for curves in an arbitrary metric space. We begin by fixing some
terminology. Let X be a metric space. For us, a curve or a path in X is a continuous
map с from a compact interval [a, b] с К to X. We say that с joins the point c(a) to
the point c(b). If ci : [αϊ, b\] -> Я and C2 : [«2, W -> Я are two paths such that
c\(b]) = 02(02)» their concatenation is the path с : [щ,Ь\ Л-Ьг — aj\ —> X defined
by c(i) = ci(i) if t e [a\,b\] and c(i) = ci(t + «2 - b\) if ί e [fri, /?i + fe - аг]·
More generally, the concatenation of a finite sequence of paths c, : [a,-, /?,] —> X, with
c,(/?,) = c,+i(a,+i) for i = 1,2,... , и — 1, is defined inductively by concatenating
Ci,... , c„-\ and then concatenating the result with c„.
1.18 Definition of Length. Let X be a metric space. The length 1(c) of a curve
с : [α, b] —> X is
n-l
/(c) = sup Y^d(c(ti),c(ti+i)),
a=to<tt<-<t„=b i=0
where the supremum is taken over all possible partitions (no bound on n) with
a = to < t\ < ■ ■ ■ < t„ = b.
The length of с is either a non-negative number or it is infinite. The curve с is
said to be rectifiable if its length is finite.
1.19 Example. An easy example of a non-rectifiable path in the metric space X =
[0, 1] can be constructed as follows. Let 0 = ίο < h < ■ ■ · < t„ < ... be an infinite
sequence of real numbers in [0, 1] such that lim^oo t„ = 1. Let с : [0, 1] —> X be
any path such that c(0) = 0 and c(t„) = Σ£=ι(-1)*+1Α· This is not a rectifiable
path, because its length is bounded below by the sum of the harmonic series.
Let Fbe the graph of с in [0, 1] x [0, 1], with the induced metric. Fis a compact,
path-connected metric space, but there is no rectifiable path in Υ joining (0, 0) to the
point (1, In 2).
We note some basic properties of length.
1.20 Proposition. Let (X, d) be a metric space and let с : [a,b] —» X be a path.
(1) /(c) > d(c(a), c(b)), and /(c) = 0 if and only if с is a constant map.
(2) If φ is a weakly monotonic map from an interval [a', b'] onto [a, b], then
/(c) = /(c ο φ).
(3) Additivity: If с is the concatenation of two paths c\ and C2, then /(c) = 1(a) +
1(a).
(4) The reverse path с : [a, b] —> X defined by ~c(t) — c(b + a — t) satisfies
1(c) = 1(c).

The Length of a Curve 13
(5) If с is rectifiable of length I, then the function λ : [a, b] —> [0, /] defined by
X(t) = l(c\[a,t]) is a continuous weakly monotonic function.
(6) Reparameterization by arc length: If с and λ are as in (5), then there is a unique
path с : [0,1] —> X such that
с ο λ = с and /(c|[o,(]) = t.
(7) Lower semicontinuity: Let (c„) be a sequence of paths [a, b] —> X converging
uniformly to a path с If с is rectifiable, then for every ε > 0 there exists an
integer Ν(ε) such that
Kc) < l(c„) + ε
whenever η > Ν(ε).
Proof. Properties (1), (2), (3) and (4) are immediate. Property (3) reduces the proof
of (5) to showing that, given ε > 0, one can partition [a, b] into finitely many
subintervals so that the length of с restricted to each of these subintervals is at most
ε. To see that this can be done, we first use the uniform continuity of the map
с : [a, b] -> X to choose 5 > 0 such that d(c(t), c(f)) < ε/2 for all t, f e [a, b] with
\t — f\ < 8. Since 1(c) is finite, we can find a partition a = t^ < t\ < ■■■ < tk = b
such that
Y^didtf), c(ti+l) > 1(c) - ε/2.
(=0
Taking a refinement of this partition if necessary, we may assume that |ί,—ί,+ 11 < Sfor
ϊ = 0,... ,k - 1, and hence d(c(t{), c(ti+])) < ε/2. But l(c\[ihil+l]) > d(c(h), c(ti+x)),
and 1(c) = £ /(c|[fi,(l.+1]) by (3). Hence
Kc) = J^Kc\i,,.ti+l]) > Y^d^td, c(fi+i) > 1(c) - ε/2,
(=0 ί=0
with each summand in the first sum no less than the corresponding summand in
the second sum. Hence, for all i we have l(c\[tht-+1]) — d(c(ti), c(ti+\) < ε/2, and in
particular l(c\[fi.,,/+l]) < ε.
(6) follows from (5) and (2). To check (7), we again choose a = i0 < t\ < · · · <
tk = b such that
*-i
l(c)<J2d(c(tilc(ti+i)) + S/2.
(=0
Then we choose Ν(ε) big enough to ensure that d(c(t), c„(t)) < ε/4k for all
η > Ν(ε) and all t e [a, b]. By the triangle inequality, d(c(ti), c(ti+l)) < 2ε/4k +
d(c„(ti), c„(ti+i)). Hence
ε *"' ε
Kc) < к— +Y'd(cn(ti), c„(i,+i)) + - < ε + l(cn).
О

14 Chapter 1.1 Basic Concepts
1.21 Definition. A path с : [a, b] —> X is said to be parameterized proportional to
arc length if the map λ defined in 1.20(5) is linear.
1.22 Remark. It follows from 1.20(2) and (5) that every path in X of length I has the
same image as a path [0, 1] —> X of length I which is parameterized proportional to
arc length. It also follows from 1.20(5) that if a path joining χ to у in X has length
d(x, y) then its reparameterization by arc length is a geodesic path joining χ to y.
An easy but useful observation is that for any (continuous) path с : [0, 1] —> X,
и-1
1(c) = sup J^ d(c(i/n), c((i + 1)/л)).
«>o ,=o
1.23 Exercice. Show that if a sequence of paths c„ : [a,b] —» X converges uniformly
to a path с and limsup„ l{c„) is finite, then с is rectifiable.
Find a sequence of rectifiable paths c„ : [0, 1] —> [0, 1] converging uniformly to
a non-rectifiable path с (see 1.19).
1.24 Constructing Metrics Using Chains. In certain situations one encounters a
numerical measure of separation between the points of a set X that does not satisfy
the triangle inequality, say ρ : Χ χ Χ —» [0, oo) with p(x, x) = 0 and p(x, y) = p(y, x)
for all jc, у e X. In such situations one can obtain a pseudometric on X by defining
n-l
dP{x, У) := inf ]P rf(jc,-, jci+I),
i=0
where the infimum is taken over all chains χ = xq, ..., x„ = y, no bound on η
(cf. (5.19) and (7.4)).
One way in which examples can arise is when one has a metric space (X, d) and
a function/ : [0, oo) —> [0, oo) with/(0) = 0; one then considers p :=f ο d.

Chapter 1.2 The Model Spaces MnK
In this chapter we shall construct the metric spaces M£, which play a central role
in later chapters, serving as standard models to which one can profitably compare
more general geodesic spaces. One way to describe M" is as the complete, simply
connected, Riemannian и-manifold of constant sectional curvature к e R. However,
in keeping with the spirit of this book, we shall first define and study the M" purely
as metric spaces (cf. [Iv92]), and defer consideration of their Riemannian structure
until Chapter 6.
For each integer n, the spaces M" fall into three qualitatively distinct classes,
according to whether к is zero, positive or negative. To simplify the notation, we
concentrate first on the cases E" = Afg, §" = Mf and W = Mn_, (we shall explain
at the end of the chapter how each M" can be obtained from one of these special
cases simply by scaling the metric). We shall give a concrete model for each of
the spaces under consideration, describe its metric explicitly and verify the triangle
inequality. In each case the triangle inequality is seen to be intimately connected
with (the appropriate form of) the law of cosines. We give an explicit description of
the geodesies in each case, and also of the hyperplanes; the latter play an important
role in our description of Isom(M").
In Chapter 6 we shall give alternative models for the metric spaces M" and we
shall describe their natural Riemannian metric.
Euclidean η-Space E"
In 1.2 we introduced the notation E" for the metric space obtained by equipping the
vector space W with the metric associated to the norm arising from the Euclidean
scalar product (x | y) = Σί=ι *'У» wnere x = (χι> ■ ■ ■, xn) and у = (уь ..., уп).
As we noted earlier, E" is a uniquely geodesic space; the triangle inequality is a
consequence of the Cauchy-Schwarz inequality, and the geodesic segments in E" are
the subsets of the form [x, y] = [ty + (1 - f)x \ 0 < t < 1}.
By definition, a hyperplane Η in E" is an affine subspace of dimension η — I.
Given a point Ρ e Η and a unit vector и orthogonal to Η (и is unique up to sign), Η
may be written as Η = {Q e E" | (Q — Ρ \ u) = 0}. Every hyperplane arises in the
following way: given two distinct points A, В е Ε", the set of points equidistant from

16 Chapter 1.2 The Model Spaces M"K
A and В is called the hyperplane bisector of A and B\ it is a hyperplane that contains
the midpoint Ρ of the geodesic segment [A, B] and is orthogonal to the vector Л — В.
Associated to each hyperplane there is an isometry r# of E", called the reflection
through H. If Ρ is a point on Η and и is a unit vector orthogonal to H, then for all
A e E" one has rH(A) = A — 2(A - Ρ | u)u. The set of fixed points of r# is H.
If Л € Ε" does not belong to H, then Η is the hyperplane bisector of A and гн(А).
Conversely, if Η is the hyperplane bisector of Л and B, then rH(A) = B.
The η-Sphere §"
The и-dimensional sphere S" is the set {x = (xu ..., x„+[) e R'!+1 | (x \ x) = 1},
where (· | ·) denotes the Euclidean scalar product. It is endowed with the following
metric:
2.1 Proposition (Definition of the Metric). Let rf:S°xS"->Kfe the function that
assigns to each pair (A, B) e S" χ S" the unique real number d(A, B) e [0, π] such
that
cosd(A,B) = (A \B).
Then d is a metric.
Proof. Clearly d(A, B) = d(B, A) > 0, and d(A, B) = 0 if and only if Л = В. The
triangle inequality is a special case of (1.14), because d(A, B) is the angle between
the segments [O, A] and [O, B] in E"+1. D
In order to harmonize our treatment of S" with that of W, we give a second proof
of the triangle inequality based on the spherical law of cosines (which can also be
deduced from (1.14)). But first we describe what we mean by a spherical triangle
in S" and give meaning to the notion of vertex angle. We must be careful about
our use of language here, because we do not wish to presuppose that d is a metric,
and therefore we should not use the definitions of geodesic, angle and triangle from
Chapter 1.
It is convenient to use the vector space structure on E"+1, which contains S". A
great circle in S" is, by definition, the intersection of S" with a 2-dimensional vector
subspace of E"+1. There is a natural way to parameterize arcs of great circles: given
a point A € S", a unit vector и e E"+1 with (u | A) = 0 and a number a e [0, π],
consider the path с : [0, a] -» S" given by c(t) = (cos t)A + (sin t)u. Note that
d(c(t), с(О) = V — f\ for all t, f e [0, a]. The image of с is contained in the great
circle where the vector subspace spanned by Л and и intersects S". We shall refer to
the image of с as α minimal great arc or, more accurately, the great arc with initial
vector и joining Л to c(a). If a = π, then for any choice of и one has c(n) = —A. On
the other hand, if d(A, Β) < π then there is a unique minimal great arc joining A to
B. If Β ψ A then the initial vector и of this arc is the unit vector in the direction of
В - (A | B)A.

The η-Sphere S" 17
By definition, the spherical angle between two minimal great arcs issuing from
a point of §", with initial vectors и and υ say, is the unique number a e [0, π] such
that cos oe = (u | υ). A spherical triangle Δ in S" consists of a choice of three
distinct points (its vertices) А, В, С € §", and three minimal great arcs (its sides),
one joining each pair of vertices. The vertex angle at С is defined to be the spherical
angle between the sides of Δ joining С to Л and С to B.
2.2 The Spherical Law of Cosines. Let A be a spherical triangle with vertices
А, В, С Let a = d(B, C), b = d(C, A) and с = d(A, B) (spherical distances). Let γ
denote the vertex angle at C. Then,
cos с = cos a cos b + sin a sin b cos γ.
Proof. Let и and υ be the initial vectors of the sides of Δ joining С to Л and В
respectively. By definition, cos γ — {u\ υ). And
cos с = (A \ B)
— ((cos b)C + (sin b)u | (cos a)C + (sina)u)
= cos a cos b (C | C) + sin a sin b(u\ v)
= cos a cos b + sin a sin b cos y,
as required. Π
2.3 The Triangle Inequality and Geodesies. For allA,B,C e §",
d(A, B) < d(A, C) + d(C, B),
with equality if and only if С lies on a minimal great arc joining A to B. Thus:
(1) (§", d) is a geodesic metric space.
(2) The geodesic segments in S" are the minimal great arcs.
(3) Ifd(A, Β) < π then there is a unique geodesic joining A to B.
(4) Any open (resp. closed) ball of radius r < π/2 (resp. < π/2) in S" is convex.
Proof. Let a = d(C, B), b = d(A, С), с = d(A, B). We assume that А, В and С
are distinct (the case where they are not is trivial). Consider a spherical triangle Δ
with vertices A,B,C. Let γ be the vertex angle at C. The cosine function is strictly
decreasing on [0, π], so for fixed a and b, as γ increases from 0 to π, the function
γ ι-* cos a cos b + sin a sin b cos γ decreases from cos(b — a) to cos(b + a). So by
the law of cosines we have cos с > cos(b + a), and hence с < b + a, with equality
if and only if γ = π and b + a < π. The conditions for equality hold if and only if
С belongs to a minimal great arc joining A to B.
The convexity assertion in (3) follows from the fact, noted above, that if d(A, B) <
π then there is a unique minimal great arc joining A to B. This arc is the intersection
of S" with the positive cone in E"+1 spanned by A and B, and hence consists of

18 Chapter 1.2 The Model Spaces M"K
Fig. 2.1 A spherical triangle in S"
points of the form λΑ + μΒ with λ + μ > 1. Suppose that A and В both lie in the
closed ball 5(P, г) с §" of radius r < тг/2 about P. By definition, С e B(P, r) if
andonlyif(C|P)> cos r. But if λ + μ > 1 then (λΑ + μΒ | Ρ) = λ(Α | Ρ) + μ(Β\
Ρ) > (λ- + /ζ) cos r > cos r. Hence the minimal great arc from A to В is contained in
B(P, r). Π
Hyperplanes in §". A hyperplane Η in S" is, by definition, the intersection of S"
with an и-dimensional vector subspace of E"+1. Note that H, with the induced metric
from S", is isometric to S"~'. We define the reflection rH through Η to be the isometry
of S" obtained by restricting to S" the Euclidean reflection through the hyperplane of
E"+1 spanned by H. Given two distinct points А, В e S", the set of points in S" that
are equidistant from A and В is a hyperplane, called the hyperplane bisector for A
and B; it is the intersection of S" with the vector subspace of E"+1 orthogonal to the
vector Л — В. If A e S" does not belong to H, then Η is the hyperplane bisector of Л
and гц(А). Conversely, if Η is the hyperplane bisector of Л and B, then rH(A) = B.
Hyperbolic η-Space H"
When approaching hyperbolic geometry from a Riemannian viewpoint, it is common
to begin by introducing the Poincare model for H" (see Chapter 6). However, from
the purely metric viewpoint which we are adopting, it is more natural to begin with
the hyperboloid model. One benefit of this approach is that it brings into sharp focus
the fact that S" is obtained from E"+1 in essentially the same way as W is obtained
from E"·1, where Е"л denotes the vector space R"+1 endowed with the symmetric
bilinear form that associates to vectors и = (щ,..., un+\) and ν = (υι,..., ν„+\)
the real number (и | υ) defined by
η
(и I υ) = -un+xvn+\ + Y^UiVi.
i=\
The quadratic form associated to (· | ·) is nondegenerate of type (и, 1).

Hyperbolic η-Space Η" 19
The orthogonal complement of ν e E"·' with respect to this quadratic form is,
by definition, the и-dimensional vector subspace vL с Е"л consisting of vectors и
such that (u \ v) = 0. If (v \ v) < 0, then the restriction of the given quadratic form
to vL is positive definite. If υ φ 0 but (υ | υ) = 0, then vL is tangent to what is
called the light cone [χ \ (χ | x) = 0}; in this case the given quadratic form restricts
to a degenerate form of rank η — 1 on v1. If (v \ v) > 0, then the restriction to υ1
of the given quadratic form is non-degenerate of type (n — 1, 1).
Fig. 2.2 The hyperbolic plane H2
2.4 Definition. (Real) hyperbolic и-space W is
{и = (щ,... ,un+l) еЕ"л | (u | u) = -1, u„+l > 0}.
In other words, W is the upper sheet of the hyperboloid {u e E"'1 | (u \ u) = — 1}.
Note that if и е Ш" then ип+] > 1, with equality if and only if щ = 0 for all
i= 1,... ,n.
2.5 Remark. We shall need the observation that for all u, υ e Ш" one has (и \ v) <
— 1, with equality if and only if и = v. Indeed,
(u | v) = ί Y^UiV,- I - un+lvn+[
-I
V=i /
^ (ί>π ίέυπ -"«+·υ«+.
= (^+1 - 1)1/2(υ„2+1 - 1)1/2 - ии+1«и+ь

20 Chapter 1.2 The Model Spaces M"K
and one checks easily that this last expression is < — 1 for any numbers un+i, vn+\ >
1, with equality if and only if un+\ = vn+\.
As an immediate consequence we see that (u — υ \ и — υ) > 0, for all u, υ e Ш",
with equality if and only if и = v.
2.6 Proposition (Definition of the Metric). Letd :Н"хН"н. R be the function that
assigns to each pair (А, В) е Ш" χ И" the unique non-negative number d(A, В) > 0
such that
cosh d(A,B)= -(A \B).
Then d is a metric.
Proof. The content of Remark 2.5 is that d is well-defined and positive definite, d is
clearly symmetric, so it only remains to verify the triangle inequality. As in the case
of §", we shall deduce this from the appropriate form of the law of cosines. Π
Again as for S", in order to state the law of cosines we must first define primitive
notions of segment, angle and triangle in W. Eventually (2.8) we shall see that
the geodesic lines in H" are the intersections of H" with 2-dimensional subspaces
of E"·'. Thus a hyperbolic segment should be a compact subarc of such a line of
intersection. More precisely, given Л e Ш" and a unit vector и е А1- с Е"·1 (i.e.,
(и I и) = 1 and (A \ и) = 0), we consider the path с : К -> И" defined by
t м> (cosh t)A + (sinhi)«. Note that d(c(t), c(0) = \t - f\ for all t, f e K. Given
a > 0, we define the image under с of the interval [0, a] to be the hyperbolic segment
joining A to c(a), and denote it [A, c(a)]. Given В е Ш" distinct from A, let и be the
unit vector in the direction of (B + (A | B)A). This is the unique unit vector и е ΑΣ
such that В = (cosh a)A + (sinh a)u, where a = u?(A, 5). We shall refer to и as the
initial vector of the hyperbolic segment [A, B].
By definition, the hyperbolic angle between two hyperbolic segments issuing
from a point of W, with initial vectors и and v, say, is the unique number a e [0,π]
such that cos a = (u \ υ). A hyperbolic triangle A in И" consists of a choice of three
distinct points (its vertices) A, В, С е И", and the three hyperbolic segments j oining
them (its sides). The vertex angle at С is defined to be the hyperbolic angle between
the segments [C, A] and [С, В].
2.7 The Hyperbolic Law of Cosines. Let Abe a hyperbolic triangle with vertices
А, В, С Let a = d(B, C), b = d(C,A) and с = d(A, B). Let γ denote the vertex
angle at C. Then
cosh с = cosh a cosh b — sinh a sinh b cos γ.
Proof. Let и and ν be the initial vectors of the hyperbolic segments joining С to A
and В respectively. By definition, cos γ = (и \ υ). And

Hyperbolic η-Space Η" 21
cosh с = - (A I В)
= — ((cosh b)C + (sinh b)u | (cosh a)C + (sinh a)v)
= — cosh a cosh b (C | C) — sinh a sinh £> (и | v)
= cosh α cosh b — sinh α sinh b cos y,
as required. D
There are many other useful trigonometric formulae in hyperbolic geometry (see,
for example, [Thu97], section 2.4).
2.8 Corollary (The Triangle Inequality and Geodesies).
For all А, В, С eW,
d(A, B) < d(A, C) + d(C, B),
with equality if and only if С lies on the hyperbolic segment joining A to B. Thus:
(1) W is a uniquely geodesic metric space.
(2) The unique geodesic segment joining A to В is the hyperbolic segment [A, B].
(3) If the intersection of a 2-dimensional vector subspace o/E"·1 with W is
nonempty, then it is a geodesic line, and all geodesic lines in W arise in this
way.
(4) All the balls in W are convex.
Proof. Let a = d(C, B), b = d(A, С), с = d(A, B). We assume that А, В and С are
distinct (the case where they are not is trivial). Consider a hyperbolic triangle Δ with
vertices A, В, С Let γ be the vertex angle at C. For fixed a and b, as γ increases from
0 to я, the function γ м> cosh a cosh b — sinh a sinh b cos γ strictly increases from
cosh(b — a) to cosh(b + a). So by the law of cosines we have cosh с < cosh(b + a),
and hence с < a + b, with equality if and only if γ = π. But γ = π if and only if
С e [A, B]. Statements (1) to (3) are now immediate from the definitions.
The proof that the balls are convex is similar to the one given in (2.3) (one notes
that a point С in [A, B] is of the form λΑ + μΒ, with λ + μ < 1 and λ, μ > 0). D
Henceforth we shall always assume that S" and Ш" are metrized as in (2.2) and
(2.8).
Exercise. Consider a geodesic triangle in H" with vertex angles α, β and 7Г/2. Let d
be the length of the side opposite the angle β. Prove that cos β = cosh d sin a.
Hyperplanes in Ш". A hyperplane Η in M" is, by definition, a non-empty
intersection of W with an и-dimensional vector subspace of E"·'. With the induced metric,
Η is isometric to И"-1. Given two distinct points А, В e W, the set of points in W
that are equidistant from A and В is a hyperplane, called the hyperplane bisector of
A and B; it is the intersection of W with the vector subspace (A — B)1- cE1,1,

22 Chapter 1.2 The Model Spaces Μ"
The reflection r# through Η is the isometry of W which is given byX н·
X — 2 (X | и) и, where и is a unit vector orthogonal to the vector space spanned by
Η (it is unique up to sign). The set of fixed points of r# is H. If A e Ш" does not
belong to H, then Η is the hyperplane bisector of Л and r#(A). Conversely, if Η is
the hyperplane bisector of Л and B, then rH{A) = B.
2.9 Proposition. 77ге spherical (respectively, hyperbolic) angle between two geodesic
segments [C, A] and [C, B] in S" (respectively, Ш") is equal to the Alexandrov angle
between them.
Proof. This proof was communicated to us by Ralph Strebel. We give the details in
the hyperbolic case; the spherical case in entirely similar.
Let a = d(A, C) and b = d(B, C), and let у denote the hyperbolic angle between
[C, A] and [C, B]. Given numbers s, t with 0 < s < a and 0 < t < b, one considers
the points As e [C, A] and Bt e [C, B] that are a distance s and ί from С respectively.
Let csj = d(As, Bt) and let ys%t be the angle at the vertex С in the comparison triangle
A(AS, Bt, С) с Е2 (notation as in 1.10). The Euclidean law of cosines yields
c2s! = s2 + i2 - 1st cos Ys,t.
The hyperbolic law of cosines relates cSyt to s and i:
(2.9-1) cosh cJ>( = cosh s cosh ί — sinh s sinh ί cos y.
But the inverse of cosh |[0ιΟΟ] is not differentiable at 0, so the behaviour of
s2 + f-c2st
COS Ys,t = ~
2st
cannot be gleaned immediately from (2.9-1). To circumvent this difficulty, we
introduce an auxiliary function А: К —> K, given by the power series
OO л
h(x) = V x>.
This power series converges on all of К and it is related to cosh by the formula
cosh* — 1 = hix2). Since /г(0) = 0 and /г'(0) = 1/2, the function h is invertible in a
neighbourhood of 0; this local inverse is given by a power series of the form
OO
h~\x) = 2x + У^ ajX*.
/=2
We continue our analysis of coshcJi(. Using (2.9-1) we can rewrite h(c2t) =
coshcJi( — 1 as
cosh с cosh t — 1 — sinh s sinh t cos γ
= (coshi — l)coshi + (coshi — 1) — sinhi sinhi cosy
= h(s2) cosh t + hit1) — sinh s sinh t cos y.

The Model Spaces M" 23
This last expression defines an analytic function g: R2 —> R with the following three
properties:
g(0, 0) = 0, g(s, 0) = й(52), g(0, t) = hit1).
Moreover the coefficient of st in the power series representation of g is — cos γ.
The first of these properties implies that the composition/ ;= h~l о g, which is
equal to c2( for small positive s and t, is defined in a small disc around the origin. It
has there an absolutely convergent power series representation that we can write in
the form
oo oo oo oo
as, t) = jy^+Σώ*** - ΚΣΣ>.*5/"1/*"1)·
j=\ k=\ j=\ k=\
By the second and third properties of g stated above, the first and second sums in this
expression for/ are equal to s2 and ί2, respectively, so if s and t are small and positive
then the term in parenthesis equals O2 + i2 - c2st)/{st). Finally, by combining the
power series expression of h~l given above with our remark about the st coefficient
of g, we see that/i = 2 cos γ. Hence
cos ySJ = =- = cos γ + 2L № f
k>\ orj>\
tends to cos γ when t, s —> 0+. Π
Remark. This proof shows that the angle between two geodesic segments issuing
from the same point in W exists in the strict sense (as defined in 1.12).
The Model Spaces MnK
2.10 Definition. Given a real number к, we denote by M" the following metric
spaces:
(1) if к = 0 then M(j is Euclidean space E";
(2) if к > 0 then M" is obtained from the sphere S" by multiplying the distance
function by the constant l/</ic;
(3) if к < 0 then M" is obtained from hyperbolic space W by multiplying the
distance function by 1/*J—k.
The following is an immediate consequence of our previous results:
2.11 Proposition. M" is a geodesic metric space. If к < 0, then M" is uniquely
geodesic and all balls in M" are convex. If к > 0, then there is a unique geodesic
segment joining x,y & M" if and only ifd(x, y) < π/\[κ. If к > 0, closed balls in
M" of radius < π/(2«/κ) are convex.

24 Chapter 1.2 The Model Spaces M\
When working with M", one constantly finds it necessary to phrase hypotheses
and conclusions in terms of the quantity π/^/κ, which made its first appearance in
the preceding proposition. For this reason, we introduce the following device.
2.12 The Diameter of M". Henceforth we shall write DK to denote the diameter of
M". More precisely, DK :— π/^/ic for к > 0 and DK := oo for к < 0.
2.13 The Law of Cosines in M". Given a geodesic triangle in M" with sides of
positive length a, b, с and angle γ at the vertex opposite to the side of length c,
for к = 0 : с2 = a2 + b2 - 2abcos(y)
for к < 0 : cosh(V—кс) =
cosh(V—к a) cosh(V—к b) — sinh(V—к a) sinh(V_ к b) cos(y)
for к > 0 : cos(V^с) = cosCV'"*) cosi^/xb) + sin(*/ica) sin(*/Kb) cos(y).
In particular, fixing a, b and к, one sees that с is a strictly increasing function of γ,
varying from \b — a\ to a + b as γ varies from 0 to π.
Note that one can pass from the formula for к > 0 to the formula for —к by
replacing */к with i«Jk = -J— Ik, since for any real t one has cos it = cosh t and
sin it = ί sinh t.
Proof These formulae are obtained from the law of cosines for S" and H" by rescal-
ing. D
Alexandrov's Lemma
In Chapter 1 we introduced the notion of a comparison triangle to enable us to
compare triples of points in an arbitrary metric space to corresponding triples of
points in the Euclidean plane. In Part II we shall have occasion to compare metric
spaces not only to E2, but also to M2 for κ ψ 0. In order to do so we shall need the
following:
2.14 Lemma (Existence of Comparison Triangles in M2.). Let к be a real number,
and let p, q, r be three points in a metric space X; if к > 0 assume that dip, q) +
d(q, r) + d(r, p) < 2DK. Then, there exist points p,q,r e M2 such that dip, q) =
dip, q), d(q, r) = d(q, 7) and d(r, p) = d(r, p).
The triangle Δ(ρ, q,r) с Wfy with vertices ρ, q, r is called a comparison triangle
for the triple ip, q, r); it is unique up to an isometry ofM2. If A C-X is a geodesic
triangle with vertices ρ, q, r, then A(p, q, 7) is also said to be a comparison triangle
for A.

Alexandrov's Lemma 25
Proof. Let a = dip, q), b = dip, г), с = diq, r). We may assume that a < b < с By
the triangle inequality, с < a + b, so in particular if к > 0 then с < π/*/κ. Hence
we may solve to find a unique γ e [0, n) which is related to a, b, с by the equation
in the statement of the law of cosines for M2 (2.12). We fix a point ρ e M2 and
construct two geodesic segments \p, q] and [p, r), of lengths a and b respectively,
meeting at an angle γ. By the law of cosines, d(q, 7) = с
The asserted uniqueness of A(p~, q, 7) is a special case of 2.15. D
It follows from Proposition 2.9 that the definition of Alexandrov angle (1.12)
would remain the same if we were to use comparison triangles in M2 rather E2.
However, the way in which the limiting angle is obtained would be different: instead
of Ζ one would consider the following quantity.
2.15 Definition. Let X be a metric space and let к be a real number. Let p,q,r e X
be three distinct points with dip, q) + diq, r) + dir, p) < 2DK. The к-comparison
angle between q and r at p, denoted Z^\q, r), is the angle at ρ in a comparison
triangle A(p, q, 7) с Μ2 for (ρ, q, r). (In particular £f\q, r) = Zpiq, r).)
Later, particularly in Chapter (II.4), we shall need to bound the geometry of
large geodesic triangles that are constructed from smaller triangles in a controlled
manner. The following technical lemma is invaluable in this regard. The reader may
find it helpful to view this lemma in the following light: take two (sufficiently small)
geodesic triangles in M2K that have one edge in common, with the sum of the interior
angles at one end of this edge, С say, at least π; delete this common edge; imagine
the remaining four edges as rigid bars, joined by hinges at the vertices; imagine
straightening this arrangement so as to decrease the interior angle at С to π. The
main content of the following lemma is that during such a straightening process the
angles at the vertices other than С increase.
We shall need the following terminology: given distinct points x, у € Μ2, with
dix, y) < DK, there is (up to reparameterization) a unique local geodesic ('line')
R —> M2 passing through χ and y; the image of this line separates M% into two
connected components; two points z, w eMj are said to lie on opposite sides of the
line if they are in different connected components of its complement.
2.16 Alexandrov's Lemma. Consider four distinct points A, B, ff, Ce M%; if к > 0,
assume that <i(C, B) + «f(C, B') + diA; B) + diA, B') < 2DK. Suppose that В and B'
lie on opposite sides of the line through A and C.
Consider the geodesic triangles Δ = Δ(Α, В, С) and Δ' = Δ(Α, Β', C). Let
α, β, γ (resp. α', β', γ') be the angles of Δ (resp. A') at the vertices А, В, С
(resp. А, В', С). Assume that γ + γ' >π. Then,
(1) diB, С) + diB', С) < diB, A) + diB1, A).
Let A be a triangle in M% with vertices А, В, В such that diA, B) = diA, B),
d(A, В') = diA, B') and d(B, В') = diB, C) + diC B') < DK. Let С be the point

26 Chapter 1.2 The Model Spaces Μ"
Fig. 2.3 Alexandrov's Lemma
of [В, в'} with d(B, C) = d(B, C). Let α, β, β' be the angles of Δ at the vertices
А, В, В'. Then,
(2) a > a + α', β > β, β > β' and d(A, С) < d(A, С); any one equality implies
the others, and occurs if and only if γ + γ' = π.
Proof. Let В' e M\ be the unique point such that d(B', C) = d(B', C) and С lies on
the geodesic segment [В', В]. Because γ + γ' > π, the angle at Cbetween [C, A] and
[C, B'] is no greater than the angle between [C, A] and [С, В']. Hence, by the law of
cosines, άφ, Α) < ά{β', A). Therefore, d{B, A) + d(Br, A) > d(B, A) + d(B', A) >
d(B, B>) = d(B, C) + d{C, &). This proves (1).
As in (1), we have d(A, В ) > d(A, B1), and by the triangle inequality, d(B, В ) >
d(B, B'); in each case equality is strict unless С е [В, В'}, that is, γ + γ' —it.
Applying the law of cosines to each of these inequalities we get a > a + a' and
β > β. Exchanging the roles of β and β' we get β > β'. A further application of the
law of cosines shows d(A, C) < d(A, C). All of these inequalities are strict except
when γ + γ' = π, in which case Δ is isometric to the union of Δ and Δ' with the
common side [A, C) deleted, so equality holds everywhere. D
The Isometry Groups Isom(M^)
Our next goal is to describe the isometry groups of the model spaces M\. A key
observation in this regard is that each of these groups is generated by reflections in
hyperplanes. If one rescales the metric on a space then one does not alter its group
of isometries, so there are essentially only three cases to consider E", S" and Ш".
The following proposition shows in particular that the group of isometries acts
transitively on M".
2.17 Proposition. Given any positive integer к and 2k points А\,...,Аь and
B\, ... ,BkinM" such that d(Ai,Aj) = d(Bh Bf)for all i,j e {1,..., k}, there exists
an isometry mapping A, to Bit for all i e {1,... ,k]. Moreover, one can obtain such
an isometry by composing к or fewer reflections through hyperplanes.

The Isometry Groups Isom(M") 27
Proof. Arguing by induction on к (the case к = 0 is vacuous) we may assume that
there exists an isometry φ such that $(A) = #г for all i e {1,... ,k — 1}, and φ
is the product of к — 1 or fewer reflections. If ф(Ак) = Bk then we are done. If
not, then let Я denote the hyperplane bisector of ф(Ак) and Bk. As usual, we let r#
denote the reflection through Я. Note that Bt e Я for г = 1, ... ,k — 1, because
d(Bh ф(Ак)) = d(<f>(Ai), ф(Ак)) = d(AhAk) = d(Bh Bk). Hence rH ο φ, which is a
product of at most к hyperplane reflections, sends Л, to Bjfor all i e {1,... , k]. D
2.18 Proposition. Let φ be an isometry ofM".
(1) If φ is not the identity, then the set of points which it fixes is contained in a
hyperplane.
(2) If φ acts as the identity on a hyperplane H, then φ is either the identity or the
reflection гн through H.
(3) φ can be written as the composition of η + 1 or fewer reflections through
hyperplanes.
Proof.
(1) If φ is not the identity, then there is a point A such that φ(Α) φ A. We claim
that the set of points fixed by φ is contained in the hyperplane bisector of Л and ф(А).
Indeed, if В is fixed by φ then d(A, Β) = ά{φ{Α), φ{Β)) = ά{φ{Α), Β), and hence
В еН.
(2) Suppose φ acts as the identity on a hyperplane Я. Given any point Л such that
φ(Α) φ A, the preceding argument shows that Η must be contained in the hyperplane
bisector of Л and ф(А). But no hyperplane is properly contained in any other, so Η is
the hyperplane bisector of Л and ф(А), and hence rH(A) = ф(А). But A was arbitrary,
so φ = rH.
(3) Fix a collection of η + 1 points A0,... ,An that is not contained in any
hyperplane. Given an isometry φ, let B0 = φ(Αο), ..., Bn = φ(Αη). According to the
preceding proposition, there is an isometry φ' which maps each Л,- to 5, and which
can be written as the composition of at most η + 1 reflections through hyperplanes.
The isometry (φ')~] ο φ fixes each of the Лг, and hence, by part (1), it is the identity.
D
We require a base hyperplane H0 С MnK. In E" we take H0 = {0} x E""1; in S"
we take H0 = §" П ({0} x E"); in W we take H0 = W П ({0} x Ε"-'-1). In each
case there is a natural identification of Ho with M"~], and we use this to define what
we mean by a hyperplane in Щ. (In the light of (2.20) this definition will become
obsolete.)
2.19 Exercise. Let Я be a hyperplane in M" and suppose that Но ф Я. If Я0 П Я is
not empty then it is a hyperplane in Ho. Conversely, every hyperplane in Hq is the
intersection of Ho with some hyperplane inM". (Because we defined hyperplanes in
S" (resp. W) as intersections of the sphere (resp. hyperboloid) in R"+1 with vector
subspaces, this is just an exercise in linear algebra.)

28 Chapter 1.2 The Model Spaces Μ"
A set of η + 1 points in M'' is said to be in general position if it is not contained
in any hyperplane. It is obvious that such sets exist.
2.20 Proposition. Let S] and S2 be subsets ofM" and letf : S] -> & be an isometry.
Then,
(1) there exists ψ e Isom(M£) such thaffy\s\ —f;
(2) the restriction of ψ to the intersection of all hyperplanes containing S] is unique.
Proof. We first consider the case where S\ = S2 = Η is a hyperplane. Because every
hyperplane is the bisector of some pair of points, the action of Isom(M") obviously
sends hyperplanes to hyperplanes. And it follows from (2.17) that the action of
Isom(M") on the set of hyperplanes in M'J is transitive. Hence there is no loss of
generality in assuming that Η = Hq.
Let Σ с Но = M"~] be a set of и points in general position and let/ : #0 —> #0
be an isometry. (2.17) gives ψ e Isom(M") such that ^Ις = /Ις· In particular
Σ is contained in the hyperplane ψ~](Ηο). Since Σ is assumed not to lie in any
hyperplane of H0, it follows that ψ~\Η0) = H0 (see 2.19). But now, applying (2.18)
to Ho = M"~], since the fixed points of ψ~]/ € Isom(#0) are not contained in any
hyperplane of H0, we have i/~]f = idHo, i.e./ = ψ\Ηο.
We now consider the general case/ : S\ —> S2. If S\ and S2 are contained in
hyperplanes H\ and Hi, then we can replace them by S\ = ^,-(5,·) С Hq, where
ψί : Η, -> Ho is an isometry. Then, by induction on the dimension of M" (the case
η = 1 is trivial) we can assume that any isometry S\ —> S'2 is the restriction of an
isometry of Ho, and the first step of the proof then applies.
Thus we may assume that S\ is not contained in any hyperplane of M". It follows
that some finite subset S0 С 5t is not contained in any hyperplane of M". By (2.17),
there exists ψο € Isom(M") such that ^olso = f\s0- If l^ols, = / then we are done.
If not, then there would exist χ e Si such that ^oW Φ /(*)> and by applying (2.17)
again we would get ψ\ e Isom(M£) such that ψ χ ls0 = /|s0 and ψχ (χ) = f(x). But this
cannot happen, because we would have a non-trivial element ψχχψο e Isom(M^)
fixing a set So that was not contained in any hyperplane, contradicting (2.18). This
completes the proof of (1).
Suppose that ψ and ψ' are both such that rfr\S[ = rfr'\sr Then ψ~~χψ' and its
inverse send any hyperplane containing S\ to another such hyperplane, and therefore
restricts to an isometry on the intersection of all such hyperplanes. Let / denote this
intersection. By making iterated use of (2.19) and (1) we see that I is isometric to
M™ for some m < n. If Si were contained in some (m — l)-dimensional hyperplane
Pel, then by making further use of (2.19) we could construct a hyperplane in M"
such that /Π Η = Ρ, contradicting the fact that / is the intersection of all hyperplanes
containing S. Therefore Si is not contained in any hyperplane of / = M™, and (2.18)
implies that ψ~χψ' restricts to the identity on /. D
2.21 Definition. Let m < η be non-negative integers. A subset Ρ С М" is called an
m-plane if it is isometric to M™.

The Isometry Groups Isom(M") 29
2.22 Corollary. If m < η then every m-plane in M" is the intersection of
(n — m) hyperplanes. Every subset ofM" is contained in a unique m-plane of minimal
dimension.
Proof. Fix m < n. There certainly exists an m-plane P0 С М" that is the intersection
of (n — m) hyperplanes. The preceding proposition shows that any other m-plane is
the image of Po by an isometry of M", and any such isometry takes hyperplanes to
hyperplanes.
The unique m-plane of minimal dimension containing S is the intersection of all
hyperplanes containing S. D
2.23 The Groups bomCM*,). Using (2.16) we can determine the full group of isome-
tries of E", S" and H". In the case of E" and S", a more pedestrian account (from a
different viewpoint) will be given in Chapter 4. A more explicit description of the
individual isometries of H" will be given in Chapter 6.
Let D(n) denote the group of displacements of E", that is, the group of affine
isomorphisms of E" that preserve the distance. The subgroup of D(n) fixing the
origin 0 e E" is, by definition, the subgroup of GL(«, K) that consists of matrices
which preserve the Euclidean metric. (The action of GL(«, K) is the usual linear
action; a matrix A = [atj] acts thus: A(x) = iy\,... ,y„) where y, = Σ/α^χ}.) A
simple calculation shows that the action of A preserves the Euclidean norm if and
only if A e 0(n), where 0(n) denotes the group of orthogonal matrices, i.e., those
real (n, «)-matrices A which satisfy Α Ά = /, where Ά is the transpose of Л and / is
the identity matrix.
D(n) also contains an abelian subgroup consisting of translations τα : χ м> χ + а.
This subgroup obviously acts transitively, so any element of D(n) can be written
uniquely as the composition of an orthogonal linear transformation and a translation.
For each orthogonal transformation φ we have φταφ~~ι = Тф^а), and hence the group
of translations is normal in D(n). The quotient of D(n) by this normal subgroup
is isomorphic to 0(n) and D(n) is naturally isomorphic to the semi-direct product
W χ 0(n) (cf. 4.13). Finally, we observe that D(n) contains all reflections through
hyperplanes in E", because 0(n) contains all reflections in hyperplanes through 0,
and D(n) (indeed its translation subgroup W) acts transitively on E". Therefore, by
part (3) of the preceding proposition, D(n) is the whole of Isom(E").
We have already noted that the natural action of 0(n + 1) on E"+1 preserves the
Euclidean norm; hence it preserves the и-sphere S". A trivial calculation shows that
the induced action of 0(n + 1) on S" is by isometries. The action is faithful, so we
obtain an injective map from 0(n + 1) to Isom(S"). But, as noted above, 0(n + 1)
contains all reflections through hyperplanes in E"+1 that contain 0, and the restriction
of such reflections to S" are, by definition, the hyperplane reflections of S". We have
shown that such reflections generate Isom(S"), so the natural map from 0(n + 1) to
Isom(S") is actually an isomorphism.
Consider the group GL(n + 1, K) (thought of as matrices, as above) with the
usual linear action on En,1(= M"+1). Let 0(n, 1) denote the subgroup formed by

30 Chapter 1.2 The Model Spaces M"K
those matrices which leave invariant the bilinear form {· | ·). A simple calculation
shows that 0(n, 1) consists of those (n + 1, η + l)-matrices A such that 'AJA = J,
where J is the diagonal matrix with entries (1,..., 1, — 1) in the diagonal. While
0(n, 1) preserves the hyperboloid {x e E"'1 | (x \ x) = — 1}, certain of its elements
interchange the two sheets. Let 0(n, 1)+ с 0(n, 1) be the subgroup of index two that
preserves the upper sheet H". This subgroup consists of those matrices in 0(n, 1)
whose bottom right hand entry is positive. It is clear from the definition that 0(n, 1)+
acts by isometries on W. A direct calculation shows that the stabilizer of the point
(0,.. -, 0, 1) is naturally isomorphic to 0(n). The group 0(n, 1)+ contains all the
reflections through hyperplanes of W, so by the preceding proposition, it is equal to
the full group Isom(H") of isometries of W.
We summarize the preceding discussion:
2.24 Theorem.
(1) Isom(E") = W χ 0(n).
(2) Isom(S") = 0(n + 1).
(3) Isom(H") = 0(n, 1)+.
In all three cases, the stabilizer of a point is isomorphic to 0(n).
Approximate Midpoints
The following result will be needed several times in Part II.
2.25 Lemma. Foreveryu el, I < DK ands > 0, there exists a constant δ(κ, Ι, ε) >
0 such that for all x,y e M\ with d(x, y) < I, ifd(x, m') and d(m', y) are both less
than jd(x, y) + δ, then d(m', m) < ε, where m is the midpoint of[x, y].
This result is a consequence of the transitivity properties of Isom(M") and the
following general observation.
2.26 Lemma. Let X be a proper geodesic space. If there is a unique geodesic
segment [jc, y] joining x,yeX, then for each ε > 0 there exists η > 0 such that
[d(x, z) + d(z, y) < d(x, y) + η] implies d(z, [x, y]) < ε.
Proof. Let S£ = [p e X \ d(p, [x, y]) = ε}. Because X is geodesic, this set is empty
only if every point of X is a distance less than ε from [χ, у]. We consider the case
where it is non-empty. Because X is proper, SB is compact. Let η be the minimum
value attained by f(z) = d(x, z) + d(z, y) — d(x, y) on Se. Note that η > 0 because
[x, y] is unique.
If d(u, [x, y]) > ε then there exists υ € Se such that d(x, u) = d(x, v) + d(v, u).
Therefored(x, u)+d(u,y) = d(x, v)+[d(v, u)+d(u,y)] > d{x, v)+d(v,y). Whence
d(x, u) + d(u, y) > d(x, y) +f(v) > d(x, γ) + η. □

Approximate Midpoints 31
Proof of (2.25). Because the action of Isom(M^) is transitive on pairs of equidistant
points, we may assume that [x, y] is the initial segment of a fixed geodesic segment
[x, y0] of length I. Let щ be a constant such that if d(x, z) + d(z, yo) < d(x, y0) + Щ
then<f(z, [x, yo]) < ε/2.
We have d(x, m') + d(m', y0) < d(x, m') + d(m', y) + d(y, y0), which is at most
d(x, y) + d(y, yo) + 25 = d(x, yo) + 25. So if 25 < η0, then d(m', ρ) < ε/2 for some
ρ e [x, yo]. Then, d(p, x) < d(p, m') + d(m', χ) < ε/2 + 5 + jd(x, y) and similarly
dip, y) < ε/2 + 5 + jd(x, y). But ρ lies on the geodesic segment [x, yo] so therefore
d(p, m) < ε/2 + 5. Thus it suffices to take 5 = ηιίη{ε/3, »7ο/2}. Π

Chapter 1.3 Length Spaces
In this section we consider metric spaces in which the distance between two points
is given by the infimum of the lengths of curves which join them — such a space
is called a length space. In this context, it is natural to allow metrics for which the
distance between two points may be infinite. A convenient way to describe this is to
introduce the notation [0, oo] for the ordered set obtained by adjoining the symbol
oo to the set of non-negative reals and decreeing that oo > a for all real numbers
a. We also make the convention that a + oo = oo for all α e [0, oo]. Having
made this convention, we can define a (generalized) metric on a set X to be a map
d : X xX ^ [0, oo] satisfying the axioms stated in (1.1).
Henceforth we shall allow metrics and pseudometrics to take the value oo.
Length Metrics
3.1 Definition. Let (X, d) be a metric space, d is said to be a length metric (otherwise
known as an inner metric) if the distance between every pair of points x, у e X is
equal to the infimum of the length of rectifiable curves joining them. (If there are no
such curves then d(x, y) = oo.) If d is a length metric then (X, d) is called a length
space.
A complete metric space X is a length space if and only if it has approximate
midpoints in the sense that for all x,y e X and ε > 0 there exists ζ € X such that
тах{ф:, г), d(z, у)} < ε + d(x, у)/2.
An arbitrary metric space gives rise to a length space in an obvious way:
3.2 Proposition. Let (X, d) be a metric space, and let d : Χ χ X -> [0, oo] be the
map which assigns to each pair of points x, у е X the infimum of the lengths of
rectifiable curves which join them. (If there are no such curves then d(x, y) = oo.)
(1) d is a metric.
(2) d(x, y) > d(x, y)for all x, у е X.
(3) If с : [a, b] -> X is continuous with respect to the topology induced by d, then
it is continuous with respect to the topology induced by d. (The converse is false
in general.)

Length Metrics 33
(4) If a map с : [a, b] —> X is a rectifiable curve in (X, d), then it is a continuous
and rectifiable curve in (X, d).
(5) The length of a curve с : [a, b] —> X in (X, d) is the same as its length in (X, d).
(6) d = d.
Proof. Properties (1) and (2) are immediate from the definition of length. (2) implies
(3), and (6) is a consequence of (4) and (5). Property (4) is a consequence of 1.20(5),
so it only remains to prove (5). Let с : [a, b] —> X be a path which has length 1(c)
with respect to the metric d, and length 1(c) with respect to the metric d. On the one
hand we have that 1(c) > 1(c), by (2), and on the other hand
/t-l _
~l(c)= sup Y^d(c(ti), c(ti+x))
a=to<tt<-<tt=b i=0
k-l
< sup ]P/(<%,_,,,,]) =/(c).
a=ro<<i<-<r*=& ,=0
Hence 1(c) = 1(c). D
3.3 Definition. Let (X, d) be a metric space. The map d defined in (3.2) is called the
length metric (or inner metric) associated to d, and (X, d) is called the length space
associated to (X, d). Note that d = d if an only if (X, d) is a length space.
The induced length metric on a subset Υ с X is the length metric associated to
the restriction of d to Υ χ Υ (which in general will not be the same as the restriction
ЮУ χ Υ of d).
3.4 Examples. Let (X, d) be a metric space. The identity map on X induces a
continuous map (X, d) —> (X, d), but this is not a homeomorphism in general. For instance,
the metric space (Y, d) considered in (1.19) is homeomorphic to [0, 1], but the
associated length space is homeomorphic to the disjoint union of [0, 1) and {1}. More
generally, the length space associated to a metric space (X, d) is connected if and
only if every pair of points in (X, d) can be joined by a rectifiable curve.
As another example, we can consider the set of rational numbers Q with the usual
metric d induced from №.. In the associated length metric d, the distance between
every pair of distinct points of Q is infinite, hence d induces the discrete topology
onQ.
3.5 Example. In the Euclidean plane we consider the complement X of an open
sector of angle a < π, namely X = {x e K2 | (x \ e\) < cos(a/2) \\x\\}, where
e\ = (0, 1), and (· | ·) is the Euclidean scalar product on K2, with associated norm
II - II-
We consider X as a length space, with the induced length metric from E2. This
is a uniquely geodesic space: if the geodesic in E2 which joins χ e X to x1 e X lies
entirely in X then this is the unique geodesic segment joining χ to x1 in X; otherwise,

34 Chapter 1.3 Length Spaces
the unique geodesic from л to x1 in X is the concatenation of the Euclidean geodesies
joining χ to 0 and 0 to x1.
3.6 Exercises
(1) Prove that the induced length metric that §" inherits from E"+1 coincides
with the metric defined in 2.1. Prove that for all κ φ 0, all χ e Μ" and all r > 0
(assuming r < π/y/ic if к > 0), the induced length metric on {3? e M£ | d(x, y) =
r\ с Μ" makes it isometric to Μ"ΓΧ 2 ~. (If к < 0 then sinir-v/ic) = sin(i/V—*0 =
i'sinh(rV—к).)
(2) Give an example of a subset У с Е2 such that the distance between each pair
of points in the induced length metric is finite but the topology which this metric
induces on Υ does not coincide with the topology given by the restriction of the
Euclidean metric.
Recall that any metric space admits a completion. In other words, every metric
space X can be embedded isometrically as a dense subspace of a complete metric
space X', which is unique up to isometry. One can obtain X' as the set of Cauchy
sequences in X modulo the equivalence relation: (xn) ~ (y„) if and only Ve >
0 3Νε > 0 such that d(xn, y„) < ε for all η > Νε. Each χ e X is identified with the
class of the stationary sequence xn = x, and the metric on X is extended to X' by:
d([x„], Ы) = l>m« d(xn, y„).
(3) Prove that if X is a length space then its completion X' is a length space. (The
converse is obviously false, e.g. (3.4).)
(4) Prove that there exists a geodesic metric space which is locally compact but
whose completion is neither geodesic nor locally compact.
(Hint: Consider the induced path metric on the following subset of the Euclidean
plane: (0, 1] x {0} U(0, 1] χ {1} UU^,{l/«} x [0, 1].)
(5) Let d\ and dj. be two metrics on a space X which are Lipschitz equivalent, i.e.
there exists a positive constant С such that
1
~f,d2{x, y) < di(x, y) < Cd2(x, y),
for each x, у € X. Show that the same inequality is then satisfied by the length metrics
associated to d\ and d2.
(6) If one has a metric space (X, d) and a positive function/ : [0, 00) —> [0, 00),
then one can apply the construction of (1.24) with ρ := f о d. Show that if X is a
length space with/(a + b) > f(a) +f(b) for all a, b > 0 and lim,_>0 f(f)/t = 1, then
dp =d.

The Hopf-Rinow Theorem 35
The Hopf-Rinow Theorem
In general, a length space need not be a geodesic space, even if the distance between
every pair of points is finite. For example, the induced metric on the Euclidean plane
minus the origin is a length metric, but there is no geodesic joining jil!- {0} to
—x. The space in this example is not complete, and in fact if one restricts to complete
locally compact spaces, then a length space in which the distance between each pair
of points is finite must be a geodesic metric space. This was known to Cohn-Vossen
[CV35a] and proved earlier in case of surfaces by Hopf and Rinow [HoRi32] (see
also the Appendix in [Rh52]). For a more general statement of the Hopf-Rinow
Theorem, see Ballmann [Ba95, p.13-14].
3.7 Proposition (Hopf-Rinow Theorem). Let X be a length space. IfX is complete
and locally compact, then
(1) every closed bounded subset ofX is compact;
(2) X is a geodesic space.
Proof. We follow the treatment of [GrLP81]. For (1) it suffices to prove that closed
balls about a fixed point a e X are compact. Given r > 0, we denote by B(r) =
{x e X | d(a, x) < r} the closed ball with centre a and radius r. Consider the set of
non-negative numbers ρ such that B(p) is compact. This is an interval containing 0;
we claim that it is both open and closed. Because X is assumed to be locally compact,
the interval contains a neighbourhood of 0. To see that it contains a neighbourhood
of each of its other points, we fix ρ > 0 such that B{p) is compact, and use the
local compactness of X to cover B(p) with finitely many balls 5(jc,-, ε;) such that each
B(x,, ε,·) is compact. There is a strictly positive lower bound, 25 say, on the distance
from any point in B(p) to the closed set X \ \J B(xt, ε,-), and hence B(p + 5) is a
closed subset of the compact set \J B(xj, ε,·).
It remains to prove that if B(r) is compact for all r < ρ then B(p) is compact. It
suffices to show that every sequence of points xn e B(p) such that d(a, xn) converges
to ρ has a convergent subsequence. We fix such a sequence (xn).
Let ερ be a sequence of positive numbers tending to 0. For each ρ and each и,
we can find a point у°п such that d(a, yPn) < ρ - ερ/2 and d(fn, xn) < ερ (to see this
one chooses a path с of length smaller than d(a, x„) + ερ/2 joining a to x„, and then
chooses a convenient point у°„ on this path). For each p, the points yy, are contained
in the compact ball B(p — ερ/2), hence we can extract from 0^)neN a convergent
subsequence (yl,)keN\ from the sequence (у2,)^е^ we can then extract a convergent
subsequence, and so on. Eventually, by a diagonal process, we obtain a sequence of
integers («,t)/t<=N such that the sequence 04)teN converges for all p. We claim that
the corresponding sequence (x„t)k<=N is Cauchy. Indeed, for a given ε > 0, we can
choose ρ such that ερ < ε/3 and then use the fact that the sequence у"Пк is convergent
(hence Cauchy) to see that for large к, к we have d(y%t,y^k,) < ε/З, and hence
d(xnt,xnk,) < d(xni,fnt) + d(fnt,/niJ) + d(fni,,xnt,) < ερ + εβ + ερ < ε.

36 Chapter 1.3 Length Spaces
We shall now prove (2). Let a and b be distinct points of X. For every integer
η > 1, there is a path c„ : [0, 1] —> X, parameterized proportional to its arc length,
such that/(c) < ύ?(α, /?) + l/n. Such a family of paths {c„} is equicontinuous; indeed
for all t, f e [0, 1] we have:
Ιί-ί'ι = г(с"Ь/]) > Л(с„(0, c(Q)
/(c) - d(a,b)+l '
hence d{c„(t), cn(f)) <eif|i — f\ < ^-frrf · The image of each path c„ is contained
in the compact set 5(2 d(a, b)). By the Arzela-Ascoli theorem (see below) there is a
subsequence of the (с„)п(=м converging uniformly to a path с : [0, 1] —> X. Finally,
by the lower semicontinuity of length (1.18(7)), we have
/(c) < liminf l(ckJ = d(a, b).
But /(c) > ύ?(α, /?), so in fact /(c) = ύ?(α, b), and therefore с is a linearly reparam-
eterized geodesic joining a to b. D
3.8 Corollary. A length space is proper if and only if it is complete and locally
compact.
The argument used to prove part (2) of the preceding proposition shows, more
generally, that if two points in a proper metric space are joined by a rectifiable curve,
then among all curves joining them there is one of minimal length (see [Hil900]).
3.9 Remark. The terminology 'proper metric space', introduced in (1.1), arises from
the fact that a metric space (X, d) is proper if and only if, given a base point jeo € X,
the function χ —» d(x, xo) is a proper map from X to Μ in the usual topological sense,
i.e., the inverse image of every compact set is compact.
Because we shall have rather frequent need of it, we include a proof of the Arzela-
Ascoli theorem. Recall that a sequence of maps/, from one metric space to another
is said to be equicontinuous if for every ε > 0 there exists 5 > 0 such that for every
η e N, ifd(x, y) < δ then d(fn(x)Jn(y)) < ε.
3.10 Lemma (Arzela-Ascoli). If X is a compact metric space and Υ is a
separable metric space, then every sequence of equicontinuous maps f, : Υ —> X has a
subsequence that converges (uniformly on compact subsets) to a continuous map
f:Y->X.
Proof. We first fix a countable dense set Q = {q\, q2, ■ ■ ■} in Y. Then we use the
compactness of X to choose a subsequence/,^) such that the sequence of points
/«(i)(#i) converges in X as η —> oo; we call the limit point f(q\). We then pass to
a further subsequence fn(2) to ensure that/„(2)(^2) converges to some point f{qi), as
η —> oo. Proceeding by recursion on k, we pass to further subsequences fn^) so

The Hopf-Rinow Theorem 37
that as n(k) tends to infinity fn(k)(qj) converges to /(<#) for all j < к. The diagonal
subsequence /,(„> has the property that limn^00/n(n)(^) =f(q) for all q e Q.
By equicontinuity, for every ε > 0 there exists 5 > Osuch that if ^(у, /) < 5 then
d(fn(y),fn(y')) < ε for all n. So taking the limit, difiqlfiq1)) < ε for all q,q' e Q
with d(q, q') < δ. Since X is compact (hence complete) it follows that/ has a unique
continuous extension Υ —> X, which again satisfies this inequality.
It remains to show that the convergence of/,(„) to/ is uniform on compact subsets.
Given ε > 0 we choose 5 as above. Given a compact subset С с У we fix N > 0
so that for every yeC there exists j(y) < N with d(y, y^) < 8, then we fix Μ
sufficiently large so that d(f„^(qj),f(qj)) < ε for all η > Μ and all j < N. Then, for
all у е С and all η > Μ we have:
d<f(y),fm(y))
< dtfifiJiqny))) + difiqj^J^qj^)) + dif^qj^J^iy))
<ε+ε+ε.
Thus/n(„) —> / uniformly on compact sets as η —> oo. D
3.11 Corollary. Τ/Τ is a compact metric space and ifc„ : [0, 1] —> X is α sequence
of linearly reparameterized geodesies, then there exists a linearly reparameterized
geodesic с : [0, 1] —> X йий? α subsequence сп(1) ^ис/г ί/ϊαί сп(1) —> с uniformly as
n(i) —> oo.
Prao/ For every и e Nandalli, f7 e [0, 1] wehaveif(c„(iX с„(0) < Z>|r — ίΊ, where
D is the diameter of X. Thus the c„ are equicontinuous. A uniform limit of geodesies
is a geodesic. D
Later we shall need the following variation on (3.11). It is important to note that
here it is not necessary to pass to a subsequence in order to guarantee convergence.
3.12 Lemma. Let χ andy be points in a proper geodesic metric space X. Suppose that
there is a unique geodesic segment joining χ to у in X; letc: [0, 1] —> Xbea linear
parameterization of this segment. Let cn : [0, 1] —> X be linearly reparameterized
geodesies in X, and suppose that the sequences of points c„(0) and c„(l) converge to
χ and у respectively. Then, cn —> с uniformly.
Proof. We fix R > 0 so that the image of each of the paths c„ lies in the (compact)
closed ball of radius R about x. If the sequence c„ did not converge to с pointwise,
then there would exist ε > 0, to e (0, 1) and an infinite subsequence c„, such that
d(cni(to), c(io)) > ε for all щ. The previous corollary would then yield a subsequence
ofthecHi converging uniformly to a linearly reparameterized geodesic d : [0, 1] —> X
joining χ = limc„,(0) to у = limc„,(l). But since d(d(t0), c(io)) > e, this would
contradict the uniqueness of с
Thus cn^- с pointwise. Using the fact that с and cn are geodesic it is easy to see
that the convergence must be uniform. D

38 Chapter 1.3 Length Spaces
Let X be a uniquely geodesic space. Let c(x, y) denote the linear reparameteri-
zation [0, 1] —> X of the geodesic segment joining χ to y. Geodesies in X are said
to vary continuously with their endpoints if c(xn, y„) —>■ c(x, y) uniformly whenever
xn —> χ and yn —> y.
3.13 Corollary. If a proper metric space X is uniquely geodesic, then geodesies in
X vary continuously with their endpoints.
The following exercise shows that the hypothesis of properness in the above
corollary is necessary.
3.14 Exercise. Let Y„ be the subspace of S2 obtained by taking a sector bounded
by two arcs of great circles, each of length π, which meet at the north and south
poles, at an angle π/A, and then removing the open ball of radius \/n about the
north pole. Yn with the induced length metric is a topological disc whose boundary
consists of three geodesic segments, two of which have length π — \; let a„ be one
of the latter segments. Let X be the space obtained by taking the interval [0, n] and,
for every integer η > 2, attaching Y„ to an initial segment of [0, π] by an isometry
cen -> [0, π - i]; endow X with the unique length metric that restricts to the given
metric on [0, π] and on each of the Y„.
Show that X is complete, uniquely geodesic, and admits a (Lipschitz-1)
contraction to a point, but that geodesic segments in X do not vary continuously with their
endpoints.
We close this section by noting a further application of the Arzela-Ascoli theorem
in the spirit of the Hopf-Rinow theorem.
3.15Definition. ByaZoo/jinametricspaceXwemeanacontinuousmapc : S1 —> X.
Such a loop is called a closed local geodesic if there exists a constant λ such that
d(c(0), с{в')) = λά{θ, Θ') for all sufficiently close Θ, Θ' e S1 (where the circle S1 is
equipped with its usual length metric).
Recall that two loops c, d : S1 —> X are said to be homotopic in X, and we
write с ~ c', if there exists a continuous map F : S1 χ [0, 1] —> X such that
F(0, 0) = с(в) and F(0, 1) = c'{0) for all θ eS'.A space X is said to be semi-
locally simply-connected if every χ e X has a neighbourhood such that each loop in
that neighbourhood is homotopic in X to a constant map.
3.16 Proposition. IfX is a compact, semi-locally simply-connected, geodesic space,
then every loop с : S1 —> X is homotopic either to a constant path or to a closed
local geodesic.
Proof. The fact that X is compact and semi-locally simply-connected implies that
there exists r > 0 such that every closed loop of length less than r is homotopic to a
constant map. So if с is not homotopic to a constant map then I = inf{Z(c') | d — c}

Riemannian Manifolds as Metric Spaces 39
is strictly positive. We choose a sequence of loops parameterized by arc length,
cn : S1 -> X which are homotopic to с whose lengths tend to I. By the Arzela-
Ascoli theorem (3.10), this sequence has a convergent subsequence; we must show
that the limit Coo, which is obviously a closed local geodesic, is homotopic to c. For
this, we choose η sufficiently large so that d(c„{6), cTO(6))) < r/A for all θ e Sl, then
we fix θο, ■.., вт so that а(с„(в{), c„(6>i+1) < r/A and d(cоо(в{), ^(θί+ι)) < r/A for
all i (indices mod m). Let pt be a path of length < r/A joining CcxXfy) to c„(0,). For
each i, one obtains a loop of length less than r by concatenating /?,, c„ | [β,,β/+ι ], then/?1+1
and Ccx,|[e,.,e,+]] with reversed orientation. A loop of such length is null-homotopic. It
follows that Coo is homotopic to cn and hence с. П
Riemannian Manifolds as Metric Spaces
We now turn our attention to the study of Riemannian manifolds from the metric
viewpoint. Recall that a Riemannian manifold is a differentiable manifold X together
with an assignment of a scalar product to the tangent space TXX to X at each point
x, such that these scalar products vary continuously with x. Such an assignment of
scalar products is called a Riemannian metric on X. More explicitly, in the classical
notation, if (jci ,..., jc„) are local coordinates (of class C1) in an open set U of X then,
at each χ e U, the scalar product on the tangent space TXX is given by a formula of
the form
η
ds2 = YJ gij(x) dxi dxj
where the gy(x) are continuous functions on U. This means that, at each point χ e U,
if ν = Σ vt(x)d/dxj and w = Σ W;(je)9/9je/ are two vectors in TXX, then their scalar
product is (v\w)= Σϊυ=ι gij(x)Vi(x)Wj(x).
If t м> c(i) is a differentiable path in X, then one writes c(t) e Τφ)Χ to denote its
velocity vector at time t, and \c(t)\ to denote its norm with respect to the given scalar
product on TC(,)X.
3.17 Definitions. If c(t) and c'(t) are two continuously differentiable paths such that
c(0) = c'(0), then the Riemannian angle between them at c(0) is the angle between
the vectors c(0), c'(0) e Τφ)Χ, namely the unique a e [0, π] such that
(c(0) I c'(0))
cos α = .
|c(0)| |c'(0)|
The Riemannian length Ir{c) of a piecewise differentiable path с : [a, b] —> X is
defined by:
Ш = f \c(t)\ dt.
J a
3.18 Proposition (The Distance Function on a Riemannian Manifold). Let X be a
connected Riemannian manifold. Given x, у e X, let d(x, y) be the infimum of the

40 Chapter 1.3 Length Spaces
Riemannian length of piecewise continuously differentiable paths с : [0, 1] —> X
such that c(0) = χ and c(l) = y.
(1) d is a metric on X.
(2) The topology on X defined by this distance is the same as the given (manifold)
topology on X.
(3) (X, d) is a length space.
3.19 Example. The simplest example of a Riemannian manifold is W with the
Riemannian metric
ds2 = J2dxl
The associated metric is the Euclidean metric, i.e., the associated length space is E".
We shall often abuse terminology to the extent of referring to the length space (X, d)
defined in (3.18) as a Riemannian manifold.
Proof of 3.18. It is clear that d is a pseudomelia To check that it is positive
definite, we fix ρ e X and consider a local coordinate system (jcj, ..., xn) defined in
a neighbourhood V of ρ such that the map q м> (x\(q),..., x„(q)) is a diffeomor-
phism of V onto an open set of W containing the closed unit ball centered at the
origin, and jc,(p) = 0 for i = 1,... , n. Let U be the set of points q e V such that
X^=i *ί(#)2 < 1· Suppose that in these local coordinates the Riemannian metric is
given by the expression
η
ds2 = Σ gij(q)dxidxj.
We want to compare it with the Euclidean metric on U given by
i
Given q e U and υ e TqX, we denote by dgip, q) the Euclidean distance Σ"=ι xi(q)2
and by |i>|e (resp. |υ|) the norm of ν with respect to the metric ds\ (resp. ds2).
Let m (resp. M) be the infimum (resp. the supremum) of |υ| over all υ e TqX
with |i>|e = 1 and q e U. By compactness, we have m > 0 and Μ < oo. And for all
ν e Tq(U), we have
™\v\e < \v\ <M\v\E.
Therefore, for each piecewise differentiable curve с in U, we have
m Ie(c) < h{c) < MlE(c)
where lE(c) denotes the length with respect to the Riemannian metric dsE.

Riemannian Manifolds as Metric Spaces 41
In particular, if с joins ρ to q, then Ir(c) > m cIe(p, q), hence dip, q) >m dgip, q).
On the other hand, if q φ U, then dip, q) > m. This shows that the pseudometric d
is positive definite. Thus we have established (1).
The balls Be(p, r) := {q e U \ dgip, q) < r} form a fundamental system of
neighbourhoods of ρ for the given topology on the manifold X. And for r < m, we
have
BE(P, r/M) с В{р, г) с вЕ(р, rim).
Thus the given topology on X agrees with the topology associated to the metric d.
To prove (3), let d be the length metric associated to d. Given p, q e X and
ε > 0, we can find a piecewise differentiable curve с joining ρ to q and such that
lR(c) < d(p, g)+e. And from the definition of length (1.18) it is clear that Z(c) < Ir(c).
Hence
dip, q) < d(p, q) < 1(c) < lR(c) < dip, q) + ε.
As ε is arbitrary, we conclude that d = d. D D
By combining the preceding result with the Hopf-Rinow Theorem (3.7) we
obtain:
3.20 Corollary. Every complete, connected, Riemannian manifold is a geodesic
metric space.
3.21 Remarks. One can prove that the Riemannian length of any piecewise
continuously differentiable curve is equal to its length in the metric constructed in (3.18).
Indeed a similar statement is true for curves in any smooth manifold endowed with a
continuous Finsler metric (see [BusM41], [Rin61]). Recall that a Finsler metric on a
smooth manifold X is an assignment to each tangent space TXX of a norm; this norm
varies continuously with x. Proposition 3.18 is also true for Finsler manifolds (with
the same proof).
In the same vein, we note that the Riemannian angle between two geodesies
issuing from the same point of a smooth enough Riemannian manifold is equal to
the Alexandrov angle between them. This will be proved in (II. LA).
3.22 Remark. A Riemannian isometry of a Riemannian manifold X is, by definition, a
diffeomorphism of X whose differential preserves the scalar product on each tangent
space. A Riemannian isometry is obviously an isometry of the associated length
space (X, d). The converse is also true if the Riemannian metric is of class C2 (see,
for example, [Hel78]).
If X is a Riemannian manifold and Υ с X is a smoothly embedded submanifold,
then the restriction to TXY of the given scalar product on TXX gives a scalar product
on the tangent space to Υ at each point χ e Y. Thus Υ inherits a Riemannian structure
from X, and we can consider the associated length space, as defined in (3.18). We
note a consequence of (3.18):

42 Chapter 1.3 Length Spaces
3.23 Proposition. Let X be a Riemannian manifold and let Υ с χ be a smoothly
embedded submanifold. Then, the induced length metric on Υ (as defined in 3.3)
coincides with the length metric associated to the Riemannian structure which Υ
inherits from X.
A similar observation holds for Finsler metrics.
Length Metrics on Covering Spaces
A continuous map ρ : Υ —> X between topological spaces is said to be a covering
map if it is surjective and every point χ e X has an open neighbourhood U such that
p~x U is a disjoint union of sets Ua such that/? restricts to a homeomorphism of each
Ua onto U. Whenever one has a covering ρ : Υ —» X, the covering space Υ inherits
all of the local structure of the base X. For example, a covering space of a manifold is
again a manifold; given a covering of a complete Riemannian manifold, one can use
the covering map to pull back a complete Riemannian metric; if the base manifold is
non-positively curved, then the pull-back metric on the covering space will also be
non-positively curved. Similarly, since measuring the length of a curve in a metric
space is a purely local process, there is a natural way to pull back the definition of
length to covering spaces. The purpose of this paragraph is to make this last idea
precise. We also give a criterion for recognizing covering spaces; this will be needed
in Chapter II.4.
3.24 Definition. Let X be a length space and let X be a topological space. Let
ρ : X —» X be a continuous map that is a local homeomorphism (i.e., every point
χ e X has an open neighbourhood U such that ρ maps U homeomorphically onto an
open set of X)4. Given a path с : [0, 1] -> X, we define its length 1(c) to be the length
of the curve ρ о с in X, We define a pseudometric on X by setting the distance d(x, y)
between two points x, у e X equal to the infimum of the length of paths in X joining
them. (One checks easily that provided X is Hausdorff, this is indeed a metric; it may
be infinite if X is not connected or if the given length metric on X is infinite.) The
metric d is called the metric induced on X by p.
3.25 Proposition. Let X be a length space, and let X be a topological space that is
Hausdorff. Let ρ : X —> Xbea continuous map that is a local homeomorphism, and
let d be the metric induced on X by p.
(1) If one endows X with the metric d, then ρ becomes a local isometry.
(2) d is a length metric.
(3) d is the unique metric on X that satisfies properties (1) and (2).
4 Such a continuous map ρ : X —> X will also be called an etale map.

Length Metrics on Covering Spaces 43
Proof. For (1), we must show that, given χ e X with image p(x) = x, if r > 0 is
sufficiently small then the restriction of ρ to B(x, r) is an isometry onto B(x, r). Let U
be an open neighbourhood of χ such that the restriction of ρ to U is a homeomorphism
onto its image p(U) = U, and let s : U —> (7 be the inverse of /?|y. Let r > 0 be
small enough so that B{x, 2r) с f/. We claim that s restricted to B(x, r) is an isometry
onto B(x, r). Indeed, given y, ζ € 5(x, r) and ε e (0, r), since X is a length space,
there exists a path с joining у to ζ in 5(jc, 2r) of length smaller than d(y, ζ) + ε;
its image under s is a path joining s(y) to j(z), hence d(s(y), s(z)) < d(y, ζ) + ε for
arbitrarily small ε > 0. But, by construction, ρ does not increase distances, therefore
d(s(y), s(z)) = d(y, z). It remains to prove that s : B(x, r) —> 5(3c, r) is sunective. But
this is clear, because ρ ο s restricts to the identity on B(x, r) and ρ does not increase
distances. This completes the proof of (1).
Because ρ is a local isometry, the length of a curve in X is the same as the length
of its image under p; assertion (2) follows immediately.
1Ы'is any metric on Xuat satisfies (1), then the identity map id : (X, d) -> (X, d')
is a local isometry, in particular it preserves the length of curves. So if <f' is assumed
to be a length metric then id : (X, d) —> (X, d) must be an isometry. D
3.26 Definition. A metric space X is said to be locally uniquely geodesic if for each
point χ e X there is an r > 0 such that every pair of points y, ζ e 5(jc, r) can be
joined by a unique geodesic in X and this geodesic lies in B(x, r).
3.27 Remarks.
(1) It is a classical theorem in differential geometry that a Riemannian manifold of
class C2, when considered as a metric space (see 3.18), is locally uniquely geodesic.
(2) If X is locally uniquely geodesic and proper then geodesies vary continuously
with their endpoints locally. That is to say, with the notation of the above definition,
geodesies in B(x, r) will vary continuously with their endpoints (cf. 3.13).
In the following proof weshall use the fact that ifXisHausdorff and if ρ :Χ^-Χ
is a local homeomorphism, then ρ has the property that 'lifts are unique': if/ and/'
are two continuous maps of a connected space Υ into X such that pof = pof, and
if /Су) = /Су) f°r some point у € Y, then/ = / on the whole of Y. To see that ρ has
this property, one simply notes that the set of points in Υ at which/ and/ coincide
is both open and closed.
3.28 Proposition. Let ρ : X —> Xbea map of length spaces such that
(1) X is connected,
(2) ρ is a local homeomorphism,
(3) the length of every path in X is not bigger than the length of its image under p,
(4) X is locally uniquely geodesic and geodesies in X vary continuously with their
endpoints locally, and
(5) X is complete.
Then, ρ is a covering map.

44 Chapter 1.3 Length Spaces
Note that conditions (2) and (3) are satisfied if/? is a local isometry.
Proof. We first prove that given a rectifiable curve с : [0, 1] -> X and a point χ e X
such that p(x) = c(0), there is a unique path с : [0, I] -*■ X that is a lift of с at x,
in the sense that c(0) = ic and p(c(t)) = c(t) for all ί e [0, 1]. Suppose that such
a lift has been constructed over an interval [0, a). Let t„ e [0, a) be a sequence of
points converging to a. By (3), we have d(c(t„), c(tm)) < Kc\[ti„ti„)) < /(с|[(„.,^). As
the sequence of numbers /(c|[o,i,]) is Cauchy, c(t„) is a Cauchy sequence in X, and
hence converges to a unique point, which has the properties required of c(a). Thus
the given lifting can be extended to the closed interval [0, a]; this shows that the
maximal subinterval of [0, 1] that contains 0 and on which a lifting с exists is closed;
it is also open by condition (2), hence it is the whole interval.
As X is (path) connected, the argument of the preceding paragraph implies that
the restriction of ρ to each connected component of X is a surjection onto X. What
remains to be proved is that for every χ e X there is a neighbourhood U of χ such
that the restriction of ρ to each component of p~x U is a homeomorphism onto its
image. We shall show that it suffices to let the role of U be played by any ball B(x, r)
which is uniquely geodesic, the geodesic segment joining χ to each point у e B(x, r)
depending continuously of its endpoints.
Given such a ball, we fix a point χ e p~x(x), and for each point у е B(x, r) we
denote by cy : [0, 1] —> B(x, r) the linearly reparameterized geodesic joining χ to y,
and we let cy denote the unique lifting of cy with cy(0) = x. Let ij : B(x, r) —» X
denote the map у м> су{\). We claim that si is a homeomorphism onto an open set
of X. By (2), it is sufficient to check that it is continuous at y.
Because ρ is a local homeomorphism, we can cover the image of cy with a finite
number of balls Bk с B(x, r), so that cy([(k - 1)/и, fc/и]) С. Вк ίοτ к = 1, ..., и,
and there exist continuous maps j* : fit —> X with /op equal to the identity on Bk
and i*(c,,(i)) = s-x{cy(t)) for all ί e [(& - 1)/и, fc/и]. As geodesic segments in B(x, r)
depend continuously of their endpoints, if 5 > 0 is small enough then for all ζ with
d(y, ζ) < δ, we have cz([(k — 1)/и, к/п\) с 5^.. And we may define a continuous
map 5 : B(y, 5) x [0, 1] -> X by: 5(z, i) = i*(cz(i)) if *€[(£- 1)/и, fc/и]; to see
that this map is well-defined and continuous, one observes that on the connected set
B(y, 5) x {tk/n} the definitions using s^\ and sk agree at (y, tk/n) and hence everywhere
(because hypothesis (2) ensures that /j has the property that lifts are unique). Since
t ы> s(cz(t)) is a lifting of cz at it, it must coincide with cz. Therefore the restriction of
ij to B(y, 5) agrees with the continuous map ζ м> s(z, 1); in particular s^ is continuous
aty.
We have shown thatp~l(B(x, r)) is the union of the open sets sz(B(x, r)) where
χ e p~l(x), and ρ restricted to each of thee sets is a homeomorphism onto B(x, r).
Finally we observe that these sets must be disjoint; for if у е s^(B(x, r))Ds?(B(x, r)),
then the lifts of cpQ) beginning at χ and x1 both end at y, hence they must coincide
and χ = x1. Therefore ρ is a covering map. D
3.29 Remark. An examination of the preceding proof shows that one can obtain the
same conclusion under the following alternative hypotheses.

Manifolds of Constant Curvature 45
(Γ) ρ is surjective,
(2') ρ is a local homeomorphism,
(3') the length of every path in X is not bigger than the length of its image under p,
(4') for every χ e X there exists r > 0 such that, for every у е B(x, r), there is
a unique constant speed geodesic cy : [0, 1] —* B(x, r) joining χ to y, and cy
varies continuously with y, and
(5') for every у e p~l(y) there is a continuous lifting cy : [0, 1] —> ^ of cy with
c,(l)=y-
3.30 Exercises
(1) Give an example of a compact geodesic space that shows that condition (4') is
strictly weaker than condition (4). (Hint: Consider a cone with a small vertex angle.)
(2) By adapting example (3.14), show that the continuity requirement in
conditions (4) and (4') is necessary, even for locally contractible spaces (if one does not
assume that the space is proper).
Manifolds of Constant Curvature
3.31 Definition. By definition, an и-dimensional manifold of constant curvature к
is a length space X that is locally isometric to M". In other words, for every point
χ e X there is an ε > 0 and an isometry φ from B(x, ε) onto a ball Β(φ(χ), ε) с М%.
3.32 Theorem. Let X be a complete, connected, η-dimensional manifold of constant
curvature к. When endowed with the induced length metric (3.24), the universal
covering ofX is isometric to M".
Proof. The following proof is due to С Ehresmann [Ehr54]. In the first part of the
proof we do not assume that X is complete.
By definition, a chart φ : U —*■ M"is an isometry from an open set U с X onto
an open set φ(ϋ) с Щ. If φ' ; {/' _> Щ is another chart and if Uil U' is connected,
then by (2.20) there is a unique isometry g e Isom(M£) such that φ and g ο φ' are
equal on U П U'.
Consider the set of all pairs (φ,χ), where φ : U —> M" is a chart and χ e U. We
say that two such pairs (φ, χ) and (φ1, χ1) are equivalent if χ = χ1 and if the restrictions
of φ and 0' to a small neighbourhood of χ coincide. This is indeed an equivalence
relation and the equivalence class of (φ, χ) is called the germ of φ at x. Let X be the
set of all equivalence classes, i.e. the set of all germs of charts. Let ρ : X —>- X and
D : X —> M" be the maps that send the germ of φ at χ to χ and 0(x) respectively.
Notice that there is a natural action of G — Isom(M") on X: if Jc is the germ of
φ at χ and if g e G, then g.ic is the germ of g ο φ at χ. The map i> : X —> M"
is G-equivariant: g.(D(x)) = D(g.x), and according to (2.20), given jc, x7 e p-1(jc),
there is a unique g e G such that g.x = x1.

46 Chapter 1.3 Length Spaces
There is a natural topology on X, called the germ topology, with respect to which
ρ is a covering and D is a local homeomorphism. The basic open sets defining this
topology are Щ, where φ : U -> M" is a chart and C/^ с X is the set of germs of φ
at the various points of U. The restriction of ρ (resp. D) to ί/ψ is a homeomorphism
onto U (resp. φ(ϋ)). Moreover, if U is connected thenp~l(U) is the disjoint union
of the open sets Ugolp, where g e G. Thus ρ : X —> X is a covering map (and in
particular X is Hausdorff). Indeed ρ : X —■► X is a Galois covering with Galois group
G (meaning that G acts by homeomorphisms on X preserving the fibres of ρ and
acting simply transitively on each fibre).
Choose a base point jeo e X and a chart φ defined at jeo. Let xo e X be the germ
of φ at xo, and let X be the connected component of X containing xq. Let ρ : X —> X
and D : X -> M" be the restrictions of ρ and D to X. Let Г С G be the subgroup of
G that leaves X invariant. Then ρ : X —> X is a Galois covering with Galois group
Γ and the map D is a local homeomorphism which is Γ-equivariant. If we endow X
with the unique length metric d such that/? is a local isometry (3.24), then D becomes
a local isometry.
Now we assume that X is complete. Then (X, d) is also complete, and (3.28) tells
us that D : X —> M" is a covering. As M" is simply connected and X connected,
D must be a homeomorphism. Thus X is simply connected (hence the universal
covering of X), and D is an isometry. D
3.33 Remark. The map D : X —> M" constructed in the proof of (3.29) is called the
developing map of X, and the group Г С Isom(A^) (which is naturally isomorphic
to 7ti(X, xo)) is called the holonomy group of X with respect to the germ (φ, χ0).
An analogous theorem with the same proof is valid for spaces with a (G, F)-
geometric structure — see ΙΠ.ί/.l.ll where we consider the more general case of
orbifolds.

Chapter 1.4 Normed Spaces
In this chapter we return to the study of normed spaces, which were introduced briefly
in Chapter 1.
Hilbert Spaces
4.1 Definition. A scalar product (or inner-product) on a real vector space У is a
symmetric bilinear map УхУ^1, written (υ, w) м> (ν \ w), with the property
that (v | v) > 0 for all ν φ 0. Associated to a scalar product one has a norm
||υ || := (ν \ v)x/1, and hence a metric. Apre-Hilbert space (or inner-product space)
is a real vector space V equipped with a scalar product; it is called a Hilbert space if
the associated metric is complete.
In order to see that the above expression really does define a norm, we must verify
the triangle inequality. This is a consequence of the Cauchy-Schwarz inequality:
\{v | w)\ < ||υ||. || w||. The validity of this inequality is clear if w = 0, and for the
general case one expands the right-most term of the expression 0 < ||υ — λιυ||2 =
(υ — λυυ | υ — λυυ) and sets the scalar λ equal to (υ | w)/(w | w).
Using the Cauchy-Schwarz inequality, we deduce the triangle inequality:
||и + w||2 = (v + w | υ + w)
= \\vf + (v\w) + (w\v)+ \\wf < \\vf + 2||V||.||W|| + |M|2.
4.2 Remark. The I1 norm on W is the norm associated to the Euclidean scalar
product (x | y) := Σ" Xiyt, where χ = (jci, ... , xn) and у = (yx, ... , yn). The
associated metric space is E". The existence of orthonormal bases (as constructed
by the Gram-Schmidt process, for example) implies that every и-dimensional pre-
Hilbert space is isometric to E".
A trivial calculation with the scalar product shows that the unit ball in a pre-
Hilbert space is strictly convex, so by (1.6) we have:
4.3 Lemma. Every pre-Hilbert space is uniquely geodesic.

48 Chapter 1.4 Normed Spaces
Not every uniquely geodesic normed space is a pre-Hilbert space. Indeed, P.
Jordan and J. von Neumann [JvN35] proved the following metric characterization of
pre-Hilbert spaces.
4.4 Proposition (The Parallelogram Law). A norm \\ ■ \\ on a real vector space V
arises from a scalar product if and only if the norm satisfies the parallelogram law:
for all v, w e V,
||«-ш||2+||« + ю||2 = 2(|И2+|М12).
If a norm does satisfy this condition, then one recovers the scalar product by setting
0>Ι«0 = ^(ΙΙ" + «ΊΙ2-ΙΙ"-«>||2).
Proof. If the norm derives from a scalar product then
||и + w||2 — ||и — w||2 = (υ + w | υ + w) — (υ — w | υ — w) = 4(υ | w).
Conversely, we show that if the norm satisfies the parallelogram law then the
following formula defines a scalar product:
(v\w)=^(\\v + w\\2-\\v-w\\2).
It is clear that (v | w) = (w | v) and (ν | υ) = \\v\\2 > 0, so it suffices to check
that when w is fixed, (υ \ w) is a linear function of v. Applying the parallelogram
law to the pairs of vectors (υ' + w, v") and (υ' — w, υ"), we get
|| υ' + ν" + wf + ||ι/ - v" + wf = 2||i/ + w||2 + 2\\v"f
|| υ' + υ" - wf + || υ' - υ" - w||2 = 2||ι/ - w||2 + 2||υ"||2.
Subtracting the second equality from the first and using the definition of the scalar
product, we find
(υ' + υ" I w) + (V - v" \ w) = 2(v' | w).
In particular if ν' = υ", since it is clear that (0 | w) = 0, we have
(2j/ I w) = 2(v' | w),
and hence the above formula can be rewritten
(υ' + ν" I w) + (V - v" | w) = (2i/ | w).
Replacing v' by 1/2(υ' + υ") and υ" by (1/2(V - υ"), we get
(υ' | w) + (v" | w) = (ν' + υ" | w) (*)
for all ν', υ", w e V.

Hilbert Spaces 49
We still have to check that (λυ | w) = λ(υ | w) for any real number λ. The
equality (*) implies that this is true for any rational number, hence by continuity for
any real number. D
There are important qualitative differences between the geometry of those norms
that satisfy the parallelogram law and those which do not. We note one important
difference concerning the notion of angle (1.12). For further differences and
characterizations of pre-Hilbert spaces see [Rob82].
Recall that the angle Z(c, d) between two geodesies c, d : [0, ε] —> X issuing
from a point ρ in a metric space X, is lim sup( (,^0 Zp(c(t), c(f)), where Zp(c(t), c(f))
is the vertex angle at ρ in the comparison triangle A(p, c(t), d(f)) С Е2.
4.5 Proposition. A normed vector space V is a pre-Hilbert space if and only if the
limit Нт,у^о ^o(c(0> с'(0) exists for all pairs of geodesic rays c, d issuing from
Oe V.
Proof. All linear segments in V are geodesies (1.6) and therefore every vector sub-
space of V is isometrically embedded. If У is a pre-Hilbert space, then every geodesic
ray с : [0, oo) —> V issuing from 0 e V is of the form t м> tu (cf. (4.3)). By
restricting our attention to the subspace spanned by any two such rays, we reduce to the
case V = E2, where it is clear that the stated limit exists.
Conversely, suppose that the above limit exists for all pairs of rays issuing from
the origin in the normed vector space V. Given any two linearly independent unit
vectors u, u' e V, we consider the rays c, d : [0, oo) —> V defined by c(t) = tu
and d(t) = tu'. We claim that if Нт,у^о ^о(с(0. c'(0) exists and is equal to a, then
a = Z0(c(0. C'(0) for all t >_0 and f > 0. Indeed, a = Нт5^0 Λ(Φή, c'(sf)),
and by the law of cosines, cos Z0(c(st), d(sf)) = ^7 (s2t2 + s2f — \\stu — ifV||2),
which is independent of s.
To complete the proof we must show that V satisfies the the parallelogram law.
It is sufficient to check it for two linearly independent vectors ν and w. Applying
the argument of the preceding paragraph to the normalized vectors и = υ/||υ|| and
и' = (υ + νυ)/\\υ + w\\, we see that the angles at 0 in the comparison triangles
Δ(0, υ,υ + νυ) and Δ(0, υ, 1/2 (υ + w)) are equal. The parallelogram law for ν and
w follows easily from this observation, by the law of cosines. D
4.6 The Hilbert Spaces £2 (S). We first consider I1 (Z), the set of bi-infinite sequences
of real numbers χ = (xn) such that Σ \χη\2 is finite. Let ||jc||2 := (Σ Ы2)'/2· We
claim that, when equipped with the term-wise operations of addition and scalar
multiplication, £2(Z) is a real vector space. That it is closed under scalar multiplication
is clear. And one sees that if χ = (x„) and у = (у„) lie in £2(Z) then so too does
χ + у = (xn + y„) by observing that the triangle inequality in E" gives a bound on
the partial sums of χ + у:
In \ '/2 / \ '/2 / \ 1/2
(Х>'- + У|2) -\Σ^Π +(έι>Ί2) <IWl2+llyll2.

50 Chapter 1.4 Normed Spaces
We define a scalar product on £2(Z), with associated norm || ■ lb, by: (x \ y) :=
^2jXi)>i- The convergence of this sum is assured by the Cauchy-Schwarz inequality
for the Euclidean scalar product:
(n \2 η η
Σ \*у<\ ζ Σ w2 Σ w2 - |W|2 iiyiil-
/=— η J ί=—η Ι=—η
Now let S be any set. We consider functions χ : S —> №.. By definition, J2s x(s)
is the supremum of the sums Σε x(s) taken over the finite subsets С С S. If ^Zs x(s)
is finite, then x(s) = 0 for all but countably many s e S (see [Dieu63] for example).
Let l2(S) be the set of functions χ such that ^Zs |x(i)|2 is finite. Arguing exactly
as in the case S = Z, we see that l2(S) is closed under the operations of pointwise
addition and scalar multiplication, and that when equipped with the scalar product
(x | y) := ^2sx(s)y(s) it is a pre-Hilbert space. The associated norm is denoted || ■ Цг-
4.7 Proposition. Let S be a set. The metric associated to the norm || ■ ||г defined
above is complete, and hence l2(S) is a Hilbert space-
Proof. Let d denote the metric associated to the norm || ■ lb- Let (x(m)) be a Cauchy
sequence in l2(S). Given ε > 0, we fix N such that for all m, m' > N,
d{x^m\ jc(m,)) = (Σ |*(т)^ - *(т,)(*)12) < ε·
Then |х(т)0) - xim'\s)\ < ε for all s e S if m, m' > N, and hence for fixed s the
sequence of numbers (jc(,n)(i)) is Cauchy, with limit x°°(s), say. This defines a function
x°° : S -> K; we claim that x°° e 12{S). From the above inequality we have that, for
all finite subset С С S
ssC
Fixing m and letting m' —> oo in this expression we get
Y^\x^\s)-x°°{s)\2 <ε2.
ssC
Hence jc(m) - x°° e 12{S), and therefore x°° = jc('"> - (jc('"> - x°°) e 12{S). Since ε
and С were arbitrary, the displayed inequality also shows that ф:(т), х°°) -» 0 as
m —> oo. D
Recall that a metric space X is said to satisfy a given condition locally (e.g., "X
is locally compact") if for every χ e X there exists ε > 0 such that the closed ball of
radius ε is (when equipped with the induced metric) a metric space which satisfies
the specified property (e.g., B(x, ε) is compact).
4.8 Example. Let S be an infinite set. Let 5i(e) denote the sequence whose only
non-zero term is in position m, and this term is ε. The infinite set {Ss(ε) \ s e S} is

Isometries of Normed Spaces 51
obviously contained in the closed ball of radius ε about 0 e 12{S) but does not have
an accumulation point, thus l2(S) is not a locally compact space.
4.9 Exercise. Let d be the metric associated to any norm on W, and let di denote
the Euclidean metric on №. Show that there exist constants Μ > m > 0 such that
m d(x, y) < d2(x, y) < Μ d(x, y) for all x, у е W. Hence deduce that every finite
dimensional normed space is proper (and hence complete).
Further to the preceding example and exercise, we note that a classical result of
F. Riesz states that a normed vector space is locally compact if and only if it is finite
dimensional (see [La68, p.37] or [Dieu63, V.9]). We prove a particular case.
4.10 Lemma. A pre-Hilbert space is locally compact if and only if it is finite
dimensional.
Proof. Suppose that V is an infinite dimensional pre-Hilbert space, with scalar
product (υ | w). Let щ,иг, ■ ■ ■ be a linearly independent set of vectors. From this we
can construct an infinite sequence of orthonormal vectors, i.e., a sequence v\, V2, ■..
such that (υ, | υ,) = 5,,; (Kronecker's delta). In order to do so, we invoke the Gram-
Schmidt procedure: i>i := ui/\\u\ || and, inductively, w„ := u„ — ^"Ji ("« I vi) апс*
v„ := w„/\\w„\\.
Now, given any ε > 0, we claim that the closed ball of radius ε is not compact.
Indeed, the vectors ευη form an infinite sequence contained in this ball, and the
sequence does not have an accumulation point, because if η φ m then ά{ε υη, ε vm)2 =
(ευ„ — ευη \ ευ„ — ευη) = 2ε2. D
4.11 Example: The Space of Finite Sequences. We have noted that all finite
dimensional pre-Hilbert spaces are actually Hilbert spaces. An easy example of a
pre-Hilbert space which is not a Hilbert space is provided by the space Co consisting
of those bi-infinite sequences of real numbers (jc„) all but finitely many of whose
entries are zero; this is a subspace of £2(Z), and one endows it with the induced
scalar product.
If χ e £2(Z) is a sequence all of whose entries are non-zero, then one can obtain a
Cauchy sequence (^m)) in C0 by definingx^ — xn if m > \n\ andx^n) = 0 otherwise.
This Cauchy sequence does not have a limit in C0. Notice that this argument shows
that Co is dense in £2(Z).
Isometries of Normed Spaces
Let У be a normed vector space. Given any a e V, the map V —> V given by χ м> х+а
defines an isometry of V; we shall refer to this isometry as translation by a. The
abelian subgroup of Isom(X) formed by such translations is clearly isomorphic to

52 Chapter 1.4 Normed Spaces
the underlying additive group of V. We introduce the notation 0{V) for the subgroup
of Isom(X) consisting of those linear transformations of V that preserve its norm.
Recall that, given two groups G and H, and an action of Η on G (i.e., a homo-
morphism Η —> Aut(G), written h ь» φ/,) one forms the semidirect product G χ Η
by endowing the set G χ Η with the operation (g, h) ■ (g', h!) = (g(Ph(g'), hh'). For
the semidirect product referred to in part (2) of the following proposition, the action
of 0(V) on V is the obvious linear action.
4.12 Theorem. Let V be a normed vector space.
(1) Every isometry of V can be expressed uniquely as a linear transformation
followed by a translation.
(2) Isom(V) = V χ 0(V).
Proof. The uniqueness assertion in (1) is clear, since the image of 0 under the given
isometry determines the translation whose existence is asserted. By composing an
arbitrary isometry φ of V with the translation χ ы> χ—φ(0), we reduce the proof of (1)
to showing that an isometry φ which preserves the origin is linear, i.e., φ(λχ) = λφ(χ)
and <p(x+y) = φ(χ)+φ(γ) for all x, у e W and all λίΐ. Mazur and Ulam [MaU132]
proved that any isometry of a normed space that preserves the origin is linear (see
[Bana32, XI.2] for a clearer proof). We shall give the proof only in the simple case
where V is uniquely geodesic.
The first of these equalities follows easily from the fact that the only geodesic
lines through 0 e V are the 1-dimensional subspaces of V, hence φ must map each
such line isometrically onto another such line. The second equality follows from
the fact that φ, as an isometry, must send the midpoint of a line segment to the
midpoint of the image segment (because such segments are the unique geodesies in
V). The midpoint of the unique geodesic segment joining χ to у is (x + y)/2, hence
φ{χ + y) = 2φ((χ + у) β) = 2φ{χ) + ф))/2) = φ{χ) + <р(у).
The decomposition in (1) allows us to express any φ e Isom(V) as the product
of a linear transformation Αφ and a translation χ м> х + φ. We claim that the
map from Isom(V) to У χ 0{V) given by φ м> (^(0), Αφ) is an isomorphism. The
uniqueness assertion in (1) assures that this map is well-defined and injective. It
is clearly surjective, so it only remains to check that it is a homomorphism. Given
φ, ψ e Isom(V), the action οίφφ on V is:
φψ(χ) = φ(Αψ(χ)+ψ(0))
= Αφ(Αψ(χ) + ψ(0)) + 0(0) = ΑφΑψ(χ) + [Αφψφ) + 0(0)].
So the image of φ ψ in V χ 0(V) is (φ(0)+Αφψ(0), ΑΦΑΨ) = (0(0), Аф) ■ (ψ(0), Αψ).
D
Of course, not all linear transformations of a normed real vector space are isome-
tries. As we noted earlier, in the case of Euclidean space an и-Ьу-и matrix A acts as
an isometry if and only if A'A = I, that is A e 0(n). Thus we have an alternative
proof of 2.24(1).

lp Spaces 53
4.13 Corollary. Isom(E") = W χ 0(η).
S" is the set of vectors in R"+1 of Euclidean norm 1. The distance between two
points A, В e S" is the Euclidean angle between [0, A] and [0, B] at 0 e W+l. Since
this angle is arcsin of one half the Euclidean distance from A to B, any isometry of
E"+1 which fixes 0 must restrict to an isometry of S". In particular this is the case for
the action of 0(ri). We obtain an alternative proof of 2.24(2) by showing that these
are the only isometries of S":
4.14 Proposition. Isom(S") = 0(n + 1).
Proof. We must show that every isometry of S" с E'!+1 can be extended to an
isometry of E"+1. Given an isometry φ of §", the desired extension φ to E'!+1 is given
by writing each χ e E"+1 in polar coordinates txvx, where vx is a unit vector and
tx > 0 is a number; one defines φ(ίν) = ίφ{υ). Hence ίφ(Χ) = tx for all χ e W+i.
Furthermore, since φ is an isometry of §", if we write a(x, y) for the angle at the
origin between the line segments [0, x] and [0, y], then for all x, у е W+l we have
а(ф(х), ф(у)) = α(φ(υχ), (p{vy)) = α(υχ, vy) = a(x, у). But then, applying the law of
cosines to the Euclidean triangle with vertices 0, x, y, we can express the Euclidean
distance between χ and у in terms of the ^-invariant quantities tx, ty and a(x, y):
d(x, y)2 = i^ + i^ — 2txtycosa(x, y).
О
4.15 Exercises
(1) Show that every linear transformation of a pre-Hilbert space V which
preserves orthogonality is a homothety, i.e., there exists a constant λ > 0 such that
ά{φ{υ), ф(ю)) = Xd{v, w) for all v, w e V.
(2) Prove the Mazur-Ulam theorem [MaU132] for finite dimensional spaces.
ip Spaces
In this paragraph we describe an important class of uniquely geodesic normed spaces
that are not Hubert spaces. Given ρ > 1 and a vector χ = (χι,..., x„) in W, the lp
norm of χ is defined by:
/" \'/P
И1Р:=(1><Л ·
The associated metric is denoted dp. It is obvious that || ■ |h is a norm, but less
obvious for ρ > 1.
4.16 Proposition. For every η e N and every real number ρ > 1, the map χ м> \\х\\р
defines a norm on W. In particular, for any two non-zero vectors x,ye W, we have

54 Chapter 1.4 Normed Spaces
ll* + y||P< 11*11, +IMIp.
Furthermore, one has equality in this expression if and only if χ is a positive multiple
ofy, hence for all I < ρ < oo, the metric space (R", dp) associated to the lp norm
is uniquely geodesic.
Proof. To check the triangle inequality, one considers the real number q > 1 defined
by
l/p+l/q=l
We claim that for any real numbers a > 0, b > 0, one has
allpbllq <a/p + b/q (1)
with equality if and only if a = b. The non-trivial case is when a > 0 and b > 0.
Then, since the function log : [0, oo) —> К is strictly concave, we have
\og(a/p + b/q) > 1/phga+l/q logb
with equality if and only if a = b. Composing both sides of this inequality with the
strictly increasing function exp, we get a/p + b/q > allpbllq, with equality if and
only if a = b.
Recall that, given two vectors χ = (xu ..., xn) and у = (уχ,..., yn) in R", we
have the Holder inequality
η
Y,\xi\\yt\<\\x\\p\\y\\q- (2)
1=1
We shall verify this inequality for the non-trivial case χ φ 0, у ф 0. Applying
inequality (1) to α,- = ^ and bt = j^, we have
Ы\р \\y\\q ~1/P\\x\\pP + 1/q]\y\\V
Summing over i = 1, ..., n, we then get the Holder inequality and see that equality
holds if and only if a, = b\ for each i.
We are now ready to prove the triangle inequality for dp. One has for each
i = 1,... ,n
\χι + yi\p < Ы \xi + УгГ1 + Ы \χι + у<Г1. (3-i)
From the Holder inequality we get
In \Vp / " \l/q
Σν^+λγ^ΙΣνί (Σ>+)'ιΐ&'~1)Ί (4)
in \ χΐρ ι η \χΐι
Σ ы \χί+yir' < ί 1>-Л (Σ><+y^~l)q) (5)

lp Spaces 55
hence, as (ρ — \)q = ρ,
In \l/p / " V/P / " \l/q
&<+^(Σ>ιρ] +(Σ>Ρ) (Σ>+»|Ρ)
which leads to the triangle inequality.
In the above triangle inequality, equality can occur if and only equality occurs
simultaneously in (3-i), (4) and (5). Equality in (3-i) implies that jc,- and y,- have the
same sign and that if χ and у are non-zero then χ + у is also non-zero. Equality in (4)
and (5) implies that
Mp __ \xi + yi\q __ Ыр
IMl£ II* + я II? \\y\\pP
for each i = 1, ..., n, hence χ has to be a positive multiple of y. Hence, by (1.6),
(W, dp) is uniquely geodesic for all 1 < ρ < oo. D
4.17 The Banach Spaces V{S). Let S be a set. Let lp{S) denote the set of maps
χ : S -> R such that J2s \X(SW is finite (cf- 4·6)· Define ||jc||p = (J2S \x(s)\p)l/p-
We claim that, when equipped with the term-wise operations of addition and scalar
multiplication, ip(S) is a real vector space. That it is closed under scalar multiplication
is clear. In order to see that if x, у e lp(S) then χ + у е lp(S), one uses the triangle
inequality in (R", dp) to bound the sums over finite subsets С С S:
(\ l/p ι \ l/p ι \ l/p
Σ \x(s) + y(s)f I < ( Σ \x(s)f I + ( J^ ШУ I < \\x\\p + \\y\\p.
s<=C I \seC J \seC /
This establishes the triangle inequality showing || ■ || to be a norm on V(S). By
arguing as in the preceding proposition one see that the associated metric space is
uniquely geodesic if and only if ρ > 1.
4.18 Proposition. For every set S and every real number 1 < ρ < oo, the metric
space associated to lp{S) is complete. In other words, lp{S) is a Banach space.
Proof. Modulo obvious changes of notation, the proof of Proposition 4.7 applies
verbatim. D

Chapter 1.5 Some Basic Constructions
In this section we describe some basic constructions that allow one to manufacture
interesting new metric spaces out of more familiar ones. We consider whether such
properties as being a geodesic space are preserved by these constructions. Each of
the constructions which we consider here will play a significant role in Part II.
Products
5.1 Definition. The product of two metric spaces X\ and X2 is the set X[ χ Χ2
endowed with the metric:
d((xux2), (у\,уг)? = d(xuyi)2 + d(x2,y2f-
For example, Ε" χ Em is isometric to E"+m.
In the proof of the next proposition we shall need the following characterization
of geodesic segments.
5.2 Exercise. Let X be a metric space. A continuous path с : / —> X is a linearly
reparameterized geodesic if and only if d(c(s), c(f)) = 2 d(c(s), c((s + fj/2)) for all
s,t e /; in other words c((s + i)/2) is a midpoint of c(s) and c(t).
5.3 Proposition. Let X be the product of the metric spaces X\ and X2.
(0) X is complete if and only if both Χι andX2 are complete.
(1) X is a length space if and only if both X[ and X2 are length spaces.
(2) X is a geodesic space if and only if both Χι andX2 are geodesic spaces.
(3) Let I с Ж be a compact interval. The map с : I —> X given bytv-* (c\ (t), c2{t))
is a linearly reparameterized geodesic if and only if the maps c\ and c2 are
both linearly reparameterized geodesies.
(4) An isometry γ e Isom(X) decomposes as a product (γι, γ2), with γχ e Isom(Xi)
andy2 e Isom(X2), if and only if for every x\ eXi there exists a point denoted
Υι(χι) e Χι such that γ({χι} χ X2) = {γι(χι)} x Xi-

Products 57
Proof. Assertion (0) is obvious. We first prove (1). The projection of X onto Xi is
distance-decreasing and hence length-decreasing, and it restricts to an isometry on
slices of the formXi χ {χ2}. From this observation it follows easily that if X is a length
space then so too is Xi (and similarly Хг)· For the converse, given {х\,хг), (УьУг) e X
and ε > 0, we fix paths c\ : [0, 1] -> Χι and cj. : [0, 1] -> X2 joining x\ to y\
and X2 to уг respectively, chosen so that l{c\)2 < d(x\,y\)2 + ε2/2 and /(C2)2 <
d{x2, У2)2 + ε2/2, and parameterized proportional to arc length. Consider the path
с : [0, 1] -> X defined by c(t) = {c\{t), сг(?))· This path joins (jq, xj) to(yi,y2)and,
according to (1.20), has length supn>0 J2"~q d(c(i/n), c((i + \)/n)). Because a and
C2 are parameterized proportional to arc length, for all η > i > 0 we have that
n2d(c(i/n),c((i+l)/n))2
= n2 d(ci(i/n), Cl((i + \)/n))2 + n2 d(c2(i/n\ c2((i + \)/n))2
< /(ci)2 + /(c2)2
< d(xi,yi)2 + ε2/2 + d(x2, У2)2 + ε2/2
= d((xi,x2),(yi,y2))2 + e2.
Hence, η d(c(i/n), c((i + \)/n)) < d{{x\,x2), (у\,Уг)) + ε for all η > i > 0, so
1(c) < d((x\, JC2), (y\, У2)) + ε. Since ε was arbitrary, this shows that X is a length
space.
Assertion (2) follows immediately from (3). A trivial calculation shows that if c\
and C2 are linearly reparameterized geodesies then so too is c. For the converse we
apply criterion (5.2). To this end, given t,s e I, we let χ = (x\, Χ2) := (ci(t), сг(0)
and у = (y\,y2) := {c\{s), C2(s)), and we denote the midpoint (ci((i+ j)/2), C2((i +
i)/2)) by w = (wi, mi). We must show that
d(muyi)= - d(xi,yi).
If two numbers a, b e [0, 1] satisfy α + fr = 1, then a2 + b2 > 1/2, with equality if
and only if a = b = 1/2. Combining this with the triangle inequality, we have
- u?(xi,yi)2 < d(xu mi)2 + d(mu yi)2,
with equality if and only if ^d(xi, yi) = d{x\, m\) = d{m\, y\). Adding this to the
similar inequality for JC2, W2, yi, we obtain:
- {d{xx,yxf + d'(x2, y2)2} < d(xumi)2 + d(x2, m2)2 + d{muy\)2 + d{m2, y2)2-
The left hand side of this inequality is equal to ^d(x, y)2 and the right hand side
is equal to d(x, m)2 + d(y, m)2. But since с is a linearly reparameterized geodesic,
d(x, m) = d(y, m) = jd(x, y). Thus all of the above inequalities must actually hold
with equality, and in particular d(m\, y\) = \d(xi, yi).
The necessity of the condition given in (4) is clear. In order to see that it is
sufficient, we assume that γ is an isometry of Χι x X2 mapping {x\} χ Χ2 onto

58 Chapter 1.5 Some Basic Constructions
ΙΥι (χι)} x Хг f°r each x\ e Χι. Notice that the map jq м> yi (*i) must be an isometry
of X\. Indeed its surjectivity follows from that of y, and it is distance preserving
because for allx\,y\ e X\
d{x\,y\) = inf{d((xl,x2), (yu η)) Ι χ2, Уг е Χι}.
If we now show that the map X2 м> yi(xi) of X2 to Хг defined by (y\ (x\), уг(хт)) =
γ(χι, хг) is independent of x\, then we can apply the preceding argument to see that
it too must be an isometry.
Suppose that γ maps (jq, хг) to {γ\ {x\), уг(хг)) and (y\, хг) to
(yiOO. Уг(х2))- We have to check that уг(хг) = у[(хг)- But,
d{x\, yi )2 = <*((*i, JC2>, (yi, хг))1
= d(y(xl,x2), у(у\,х1))г
= diydxi), ПЫ))2 + d(y2(x2), y&i))2
= d(x{,yi)2 + d(y2(x2), Yiixi))2-
Hence d(y2(x2), У2{хг)) = 0, as required. D
5.4 Remark. Given any finite number of metric spaces (X\, d\), ..., (X„, d„), one
can consider their product Y["=l X' w*m me metric in which the distance from χ =
(x\, .. ■, x„) to у = (yi, ..., y„) is given by the formula
η
d{x,yf = Yjdl(xi,yi)2.
;'=1
If 1 < m < η then the natural bijection Πί=ι·^ί x Π/Lm+i·^/ ~* YTk=\ Xk *s an
isometry. And by induction on η (or directly) one can prove the analogue of (5.3) in
this more general setting.
5.5 Exercises
(1) One might describe the space constructed in (5.4) as the £2-product of the
spaces (Xit di). More generally, for any 1 < ρ < oo, one can define an £p-metric on
Π"=ι Χι by the formula
η
d(x,yy = J2di(xi,yif.
i=l
Using the properties of the £p-norm in W, prove that d is a metric and that the
analogues of (5.3) parts (0) to (3) are valid if 1 < ρ < oo.
(2) Consider now the product Π;<=Ν-^ί of countably many metric spaces (X), dt).
Prove that the function which associates to χ = (χ,), у = (л) е Π;<=ν X' me number
d(x, у) е [0, oo] defined by
oo
d(x,y)= Y^di(xi,yif
/=1

«-Cones 59
is a metric on Π;<=ν ^- ^n order to get a metric with values in [0, oo) one must choose
a sequence of basepoints jc,- e X,- and consider only those sequences (y,·) that are a
finite distance from (jc,). This subspace is called the i2 -product of the pointed spaces
(X,·; jc,). Which parts of (5.3) have valid analogues in this setting?
Check that if X,· = К for each i, and we choose 0 e X,· as the basepoint for each
i, then then £2-product is isometric to £2(N).
re-Cones
The purpose of this section is to describe an important construction which is
essentially due to Berestovskii [Ber83] (see the article by Alexandrov, Berestovskii
and Nikolaev [AleBN86]). In order to offset the rather technical appearance of this
construction, we first recall the law of cosines in the model space M" (see 2.12);
this indicates the origins of the definition that follows. We also draw the reader's
attention to (5.8).
Let γ be the angle between two geodesic segments [jco, jq] and [jc0,jc2] in M£.
Let a = d(xo,xi), b = d(xo,xj) and с = d(x\,xj). If к > 0 then assume that
a + b+ с < 2DK. With this notation:
for к = 0 : с2 = a2 + b2 - lab cos(y)
for к < 0 : cosh(V—fc) =
cosh( V—f a) cosh{y/—кЬ) — sinh(V—к a) sinh(V—к b)cos(y)
for/c > 0 : cos(^c) = cos(^a)cos(^/tcb) + sin(^/lca)sm(^/lcb)cos(Y).
5.6 Definition (The к -Cone over a Metric Space). Given a metric space Υ and a real
number к, the к-сопе X = CKY over Υ is the metric space defined as follows. If
к < 0 then, as a set, X is the quotient of [0, oo) χ Υ by the equivalence relation given
by: (f, y) ~ {if, У) if (i = f = 0) or (t = f > 0 and у = у1). If к > 0 then X is
the quotient of [0, DK/2] χ F by the same relation. The equivalence class of (t, y) is
denoted ty. The class of (0, y) is denoted 0 and is called the vertex of the cone, or the
cone point.
Let d„(y, /) := min{7r, d(y, /)}. The distance between two points χ = ty and
x1 = ify' in X is defined so that d{x, χ1) = t if x1 =0 and so that cos(Zq (jc, χ1)) =
dniy, У) if i, f7 > 0. This is achieved by defining:
for к = 0:
φ:, jc')2 = ί2 + ί'2 - гя'сов^О', /)),
for /с < 0:
cosh(V^c d(x, jc'))
= cosh(V—кt)cosh(V—f О — sinh(V—/ci)smh(V—Kl!)cos{d„(y, y1)),

60 Chapter 1.5 Some Basic Constructions
and for к > 0:
d{x,x) < DK and
соъ(^/ка(х, x')) = cos(^/Kt)cos(«Jlct') + sin(^/ict) sin(^/ict')cos(dn(y, y')).
For к = 0, the /c-cone over Υ is also called the Euclidean cone over Y. We shall
prove in Proposition 5.9(1) that these formulae do indeed define a metric on X (it is
only the triangle inequality that is not obvious).
5.7 Remarks. In the notation of the preceding definition: if t, f > 0, then d(x, x1) =
t + f if and only if d(y, /) > π. This observation has two consequences. First, if
one replaces the given metric d on Υ by the metric d„, then the induced distance
function on X = CKY remains unaltered. Notice that one can recover the metric dn
on Υ from the distance function on X, because if χ = ty, x1 — fy' with t, f > 0, then
d„(y, У) = Zq (x, x1)- It follows that, since the length metrics d and dn agree on Y,
one can recover the metric on Υ from the distance function on X under the added
hypothesis that Υ is a length space.
Secondly, a path с : [—h, t2] -> CKY, where ti,t2 > 0 and c(0) = 0, is a
geodesic if and only if there exist у\,уг € Υ such that d(y\, y2) > я, c(—s) = syi
for s e [0, ii] and c(s) = sy2 for s e [0, t2].
The above definition is motivated in part by the following example.
5.8 Proposition. If Υ is the sphere S"_1, then X = CKY is isometric to MnKific< 0,
and a closed ball of radius DK/2 in M" if к > 0.
Proof. Let о be a point in M", identify S"_1 to the unit sphere in the tangent space
T0M"K and let exp0 : T0{M"K) -> M"K be the exponential map (see 6.16). Then the
map CKS"~l —» M" which sends the class of (i, y) to exp0(fy) is an isometry onto
M" if к < 0, and an isometry onto the closed hemisphere centered at о if к > 0.
This follows from the definition of the metric on CKS"~l and the law of cosines in
MnK. D
5.9 Proposition.
(1) The formulae in (5.6) define a metric onX = CKY.
(2) Υ is complete if and only ifX is complete.
Proof. (1) Consider three points *,· = i,y,·, i = 1, 2, 3 in X. We want to prove that
d(x\, x-s) < d(x\, x2) + d(x2, хз). This is easy to check if one of the и is 0, because in
that case the assertion follows from the triangle inequality in M\.
Assume that i,· > 0 for i = 1, 2, 3. We consider two cases.
Case 1: d(yuy2) + d(y2, y3) < π.
It follows from the triangle inequality in Υ that d(y\, уъ) < π. Consider three
points yl, y2, Уз in S2 such that d(yi, yf) = d(yh yj), for i,j e {1, 2, 3}. As in (5.8), the

«-Cones 61
subcone Ск{у\,у2, Уг\ is isometric to the subcone CK{y{, y2, y$} !Ξ CK(S2) с M\, so
we may appeal to the triangle inequality in M\.
Case 2: d{y\, y2) + d(y2, Уъ)>я.
Consider three points У\,у2, Уъ occurring in the given order on the circle §' such
that d(yl,y2) = d„(yuy2) and d(y2,y3) = dn(y2,y-$). We identify CKS{ to M2 if
к < 0 and to a closed hemisphere in M2 if к > 0. If 5c,- = by,- for i = 1, 2, 3, then
d(x\,x2) = d(xi,x2) and d(x2, x$) = d(x2, X3) and d{x\, x$) < t\ + t$. As d(y1,y2) +
а(У2>Уъ) - я,Ьу Alexandrov'slemma(2.16) wehavei!+ί3 < d(x\,x2) + а(х2,хъ).
Therefore, d(x\, x-$) < d{x\, x2) + d(x2, x^).
(2) Assume that X is complete. Given a Cauchy sequence (yn) in Y, \&ixn = t0y„,
where ίο > 0 and to < DK if к > 0. From the definition of the metric on X, one sees
immediately that (xn) is a Cauchy sequence in X, and hence converges to some point
a distance i0 from 0, let us say io^· Then, у = limy„.
Conversely, assume that Υ is complete. Let xn = tnyn be a Cauchy sequence in
X which does not converge to 0 e X. Since it is Cauchy, this sequence is contained
in a ball of finite radius about 0 e X, in other words the t„ are bounded. Passing to a
subsequence, we may assume that they converge, to t φ 0 say. For t„, tm sufficiently
close to t, one can estimate d(y„, ym) in terms of d(xn,xm), thus (y„) is Cauchy, and
hence converges, to у say. It follows that a subsequence of the Cauchy sequence
(t„yn) converges to ty. Π
5.10 Proposition (Characterization of Geodesies). Let χ ι = tiyi andx2 = t2y2 be
elements of CK Y.
(1) If t\,t2 > 0 and d(yi,y2) < π, then there is a bisection between the set of
geodesic segments joining y\ to уг in Υ and the set of geodesic segments joining
x\ to x2 in X.
(2) In all other cases, there is a geodesic segment in X joining x\ to x2; this segment
is unique, except possibly in the case where к > 0 and d(x\, x2) = DK.
(3) Any geodesic segment joining x\ to x2 is contained in the closed ball of radius
max{ii, t2} about the vertex 0 e CKY.
Proof. (1) Consider a geodesic segment \y[, y2] с F joining y\ to y2. The subcone
CK\y\, Уг\ £ CKY is isometric to a sector in M2. which is convex, hence there is a
geodesic segment joining jc] to хг which is contained in CK \y\, y2]-
For the converse, we consider a geodesic segment [x\, x2] с X joining x\ = t\y\
to x2 = t2y2; let χ = ty be an arbitrary point of [x\, x2]. Notice that t > 0, for
otherwise d{x\, x2) = t\ + t2, and this would imply d(yi, y2) > n, contrary to the
hypothesis. Therefore the projection ty м> у of [x\, x2] into Υ is well-defined; we
want to prove that the image of [x\, x2] under this projection is a geodesic segment
joining у ι to y2 in Y. For this it is sufficient to prove that d(y\, y)+d(y, y2) = d(y\, y2).
To check this equality, we consider in M2 comparison triangles Δι = Δ(0, χ, χ\)
for (0, χ, x\) and Δ2 = Δ(0,χ,x2) for (0,x, хг), and we assume that these triangles

62 Chapter 1.5 Some Basic Constructions
are arranged so that x\ and x2 are on opposite sides of the common segment [0, x]
(cf. Alexandrov's lemma). Note that the vertex angle of Δι atO is equal to d(y\, y) and
the vertex angle of Δ2 at б is d(y, y2). As d(I], x) + d(x, x2) = d{x\, x2) < h +t2 =
d(0, xy) + d(0, хг), we have d(y\, y) + d(y, y2) < it. Let Δ(0, χ\, χ2) be a comparison
triangle in M\ for (0, x\, хг). The angle at 0 is d(y\, y2). By the law of cosines 2.13,
since d(x\,x2) < d(xi,x2), we have d(yi,y) + d(y,y2) < d(y\,y2). The reverse
inequality is simply the triangle equality in Y, hence d(y\, y2) = d(j>], y) + d(y, y2).
(2)Ifwearenotincase(l),thend,(jci,ji:2) = ii+i2andthepathc : [0, ?1+?г] —> X
that sends t e [0, t\ ] to (ii — t)y\ and t e \t\, t{[ to (t —1\ )y2 is a geodesic path joining
x\ to X2- We must prove that this is the only geodesic segment joining x\ to X2, if
t\ + t2 < π Ik . If t\ = 0 or t2 = 0, it is easy to see that this is true, so we need only
consider the case where t\, t2 > 0 and d(y\, y2) > it.
It is sufficient to prove that if χ = ty satisfies d{x\, x) + d(x, x2) = d(x\, x2), and
d(x\ ,x) < t\, then у = у ι. As in the proof of (1), we construct the two comparison
triangles Δι and Δ2 in M^. The vertex angles at 0 of Δι and Δ2 are respectively
djtiyx, y) and djtiy, y2), and so the sum of these two angles is not less than it. But
d(x\, 0) + d(0, x2) = d{x\, x2) = d(x\, x) + d(x, x2) (and this sum is smaller than
2DK if к > 0). Such an equality is only possible if both of the triangles Δι and Δ2
are degenerate, in which case χ = tx\.
Part (3) follows from the convexity of balls in M\ (see the beginning of the proof
of(l)). D
As with the other constructions considered in this section, it is natural to ask
whether the cone over a geodesic space is again a geodesic space; conversely, one
might ask whether knowing that the cone over a space is geodesic, one can deduce
that the space itself is geodesic. In the latter case, one sees immediately that the
answer is no, because as we noted earlier, the isometry type of CK Υ is not changed if
one truncates the metric on Υ at any value > it. However, one can obtain a positive
result by taking account of this phenomenon.
Recall that a subset Υ of a metric space X is said to be convex it every pair of
points in Υ can be joined by a geodesic segment and every such geodesic segment is
contained in Y.
5.11 Corollary. Let X be the к-сопе CK Υ over a metric space Y. Then the following
conditions are equivalent:
(1) X is a geodesic space;
(2) any ball in X centred at the vertex of the cone is convex;
(3) there is an open ball centred at the vertex of the cone which is convex;
(4) Υ is it-geodesic.
The equivalence of (1) to (4) remains valid if one replaces "geodesic" by
"uniquely geodesic" and "convex" by "convex and uniquely geodesic".
Proof. Part (3) of the preceding proposition shows that (1) =φ· (2) =Φ· (3). If we
know that a ball of radius 2i > 0 centred at the vertex of the cone is convex, then

Spherical Joins 63
УиУг € У with d(y\, уг) < π, then any points of the form x\ = ty\ and хг = tyi
can be joined by a geodesic in X and hence, by part (1) of the preceding proposition,
there is a geodesic joining}'i to уг in F. Thus (3) =Φ· (4). Part (1) of the proposition
also shows that the ball of radius t about the cone point is uniquely geodesic if and
only Υ is uniquely я-geodesic.
Assume that Υ is 7r-convex. Consider two points x\ = tiyi and X2 = Нуг of
X. If t\ > 0, ?2 > 0 and diy\ ,уг) < π, then according to part (1) of the preceding
proposition, each geodesic segment joining y\ to уг corresponds to a geodesic joining
x\ to хг- And in all other cases, а{х\,хг) = t\-\-h and part (2) of the preceding applies.
Thus (4) => (1). D
5.12 Exercises
(1) Show that if Υ is a length space then CKY is a length space.
(2) Let (F, <f) and (F', d") be length spaces. Prove that every isometry/ : CK Υ —>
Ск F' that sends the cone point to the cone point, induces an isometry (F, d) —> (F, d')·
Spherical Joins
The product construction allows one to obtain E"+m from the pair of spaces (Ε", Em).
In this paragraph, we describe a method for combining pairs of metric spaces which
when applied to the pair (§", Sm) yields §"+m+1. This construction, which is called
the spherical join, is due to Berestovskii [Ber83] (see also [CD93]).
5.13 Definition. Let (Y\,dl) and (F2, d2) be two metric spaces. As a set, their
spherical join Y\ * Yi is [0,7r/2] χ Υ\ χ Υ% modulo the equivalence relation
which identifies {в,у\,уг) to (Θ', У[,У2) whenever [(θ = Θ' = 0) and y\ = y\]
or [(Θ = Θ' = Л-/2) and y2 = γ2] or [θ = θ' <£ {0, тг/2} and yx = у, у2 = у'2].
The equivalence class of (θ, У\,уг) will normally be denoted (cos θ y\ + sin θ y2).
Sometimes we shall denote the class of (0, y\, уг) (resp. (тг/2, у\, y2)) simply by y\
(resp. уг), thus implicitly identifying Y\ and F2 to subsets of Y\ * Yi-
We define a metric d on F] * F2 by requiring that the distance between the points
у = (cos θ y\ + sin θ уг) and У = (cos б'У, + sin 6)'У2) be at most π, and that d satisfy
the formula
cos(d(y, y')) = cos θ cos θ' cos(dl(y], y[)) + sin θ sin 0' cos(rf^(y2, У2)).
(1) If one equips F] and F2 with the truncated metrics d\ and d^, then the natural
inclusion of each space into Y\ * Уг is an isometry onto its image.
(2) If Y\ is a single point then F] * F2 is naturally isometric to the cone C\ F2.

64 Chapter 1.5 Some Basic Constructions
The fact that the formula in (5.13) does indeed define a metric on Y\ * Y2 is
implicit in the following proposition.
5.15 Proposition. For any metric spaces Y\ and Y2, there is a natural isometry of
C0(Yi * Yi) onto C0Yi x C0Y2.
Proof. Consider the map Φ : C0(Fi * Y2) -> CqY\ χ C0Y2 defined by
<E>(i(cos# yi + sin θ y2)) = (tcos6 yu ίήηθ y2).
We claim that Φ is an isometry. Given two points χ = i(cos θ у ι + sin θ y2) and
χ1 = f'Ccosв'у\ + sine'y'2) in C0(Yi * Y2),
d(x, x1)2
= f + t'2 -2ii'(cos6> cose'cos(dl(yuy\)) +sine sine'cos(d2K(y2, y'2)))-
On the other hand,
(Ф(х), Ф(^))2 = ^cos2^ +i'2cos26i'-2ii'cos6i cos^'cos^Ob y\))
+ t2 sin2θ + t'2 sin2 Θ' - 2tf sin6l sin0' cos(dl(y2, y'2))-
These expressions are obviously equal. D
5.16 Corollary. There is a natural isometry from S" * Sm to §"+m+1.
Proof Υ = S"+m+1 is the only geodesic space with C0Y = En+m+2 = C0§" x C0Sm.
О
5.17 Exercise. Let X and Υ be non-empty metric spaces. Prove that X * Υ is path-
connected and give an example where it is not locally connected. Prove that if X is
path-connected and Υ is non-empty then X * Υ is simply-connected.
Quotient Metrics and Gluing
A natural way to construct interesting new metric spaces is to take a disjoint collection
of known metric spaces and glue them together. The purpose of this section is to give
a precise meaning to this idea. The two basic ingredients of the discussion are the
notions of disjoint union and quotient pseudometric.
5.18 The Disjoint Union of Metric Spaces. Let (Χλ, dx)xSA. be a family of metric
spaces. Their disjoint union X = JJX Χλ is the metric space whose underlying set
is the disjoint union of the sets Χλ (i.e. the set of pairs (χ, λ) with λ e Λ and
χ e Χχ) and whose distance function d is defined by d((x, λ), (χ1, λ)) = dx(x, χ1) and
d((x, λ), (χ/, λ')) = οο if λ φ λ'.

Quotient Metrics and Gluing 65
Most of the time we shall simply declare that the spaces Χχ are disjoint, meaning
that we implicitly identify each Χλ with a subset of the disjoint union by the map
χ м> (χ, λ).
5.19 Quotient Pseudometrics. Let X be a set with an equivalence relation ~, let
X = X/~ be the set of equivalence classes and let ρ : X —» X be the natural
projection. Associated to any metric donX there is a pseudometric d onX defined
by the formula:
η
2(x, y) = inf ]P d{xt, yi),
i=l
where the infimum is taken over all sequences С = (x\, у ι, хг, уг, ■ ■ ■, хп, У η) of
points of X such that x\ &x,yn e y, and y, ~ xi+i for i = 1,..., и — 1. Such a
sequence will be called an η-chain joining χ to у and its length 1(C) is defined to
be Σ"=] d(x-„ yi). Using the triangle inequality for d, we see that there is no loss of
generality in assuming у,- φ xi+\.
It is obvious that d is symmetric and satisfies the triangle inequality, but in
general d is only a pseudometric rather than a metric (for instance if there is an
equivalence class which is dense in X then d is identically zero). We call d the
quotient pseudometric associated to the relation ~. Note that d(x, y) < d(x, y) for all
x, у е X.
5.20 Lemma. Let (X, d) be a length space, let ~ be an equivalence relation on X
and let d be the quotient pseudometric on X = X/~. If d is a metric then (X, d) is a
length space.
Proof. Suppose that d(x, y) is a finite positive number a. Given ε > 0, one can find an
и-chain joiningxto}' such that Σ"=ι d(xi,yi) < a+ ε/2. Because d is a length metric,
for each i = 1,..., η there exists a continuous path c,- : [0, 1 ] —> X of length smaller
than d(xi, y0 + ε/η such that c,(0) = jc, and c,-(l) = y,-. Because d(x, y) < d(x, y)
for all x, у e X, the length of the path с in X which is the concatenation of the paths
ρ о с,· is a curve in X of length smaller than a + ε joining χ toy. Π
5.21 Examples
(1) If X is a closed interval in Ж of length a and ~ is the equivalence relation
identifying the extremities of this interval, then the quotient (X, d) is isometric to a
circle of length a.
(2) Let X = JJZ /„ where each /„ is isometric to the the unit interval [0, 1]. Let
~ be the equivalence relation that identifies the initial point of /„ with the terminal
point of /„_i for each η e Z. The quotient (X, d) is isometric to K.
(3) (Metric graphs.) We use the notation of (1.9). The metric graph associated to
a combinatorial graph Q with edges £, vertices V, endpoint maps 30, 3i : £ -> V,
and length function λ : £ -> (0, oo) is the quotient of V U [Jes£([0, λ(ε)] x {e}) by
the equivalence relation generated by (0, e) ~ 30(e) and (X(e), e) ~ 3i(e).

66 Chapter 1.5 Some Basic Constructions
(4) Let X be a sector in the Euclidean plane (resp. the hyperbolic plane) which
is the convex hull of two geodesic rays с and d with c(0) = c'(0) = xo that form
an angle a < π. On X we consider the equivalence relation generated by c(i) ~
c'(i), Vi e [0, oo). Then the set X of equivalence classes with the quotient metric is
isometric to the Euclidean cone (resp. the hyperbolic cone) over a circle of length a.
(5) Let X be a length space and let (Ux : λ e Λ) be a family of open subsets such
thatX = |Ja ^λ-Letix : Ux ^ Xbe the natural inclusion and let i: Цл Ux -> Xbe
the induced map on the disjoint union. Then X is naturally isometric to the quotient
of Цл Ux by the equivalence relation [χ ~ xf iff i(x) = ϊ(λ^)]. (It is easy to see that
the natural map from the quotient to X preserves the length of paths.)
(6) Given an action of a group Γ by isometries on a metric space X one can define
an equivalence relation by χ ~ у if and only if there exists γ e Γ such that γ.χ = у.
The quotient space X is the set of Γ-orbits and the quotient pseudometric is given
by the formula: d(x, y) = inf{d(x, y) \ χ ex,y ey}. In other words, one need only
consider 1-chains. Indeed any и-chain С = (χι, yi, хг, уг,..., х„, у„) with n > 1 and
yi = γ.χ2 can be replaced by the (и — l)-chain(xb γ-уг, у-хз, у-Уз, · · ·, y-xn, У-уп)
whose length is not bigger than 1(C).
If the action is free and proper (see 8.2), then the projection ρ : X —> Г\Х = X
is a covering and a local isometry.
(7) In (3.24) we described how, given a length space (X, d) and a local homeomor-
phism ρ : Υ —» X, one can construct an induced length metric d on Y. The quotient
of Υ by the equivalence relation [y ~ y' iff /?(y) = /?(y')] 's naturally isometric to X.
In symbols, d= d.
5.22 Exercises
(1) In the following examples, describe the geodesies and the balls in the
corresponding quotient metric space.
(i) Consider a circle S in E2 and the equivalence relation on E2 generated by [χ ~ у
if x, у е S].
(ii) Let S be as above but now consider the equivalence relation generated by [χ ~ у
if jc and у are antipodal points on S].
(2) Let Г be a group acting by isometries on a metric space X. Assume there is
a closed convex set F whose translates by Г cover X and assume also that for each
χ e F there exists an ε > 0 such that {y e T\y.F Π Β(χ, ε) φ 0} is finite.
Consider on X the equivalence relation ~ whose classes are the orbits of Γ.
Let ~/r be the restriction of ~ to F. Show that the natural bijection from F/~/r
to X/~ = Γ\Χ is an isometry (when each is endowed with its respective quotient
metric). (Hint: Show that any path in X is contained in the union of a finite number
of translates of F.)
As an example, let Γ be the subgroup of Isom(E2) generated by two linearly
independent translations χ ы> χ + a and χ н» х + b. Consider the parallelogram
F = {ta + sb\t, s e [0, 1]}. Then Γ\Ε2 is isometric to the quotient of F by the

Quotient Metrics and Gluing 67
equivalence relation generated by [ta ~ ta + b, sb ~ b + sb], which is a torus locally
isometric to E2.
(3) Let ~ be an equivalence relation defined on a metric space F and let ~K be the
equivalence relation on its /c-cone X = CKY defined by [ty ~K ί7/ iff у ~ 3/ and ί =
ί7]. Let F and X be the quotients of Υ and X by these equivalence relations. Prove
that CK Υ is naturally isometric to X.
(4) For i = 1,2, let F, с X, be a subspace. Let/ : Y\ —> F2 be a bi-Lipschitz
homeomorphism and let X be the quotient of the disjoint union of X\ and X2 by the
equivalence relation generated by у ~ /(y) for all у e Fi. Show that the natural map
Xi -> X is bi-Lipschitz. Show also that if each X, is proper and each F, с X, is
closed, then X is proper.
Gluing Along Isometric Subspaces
The most obvious way of gluing metric spaces is by attaching them along isometric
subspaces.
5.23 Definition. Let (Χλ, dx)xs^ be a a family of metric spaces with closed subspaces
Αχ с Χχ. Let A be a metric space and suppose that for each λ e Λ we have an isometry
ix : A —> Ax.LetXdenotethequotientof the disjoint union JJAXx by the equivalence
relation generated by [ix(a) ~ ίχ'(α) Va e Α, λ, λ' e Λ]. We identify each Χλ with
its image in X and write
χ=|_μλ.
A
X is called the gluing (or amalgamation) of the Χχ along A.
5.24 Lemma. L«i X = |_Ja^· In the quotient pseudometric on X, the distance
between χ e Χχ and у е Χχ< is given by the formula:
d(x, y) = dx(x, y) ι/λ =λ'
rf(jc, у) = infect, ίλ(α)) + άλ·(ϊλ·(α), у)} ι/λ /λ'.
And:
(1) ύ? и α metric on X.
(2) /fA is finite and each Χλ ίί proper, then X is proper.
(3) If each Χλ is α geodesic space and A is proper, then X is a geodesic space.
(4) If each of the Χχ is a length space then X is a length space, moreover d is the
unique length metric such that the induced length metric on each Χλ С Xis άχ.
Proof. To see that d is the quotient pseudometric note that, by the triangle inequality
in Χλ, any w-chain С in JJA Χλ joining χλ e Χχ to χχ e Χχ can be replaced by
a 2-chain whose length is no greater than 1(C). And if λ = λ' then (again by the
triangle inequality) one can actually take a 1-chain.

68 Chapter 1.5 Some Basic Constructions
The content of (1) is that d is positive definite. This follows immediately from
the fact that Αχ is closed in Χχ for every λ e Λ.
In order to establish (2) one simply observes that, since άχ = d on Χλ χ Χχ, the
intersection of Χχ with any closed bounded set С С I is closed and bounded in Χλ.
For then if each (Χχ, άχ) is proper and Λ finite, С is a finite union of compact sets,
hence it is compact.
An easy compactness argument shows that if A is proper, then for every χ e
Xx, у e Χχ' with λ φ λ', there exists a e A such that d(x,y) = d(x, ix(a)) +
d(ix'(a), y). Part (3) follows immediately from this observation. We leave the (easy)
proof of (4) as an exercise. D
5.25 Exercises. Let (Χχ, άχ) and X be as in (5.23). Prove the following statements.
(1) If each Χχ is complete, then X is complete.
(2) If each Χχ is locally compact and Λ is finite, then X is locally compact.
(3) Give examples to show that, for Λ = {1, 2}, it may happen that X\ and X2 are
complete (resp. locally compact) geodesic spaces but X is not a geodesic space.
(Hint: In the complete case, you may wish to consider a metric graph with countably
many edges emanating from a single vertex, the и-th having length 1 + -, and then
double this space along the set of free endpoints. In the locally compact case, you
may wish to consider a bounded region in the Euclidean plane.)
(4) Is the metric constructed in (5.24) determined by the fact that ά\χίΧχχ = άχΐ
What about the associated length metric?
5.26 Successive Gluing. Let (Xj, dj),j = 1, 2..., η be a sequence of metric spaces;
assume that an isometry/2 from a closed subspace A2 of X2 onto a closed subspace
/2(A2) of X\ is given and form the amalgamation Y2 := Χι uAl X2 as in (5.23); assume
that an isometry/з of a closed subspace A3 of X3 onto a closed subspace/3(A3) of F2
is given and form the metric space F3 := (X\ UA, X2) LU, X3; proceeding inductively,
suppose that for every j > 2 an isometry^ of a closed subspace A,- of Xj onto a closed
subspace of Yj-\ is given and define Yj = ζ_ι uA. Xj. We say that Y„ is obtained
from the sequence Χχ,... ,Xn by successive gluing. Note that Yj is isometrically
embedded in Yj+\. Because of this, one can glue an infinite sequence of spaces in the
same manner.
Note also that, for each i, the natural inclusion pi : X, -> Yn is an isometric
embedding and that Yn is the quotient of the disjoint union JJ,'=1 X-, by the equivalence
relation which identifies two points if they have the same projection in Y„.
When one pictures ways of gluing spaces together, one quickly thinks of natural
processes that do not conform to the simple templates described in (5.23) and (5.26)
— see (5.21) or (5.29) for instance, and Chapters 7 and II. 11 for many more examples.
Often one wants to take a more local approach to gluing, identifying subspaces by
means of (not necessarily injective) local isometries, for example. The following
lemma is very useful in this regard (see II. 11.4).

Quotient Metrics and Gluing 69
5.27 Lemma. Let X be a metric space and let (X, d) be the quotient ofX by an
equivalence relation ~. Let χ e X and suppose that there exists ε(χ) = ε > 0 such
that:
(i) for all x, x1 e χ and ζ e B(x, ε), ζ' e B(x/, ε) with ζ ~ z!, we have d(x, z) =
d(xf, z'); and
(ii) X' = υ^εϊ Β(χ, ε) is a union of equivalence classes.
Then,
(1) for every у e X with d(x, y) < ε, there exists χ ex such that d(x, y) = d(x, y),
(2) for every χ ex and у e B(x, ε) we have d(x, y) = d(x, y);
(3) let d! and ~' be the restrictions ofd and ~ to X' and let (X, a) be the quotient
ofX' by the equivalence relation ~'.· then the inclusion ofX' into X induces an
isometry ofB(x, ε/2) onto B(x, ε/2), where x' is the ~' equivalence class of
χ ex.
Proof. We first prove (1). If d(x, y) < ε, then the definition of d yields an m-chain
С = Οι, у],..., xm, ym) joining х\ e χ to у = ym with 1(C) < ε. It is sufficient to
prove that if m > 1 then there is an (m — l)-chain C" joining an element of χ to
у with 1(C) < 1(C). As y\ e B(x\, ε) and y\ ~ xi, using (ii) we can find x'] e χ
such that x2 e B(x/l, ε). By (i) we have d(x/l,x2) — d(x\,y\), hence d(x/l,y2) <
d(xf], x2) + d(x2, уг) = d(x\ ,y\) + d(x2, y2). Thus we can replace С by the (m — 1 )-
chain C" = (x/2, y2,..., xm, ym) and 1(C) < 1(C).
To prove (2), note that by the first part of the proof we have d(y, x) = d(y, χ1) < ε
for some x1 ex, and by (i) (with ζ = ζ' = у) we have d(x, y) = d(y, x1).
Finally (ii) implies that the inclusion X' -> X induces an injection X -> X; and
(1) and (2) imply that for each 5 < ε the ball B(x, δ) is mapped bijectively onto the
ball B(x, δ). To measure the distance between two points of B(x , ε/2) we can use
chains of length < ε, and such a chain is contained in X': this proves (3). D
5.28 Corollary. If for each χ e X there exists ε(χ) > 0 as in (5.27), then d is a
metric, and for all x, у e X with χ ex and d(x, y) < ε(χ) we have d(x, y) = d(x, y).
5.29 Exercises
(1) Let Ρ be a convex polygon in И2 with 4g sides. Proceeding clockwise around
P, we orient its edges and label them (in order) a\, b\, a\, b\,..., ag, bg,a', b'g.
Suppose that l(at) = l(a'J and l(bt) = l(b\) for all i, where I denotes length. Let7,
(resp. hi) be an isometry from a, to a\ (resp. £>, to Щ that reverses orientation. Let ~ be
the equivalence relation on Ρ generated by: [x ~ ji(x)Vx e a,]and[jc ~ ht(x)Vx e b{\,
for i = \, ..., g. Prove that the quotient pseudometric on S = P/~ is a metric (using
Lemma 5.27) and that S is a compact surface (of genus g). Show that if the sum of
the angles at the vertices of Ρ is 2π, then S is locally isometric to the hyperbolic
plane. Deduce from this and (3.32) that Γ = niS acts by isometries on И2 and that
the quotient metric on Г\И2 (in the sense of 5.21(6)) is isometric to S.
(2) (Gluing with a tube.) Let X\ and X2 be two disjoint metric spaces, let С be
a circle of length I and let i\ : С -> Χι and i2 : С ->■ X2 be two local closed

70 Chapter 1.5 Some Basic Constructions
geodesies. Let X be the quotient of the disjoint union of X\ U (C χ [0, 1]) U X2 by
the equivalence relation generated by ι'ι (ί) ~ (t, 0) and i2{t) ~ (t, 1). Show that the
quotient pseudometric is a metric. (Hint: Use Lemma 5.27 and successive gluing.)
Give a necessary condition on I so that the natural inclusions of X\ and X2 into
X are isometric embeddings.
Limits of Metric Spaces
In this final section of the chapter we shall explore some notions of limit for sequences
of metric spaces, and consider spaces whose points are themselves metric spaces.
The idea of taking the limit of a sequence of spaces has played a central role in many
recent advances in geometry and geometric group theory. Hausdorff introduced this
idea in the case where the sequence of spaces concerned is embedded in a fixed
ambient space, but the range of recent applications rests upon the idea of taking
limits in the absence of any obvious ambient space, and this innovation is due to
Gromov.
The first part of this section is based on [BriS94].
Gromov-Hausdorff Convergence
We begin by considering the classical construction of Hausdorff, partly for
completeness but also because the basic structure of the argument in the following proof serves
as a blueprint for future proofs. Two particular features to note are: first, compact
sets are approximated by finite sets in a uniform way; secondly, a diagonal sequence
argument is used to construct a limit object as (the closure of) an increasing union
of finite sets.
5.30 Definition. Let X be a metric space and let Ve(A) denote the ε-neighbourhood
of a subset А С X. The Hausdorff distance between A, В с X is denned by:
dH(A, Β) = Μ{ε | А с νε(Β) and В с Ve{A)}.
5.31 Lemma. Let X be a compact metric space and let CX be the set of closed
subspaces ofX. Then, {CX, dn) is a compact metric space.
Proof. The only nontrivial point to check is that CX is compact.
Consider a sequence C, in CX. We must exhibit a convergent subsequence. First
notice that given any ε > 0 there exists an integer Μ (ε) such that, in its induced
metric from X, every A e CX can be covered by Μ(ε) open balls of radius ε. Indeed,
because X is compact one can cover it with Μ(ε) balls of radius ε/2, then for each
such ball which intersects A one chooses a point in the intersection and takes the ball
of radius ε about that point. In this way, for every positive integer η and every C„ by

Limits of Metric Spaces 71
taking duplicates if necessary, we may assume that C, is covered by precisely M( 1 /и)
balls of radius l/и. We wish to retain the centres of the balls from the и-th stage
of this construction at subsequent stages. Thus we define N(l/n) = 5Zm<„M(l/m)
and cover C, with N{\/n) balls of radius l/и with centres xn(i,j) e C, for j =
I,... ,N{\/n), where xn+i(i,j) = xn{Uj) if j < N(l/n). This allows us to drop
the subscript и from x„(i,j). Let E(i, и) := {x(i,f) : j < N(l/n)} and note that
άΗ(0„Σ(ί,η))< l/и for alii.
С, э *(1,1),*(1,2),... ,*U,&...
C2 э x(2, 1),jc(2,2), ... ,jc(2,j).-■■
С,- э x(i, l),x(i, 2),... ,jc(i,7), ...
Because X is compact, we may pass to a subsequence of the C, in order to
assume that the sequence x(i, 1) converges in X, to χ(ω, 1) say. Let C/ denote this
subsequence. Inductively, we may pass to further subsequences C* in order to assume
that for j = 1,... ,k each of the sequences (x(i,j))j converges in X to x(coJ). Let
Сш be the closure in X of {χ(ω,7) ξ j e N}. We claim that the diagonal sequence Ckk
converges to Сш in CX. To simplify the notation, we write CV in place of C\.
Let η > 0 be an integer. Given any integer I > 0, since й?я(С,, E(i, и)) <
\/n for all i, in particular x(i, I) is a distance at most l/и from E(i, и), and hence
χ(ω, ΐ) is a distance at most l/и from Σ(ω, η) := {x{w,j) : j < N{\/n)}. Thus
ан(Сш, Σ(ω, и)) < l/и. And if к? is large enough to ensure that d(x(k',j), χ(ω,β) <
l/и for ally <N(l/и), then:
d//(CV, Сщ)
< ^(^,Σ(^,π))+^(Σ(^',π),Σ(ω,π)) + ί?Η(Σ(ω,π), Сш)< 3/и.
D
One can rephrase Hausdorff convergence in terms of convergence of sequences
of points:
5.32 Lemma. C„ converges to С е СХ if and only if:
(1) for all χ & С there exists a sequence xn e C„ such that xn —¥■ χ in X, and
(2) every sequence y„^ e C„(,·) wii/г и(г') -> oo «ω α convergent subsequence
whose limit point is an element ofC.
Proof. Exercise.
D

72 Chapter 1.5 Some Basic Constructions
We wish to consider what it means for a sequence of compact metric spaces to
converge to a limit space when there is no obvious ambient space containing the
sequence. For this we need the following definition.
5.33 Definition. A subset S of a metric space X is said to be ε-dense if every point
of X lies in the ε-neighbourhood of A. (Under the same circumstances, S is called an
ε-net in X.)
An ε-relation between two (pseudo)metric spaces Χι and X2 is a subset R с
Χι χ X2 such that:
(1) for i = 1, 2, the projection of R to X,- is ε-dense, and
(2) if (jq, x2), (y!x,x!j) e R then |ί/χ,(*ι, .ή) — dx2(x2,x'2)\ < ε. The relation is said
to be surjective if its projection onto each X,- is surjective. If there exists an ε-
relation between X\ and X2 then we write X\ ~ε X2, and if there is a surjective
ε-relation then we write X\ ~ε X2.
We define5 the Gromov-Hausdorff distance between Χχ and X2 to be:
DH(XltX2):=MiB\Xi~eX2}.
If there exists no ε such thatXi ~ε Χ2, then DH(X\,X2) is infinite.
Sometimes, instead of writing "(x, y) e R" we shall write "χ is related to y" or
"jc corresponds to y".
5.34 Lemma.
(1) lfXx~eX2thenXx ~3εΧ2.
(2) D# satisfies the triangle inequality and hence defines a pseudometric (which
may take the value сю) on any set of metric spaces.
Proof. Exercise. D
Terminology: We say that a sequence of (pseudo)metric spaces X„ converges to X in
the Gromov-Hausdorff metric (or, in Gromov-Hausdorff space), and write X„ —> X,
if and only if DH(Xn, X) ->■ 0 as η -> oo.
5.35 Remarks and Exercises
(1) The graph of/ : Xi —> Хг is a 0-relation if and only if/ is an isometry.
(2) The following alternative notion of distance between metric spaces is the
one used by Gromov in [Gro81b]. Given two metric spaces Xi and X2, consider all
metrics d on the disjoint union X! JJ X2 that restrict to the given metrics on Xi and
X2, and define
D'H(XbX2):=infrfH(X,,X2),
a
where dH is as in definition (5.30).
5 Some authors prefer to use ~e instead of ~e in this definition; the difference is not
significant, 5.34(1).

Limits of Metric Spaces 73
(3) If X\ be a space with only one point and X2 is a space consisting of two points
a distance one apart, then DH{XX,X2) = 1 andD^(XbX2) = 1/2.
(4)LetXi and X2 be metric spaces. Show thatD^(XbX2) < eifandonlyifthereis
a metric space Υ and isometric embeddingsy, : X,- ^ Fsuchthatd//(/i(Xi),72(X2)) <
ε. (Hint: Given F and ?? > 0, perturbyi(Xi) in Υ χ [0, η] to make it disjoint from
л№).)
(5) Let Xi and X2 be metric spaces. Prove that 2Ό'Η{Χχ, X2) = ®н{Х\> Χι)-
(Hint: To see that 2D,H(Xl,X2) < D#(Xi,X2), given a surjective ε-relation R С
Χι χ X2, let Ζ be the disjoint union of a family of copies of [0, ε] indexed by R and
consider a quotient of Xi ЦХ2 Ц Ζ.)
The Gromov-Hausdorff distance between a metric space and any dense subset of
it is zero, and hence limits of sequences of spaces are not unique in general. However:
5.36 Proposition. Let A and В be compact metric spaces. A and В are isometric if
and only ifDH(A, B) = 0.
Proof. We shall show that if DH(A, B) — 0 then A and В are isometric, the other
implication is trivial. Let {an}„ be a countable dense subset of Л and let Rm be a
surjective (l/m)-relation between A and B. We choose bm_„ e В so that (a„, bm_n) e
Rm. By passing to a subsequence of {bm_ \)m we may assume that bm_\ -> b\ in 5. By
passing to a further subsequence we may assume that bm-2 -> b2, and so on. Thus
for all n, ri and infinitely many m we have that \dA(an, a„-) — dB(bm_n, b„ui>)\ < 1/m,
and hence ал(а„, a„>) = dB(b„, bn·). The desired isometry A -> В is the unique
continuous extension of a„ м> b„. D
(5.36) does not extend to proper metric spaces:
5.37 Exercise. Consider the following two metric graphs: each is constructed by
attaching a segment to integer points of the real line; in the first case the segment
attached at m has length | sinm| and in the second case it has length | sin(m + ^)|.
Prove that the Gromov-Hausdorff distance between these spaces is zero but that they
are not isometric. (Hint: Given ε > 0, let 5 > 0 be such that |jc — y| < 5 implies
| sinx—sinyl < ε. Note that there exist integers r and s such that | (r—^) — 2sn \ < δ.)
5.38 Proposition. LetX be a complete space that is a Gromov-Hausdorff limit of a
sequence X„.
(1) If each Xn is a length space, then X is a length space.
(2) If each X„ is proper, then X is proper.
(3) If each Xn is a proper geodesic space, then so too is X.
Proof. In order to prove (1), since X is complete it is enough to show that X has
approximate midpoints. Suppose that x,y eX and ε > 0 are given.

74 Chapter 1.5 Some Basic Constructions
If и is sufficiently large then there is a surjective ε-relation Rn с Χ χ Xn. Choose
x„, уn e X„ such that (x, x„) e /?„ and (у, >>„) е /?„, and let гп е X„ be such that
i№x{d(xn, z„), d(zn, yn)} < d{xn, y„)/2 + ε. Choose zeX with (z, z„) e Rn. Then,
1 1 5
d(x, z) < d(xn, z„) + ε < - d(x„, yn) + ε + ε < - d(x, у) + -ε.
Similarly d(y, ζ) < d(x, у) + 5ε/2. This proves (1).
Each closed ball in X is the limit of balls in the X„, and if a complete space is
the limit of compact spaces then it is compact (5.40). This proves (2). Part (3) is
immediate from (1), (2) and the Hopf-Rinow theorem. D
The interested reader should be able to think of many more properties that are
preserved under the taking of Gromov-Hausdorff limits. We should mention one
property that is not preserved: if the spaces concerned are not proper, then a complete
limit of geodesic spaces need not be geodesic. To see this, consider the metric graph
X with two vertices and edges {em | m e Z+}, where em has length (1 + 1/m). This
is not a geodesic space, but it is the limit of the geodesic spaces X„, where X„ differs
from X only in that its и-th edge has length 1.
In the proof of (5.31) we relied on the fact that closed subspaces of a fixed
compact metric space are uniformly compact in the following sense.
5.39 Definition. A family (Сх)л<=л of metric spaces is said to be uniformly compact
if there is a uniform bound on their diameters and for every ε > 0 there exists an
integer Ν(ε) such that each Cx can be covered by Ν(ε) balls of radius ε.
5.40 Exercise. If a sequence of compact spaces converges in Gromov-Hausdorff
space, then it is uniformly compact and the completion of the limit is compact.
(Hint: Recall that a metric space is compact if and only if it is complete and totally
bounded6.)
The following was an ingredient in Gromov's proof of his polynomial growth
theorem (see [Gro81b] and (8.37)).
5.41 Theorem (Gromov). If a sequence of compact metric spaces C, is
uniformly compact, then it has a subsequence that converges in the Gromov-Hausdorff
metric.
Proof. We follow the proof of (5.31). Thus we first construct a sequence of points
(x(i,j))jin each C,, with the property thatforallw e N the balls of radius l/«aboutthe
first N{\/n) terms in the sequence cover C,. Let S(i, n) = {x(i, 1), ..., x(i, N(i/n)}.
6 A metric space is totally bounded if, for every ε > 0, it is the union of finitely many balls
Β(χ,ε).

Limits of Metric Spaces 75
С, э x(l,l),x(l,2) x(lj),...
C2 э x(2,l),x(2,2),... ,x(2J),...
С, э x(i, 1), x(i, 2),... , x(i,j),...
Let φ denote the metric on C,. By passing to a subsequence Cf of the C,, we may
assume that the numbers dt{x{i, 1), x(i, 2)) tend to a limit, 5(1, 2) say, as i -> oo. By
passing to increasingly rarer subsequences Cf, given any integer к we may assume
that di(x(i,j), x(i,f)) ->· 8(j,f) as i ->· oo for all 1 < j < / < A:. We define 5 :
Ν χ Ν -> [0, oo) recursively in this manner, and claim that the diagonal subsequence
C£ is convergent.
Let / be the set of i indexing this diagonal subsequence of spaces.
We shall construct the desired limit as the completion of a countable space. To
this end, we take a countable set COo = {xi, xi, ■ ■ ■} and define a pseudometric on COo
by setting dx,(xj, хй) = S(j, k). We then take equivalence classes under the relation
that identifies points which are a distance zero apart, and define Cx, to be the metric
completion of this space. It is convenient to continue to write xj for the image of Xj
in См·
Fix an integer η > 0. Because S(i, n) is a (l/«)-net for C,, given any I > N(l/n)
and i e N we have dt{x{i, I), x(i,j)) < \/n for some j < N(l/n). If we fix I and let
the index i tend to infinity through the diagonal indexing set /, then the same choice
of У must recur infinitely often, and for this value we have d^Xj, xi) < \/n. Thus
5(oo, n) :— {xi,..., хщ\/„)} is (l/«)-dense in COo and hence in C^.
If г e / is sufficiently large, then by definition R„ := {{x(i,j),Xj} с S(i, η) χ
5(oo, и)} is a surjective (l/«)-relation. Moreover S(i, n) is (l/«)-dense in C, and
5(oo, n) is (l/«)-dense in CM, therefore Rn is an (1/«)-relation. Since η > 0 is
arbitrary, this proves the desired convergence.
Finally we note that Coo is totally bounded (indeed we have described a finite
(l/«)-covering for every η > 0), so since it is complete by construction, Coo is
compact. D
In [Gro81b] Gromov established the above compactness criterion by a different
argument, realizing the C, as compact subspaces of a fixed compact space.
5.42 Remarks
(1) Let d denote the usual metric on ШР1, let5„ be the closed ball of radius η about
a fixed point and consider the sequence of metric spaces Xn = (Bn, ^d). We claim
that {Xn | η e N} is not uniformly compact. To see this, we note that the volume
V(n) of a ball of radius η in ШР1 grows like e", so if one could cover Xn by Ν(ε) balls
of radius ε > 0, then one could cover (B„, d) by Ν(ε) balls of radius ηε, and hence
V(n) < Ν(ε).ν(εη). If we fix ε and let η -¥■ oo, this inequality becomes absurd.

76 Chapter 1.5 Some Basic Constructions
It follows from (5.40) that the sequence Xn does not have a subsequence that
converges in Gromov-Hausdorff space.
(2) By way of contrast, note that in Em the rescaled balls (B„, - d) are all isometric;
in particular they are uniformly compact. The previous inequality regarding volumes
is of course not absurd in this case, because V(n) = cnm, where с is a constant
depending on m. Polynomial growth is the real key here: in [Pan83] Pierre Pansu
proved some remarkable results concerning uniform compactness and convergence
of nilpotent groups.
(3) There are many applications of Gromov's compactness criterion (5.41) in
differential geometry, but to expose even a part of this literature would take us
well beyond the scope of this book. For an introduction to convergence theorems in
Riemannian geometry see [GrLP81], [B1W97] and [Pet96]. We mention one result to
give a flavour of some of the applications: For all η e N, allD > 0, allvo > 0 and all
k eM., the class M"(k, vq, D) of η-dimensional Riemannian manifolds with volume
> Vq, diameter < D and sectional curvature < k isprecompact in Gromov-Hausdorff
space (and in the C1·" topology1 for every a < I). Moreover, Л4"(к, vo, D) contains
only finitely many different manifolds up to diffeomorphism. In the form stated, this
result is due to Grove, Peters and Wu [GPW90]. There are similar results by a number
of authors, beginning with the work of Cheeger and Gromov [ChGr86].
Convergence of Pointed Spaces
Gromov-Hausdorff convergence works well in contexts where one wishes to
consider sequences of compact metric spaces, but it is a less satisfactory concept for
convergence for non-compact spaces. One obvious disadvantage is that the distance
between a compact space and an unbounded space is always infinite. Thus, for
example, Gromov-Hausdorff convergence is insufficient to capture the intuitive notion
that as the radius of a sphere of constant curvature tends to infinity, to an observer
standing at the north pole, the sphere looks increasing like Euclidean space.
The key here is that the intuitive sense of convergence comes from observations
taken at a fixed point.
5.43 Definition. Consider a sequence of metric spaces Xn with basepoints x„ eX„.
The sequence of pointed spaces (X„, x„) is said to converge to (X, x) if for every r > 0
the sequence of closed balls B(x„, r) (with induced metrics) converges to B(x, г) с X
in the Gromov-Hausdorff metric.
Under the same circumstances we say that X is a pointed Gromov-Hausdorff
limit of the Xn and that (Xn, x„) converges to (X, x) in the pointed Gromov-Hausdorff
sense.
Just as (unpointed) Gromov-Hausdorff convergence is most suitable for
sequences of compact metric spaces, so pointed convergence is most suitable for the
study of proper spaces. In particular:
7 For the definition of the ClQ topology see [Pet96].

Ultralimits and Asymptotic Cones 77
5.44 Lemma. If a complete space is the (pointed) Gromov-Hausdorff limit of a
sequence of proper spaces, then it is itself proper.
And Gromov's pre-compactness criterion becomes:
5.45 Theorem. Let (Xn,xn) be a sequence of pointed metric spaces. If for every
r > 0 and ε > 0 there exists an integer N(r, ε) such that B(xn, r) can be covered
by N(r, ε) ε-balls, then a subsequence of(X„, x„) converges in the pointed Gromov-
Hausdorff sense.
We leave the reader to formulate and prove the analogue of (5.38) for pointed
convergence.
5.46 Remark (Related notions of convergence). There are a number of situations
(notably in low-dimensional topology and geometric group theory) where one has a
natural degeneration of metric structure in which proper spaces approximate spaces
that are not proper. Such a situation arises, for example, in the study of degenerations
of hyperbolic structures [Sha91]. In light of (5.44), we need a weaker definition of
convergence to cover such cases. A suitable notion was introduced by Paulin [Pau88]
(see also, Bestvina [Bes88]). The idea of this generalization is that finite subsets in
the limit space should be approximated by finite subsets of the limiting sequence
(cf. (II.3.10)). There is a natural extension of this situation where one has a fixed
group acting on a sequence of spaces and one wishes to take an equivariant limit.
Such equivariant limits have played an important role in the study of 3-manifolds (see
[Sha91]) and many recent developments in geometric group theory, e.g. [Sela97].
(See section 4 of [BriS94] for further remarks.)
Ultralimits and Asymptotic Cones
Many of the key arguments in the preceding section are characterized by the fact
that one takes repeated subsequences to obtain the desired limiting space. A
considerable clarification of the argument would result if one could extract a convergent
subsequence all at once; non-principal ultrafilters provide a tool for doing so. They
also provide a means of constructing a limiting object (ultralimit) in cases where no
Gromov-Hausdorff limit of a sequence of spaces exists. For example, given a finitely
generated group Γ with a word metric d, it is natural to consider the sequence of
metric spaces Xn — (Γ, ^d); the identity element serves as a basepoint. One expects
limit points of this sequence to contain information about the asymptotic properties
of the group Γ. If Γ has polynomial growth thenX„ converges in the pointed Gromov-
Hausdorff topology ([Gro81b], [Pan83]). In general though, Xn will not even contain
a convergent subsequence. One remedies this by passing to ultralimits (5.50), the
significance of which were brought to the fore in this context by van den Dries and
Wilkie [DW84].

78 Chapter 1.5 Some Basic Constructions
In order to motivate the definition of ultrafilters and ultralimits, let us pursue
remark (5.46) and reflect that one might define a metric space (Y, d) to be a limit of
a sequence of metric spaces (Y„, dn) if for every finite set of points \p\, ... ,p,} d Υ
and every ε > 0, for infinitely many η one can find subsets \p", ... ,p"} С Y„ such
that idnip",^) — d(pj,pk)\ < ε for all 1 < j, к < i. This is too weak a notion of
limit for most purposes (but not all (II.3.10)). In order to obtain a stricter notion of
convergence one should replace "for infinitely many n" by "for almost all n", and
one should also require that the points of Υ account for all sequences of points in the
approximating sequence that might be said to converge. In order to quantify "almost
all" we need a measure:
5.47 Definition. A non-principal ultrafilter on N is a finitely additive probability
measure ω such that all subsets S С N are ω-measurable, w(S) e {0, 1} and ω(Ξ) = 0
if S is finite.
5.48 Exercise (Existence of Non-Principal Ultrafilters). Let N be a set and let VN
denote the set of its subsets. A filter for N is a map μ : VN -> {0, 1} such that:
μ(0) = 0, μ(Ν) = 1; if S С Τ then μ(5) < μ(Γ); and if μ(5) = μ(Γ) = 1 then
μ(5 Π Τ) = 1. And μ is called an ultrafilter if in addition μ(5) + μ(Ν \ S) = 1 for
all S e VN. (One thinks of μ as labelling subsets S as large if μ(5) = 1 and small if
μ(5) = 0.)
There is a natural ordering on the set of filters for any set N, namely μ < μ' if
μ(5) < μ'(^) for all S e VN. Given S e VN, one obtains an ultrafilter by defining
μ.ς(Τ) = 1 if and only if S с Т. Such an ultrafilter is called principal.
(1) Prove that a filter is ^-maximal if and only if it is an ultrafilter.
(2) If N is an infinite set then the map μ/ : VN —> {0, 1} which assigns the value
1 to a set if and only if its complement is finite is a filter. Prove that an ultrafilter ω
on N is not principal if and only if μ^ < ω.
(3) By applying Zorn's lemma to the filters μ with μ/ < μ, deduce that there
exist non-principal ultrafilters in the sense of (5.47).
Non-principal ultrafilters pick out convergent subsequences:
5.49 Lemma. Let ω be a non-principal ultrafilter on N. For every bounded sequence
of real numbers a„ there exists a unique point I e Ж such that ω{η : \a„ — l\ < ε} = 1
for every ε > 0. One writes I = Нтш а„.
Proof Exercise. D
One should be aware that linv an depends very much on the choice of ω. For
example, given two convergent sequences (bn) and (c„), if one defines a„ to be b„
if η is even and c„ if η is odd, then Итш ап will equal lim b„ if the even integers
have ω-measure one, and lim c„ if the odd integers have ω-measure one. A similar
dependence on the choice of ω is built into the following definition.

Ultralimits and Asymptotic Cones 79
5.50 Definition (ω-Limits and Asymptotic Cones). Let ω be a non-principal ultra-
filter on N. Let (X„, dn) be a sequence of metric spaces with basepoints pn and let IM
denote the set of sequences (xn), where xn e Xn and dn(x„,pn) is bounded
independently of n. Consider the equivalence relation [(x„) ~ (y„) iff lima, dn(xn, yn) = 0],
and let Χω denote the set of equivalence classes. Endow Χω with the metric
da>((xn), (Уп)) = HnviinCxn, y„). One writes (Χω, άω) = Ιΐπιω(Χη, d„) (if the choice
of basepoints is not important).
One may be interested in the case of a fixed metric space X with a basepoint and
a metric d, and X„ = (X, ^d). In this caseXa, is called an asymptotic cones of X and
is denoted Сопеш(Х).
5.51 Remark. One can construct metric spaces X (subsets of the line even) and non-
principal ultrafilters ω φ ω' so that Сопеш(Х) and СопеШ'(Х) are not homeomorphic.
Simon Thomas and Boban Velickov recently showed that this can happen even when
X is the Cayley graph of a finitely generated group.
Ultralimits are related to Gromov-Hausdorff limits by the following exercise.
5.52 Exercise. Suppose that the pointed space (Χοο,Ροο) is the Gromov-Hausdorff
limit of a sequence of proper spaces (Xn,pn). Show that for every non-principal
ultrafilter ω on N, the ultralimit (Χω, ρω) is isometric to (IM, p^). (See (5.55) with
regard to the need for properness.)
5.53 Lemma. The ultralimits of all sequences of metric spaces are complete.
Proof. We sketch the proof and leave the details to the reader. LetXo, = Нтш Хп. Let
d„ denote the metric on X„. Let (xO,)j be a Cauchy sequence in Χω and represent each
entry xOj by a sequence (jc^), where x'j e X,. Let Л о = N and for к е N inductively
define a strictly decreasing sequence of subsets А* С A^-i with w(Ak) = 1 so that
for all г and ally between 1 and ί: we have |й?ш(Уш,х^) —4(^,^)1 < 1/2*. Set у,- = х\
for all i e Ak-\ ν Ak and prove that the sequence (y,·) defines a point уш е Χω such
that χΌ) -> уш asj -> oo. D
5.54 Exercise. Show that any ultralimit of length spaces is a length space, and any
ultralimit of geodesic spaces is a geodesic space.
5.55 Remarks. If X is a proper metric space, then the map which sends each χ e X to
the constant sequence at χ defines an isometry fromX to the ultralimit of the constant
sequence X„ = X. However, this is not true of complete metric spaces in general.
Indeed, if X is a countably infinite set endowed with the discrete metric, d(x, y) = 1
for all χ ψ у, then for any non-principal ultrafilter ω, the ultralimit of the constant
sequence Xn — X is an uncountable set with the discrete metric.
Some authors, e.g. [K1L97], find it useful to allow sequences of scaling factors other than
(1Jn) in the definition of an asymptotic cone.

80 Chapter 1.5 Some Basic Constructions
Asymptotic cones and ultralimits have played a significant role in recent proofs
of rigidity theorems (e.g. [K1L97]). This follows the influence of Gromov [Gro93].
The asymptotic cone of a finitely generated group was introduced by van den Dries
and Wilkie [DW84] as a device for streamlining Gromov's proof of his polynomial
growth theorem. See [Gro81b], [Dru97], [Paps96] and [Bri99a] forresults connecting
the geometry and topology of asymptotic cones of groups to algebraic properties of
groups.

Chapter 1.6 More on the Geometry of MnK
In this chapter we return to the study of the model spaces M". We begin by describing
alternative constructions of W — M"_, attributed to Klein and Poincare9. In each
case we describe the metric, geodesies, hyperplanes and isometries explicitly. In the
case of the Poincare model, this leads us naturally to a discussion of the Mobius
group of S" and of the one point compactification of E". We also give an explicit
description of how one passes between the various models of hyperbolic space. In the
final paragraph we explain how the metric on M" can be derived from a Riemannian
metric and give explicit formulae for the Riemannian metric.
The Klein Model for Mn
In Chapter 2 we constructed H" as a subset of E"1 = W+l. This subset does not
contain 0 e R"+1, and each 1-dimensional subspace of K"+1 intersects it in at most
one point. Therefore, the natural projection from K"+1 χ {0} to the real projective
space P" of dimension η restricts to an injection on H". One way to obtain the Klein
model for H" is to simply transport the geometry of H" С Ε"·' by the map p. We
shall explain this in some detail.
Let P" denote real projective и-space, i.e. the set of 1-dimensional vector sub-
spaces of W+l. Let ρ : W+l \ {0} ->■ P" be the natural projection, which associates
to each point X e W+l χ {0} the line passing through 0 and X. Let χ = p(X). The
coordinates of X are called homogeneous coordinates for x; thus "the" homogeneous
coordinates of χ are well defined up to multiplication by any non-zero real number.
The projective transformations of P" are, by definition, the transformations of P"
induced by the linear transformations of W+'; they form a group PGL(« + 1, K)
naturally isomorphic to the quotient of the group GL(« + 1, K) by the normal subgroup
formed by non-zero multiples of the identity matrix. A projective line (more
generally a projective /j-plane) is a subset of P" formed by the 1-dimensional subspaces
contained in a 2-dimensional (resp. a (p + l)-dimensional) subspace of W+l.
The projection ρ sends the set of points Χ φ 0 of E"·1 = K"+1 such that (X |
X) = 0 onto the set of points in P" whose homogeneous coordinates satisfy the
9 The basic properties of real hyperbolic spaces were first established by Lobachevski in
the nineteenth century. See [Mil82] for an account of the subsequent developments of the
subject.

82 Chapter 1.6 More on the Geometry of M"
quadratic equation 5Z"=] Xf — X^+l = 0; the set of such points is a quadric, and
shall be denoted Q. The map ρ restricts to a bijection from H" to the interior of
the quadric Q, i.e. the connected component of P" — Q which is the set of points
whose homogeneous coordinates satisfy 5Z"=1 Xf — X^+i < 0. Hence ρ induces an
isomorphism from the group 0(n, 1)+ of isometries of H" to the group of projective
transformations of P" that preserve the interior of the quadric Q. The geodesic lines
in W are the intersections of H" with the vector subspaces of E"'' of dimension 2;
they are mapped by ρ onto the non-empty intersections of the interior of Q with the
projective lines of P".
The projective form of the Klein model for hyperbolic и-space is the interior of
the quadric Q endowed with the unique metric such that d(p(X), p(Y)) — d(X, Y)
for all Χ, Υ e Ш" с Ε"1. This metric admits an elegant description in terms of
projective geometry: given two distinct points χ and у in the interior of Q, let Χχ
and у χ be the points of intersection of Q with the projective line through χ and y,
arranged so that Χχ,χ, у, Ух occur in order on the projective line through χ and y.
6.1 Lemma. The distance between two points x, у in the projective Klein model is
given by the formula
d{x, У) = ~ logO, У, Χοο, 3Όο)>
where (x, y, Χχ, ух) is the cross ratio of the four aligned point x, y, Χχ, ух.
Proof. Recall that the crossratio(x, y, v, w) of four aligned distinct points x,y, v, w e
P" is defined as follows. Let X and Υ be points of K"+' - {0} projecting by ρ to χ and y.
Then there are pairs of real numbers (λ, μ) and (λ', μ') such that ρ(λΧ+μΥ) — υ and
ρ(λ'Χ+μ'Υ) = w. By definition, (x, y, v, w) =■ λμ'/μλ' e [0, oo]. This definition is
independent of the various choices. An alternative description of (x, y,v,w) is to say
that it is the cross ratio (X, Y, V, W) (in the sense of 6.4(3) below) of any four aligned
points X, Y, V, W e R"+1 - {0} such uieXp(X) = x, p(Y) = y, p(V) = υ, p(W) = w.
If h is a projective transformation, then {h(x), h(y), h(v), h(w)) = (x, y, v, w).
To prove the lemma, we use the invariance of the cross ratio under projective
transformations and the fact that Isom(H") acts transitively on pairs of points which
are a fixed distance apart. This enables us to reduce to the case where χ and у are the
images under ρ of points X = (Xu ..., X„+)) and Υ — {Y\, ..., Yn+\), all of whose
coordinates are zero except Xn+\ = 1, and Y\ = sinhr and Fn+) = coshr, where
r = d(x, y). Then Χχ — (cosh r + sinh r)X - Υ and Υχ — (cosh r — sinh r)X — Υ are
sent by ρ to Χχ and yx respectively. One calculates that (x, y, Χχ, ух) = e2r. D
One can obtain a more specific parameterization of the Klein model by replacing
the interior of the quadric Qcp with the open unit ball B" — {x e W : ||jc|| < 1},
where ||*|| denotes the Euclidean norm of x. This is essentially done by identifying
B" with the disc Β" χ {1} cl"+l and noting that there is exactly one point of this
disc on each line in W+l that represents a point in the interior of the quadric Q.
The following proposition describes this construction more precisely.

The Klein Model for H" 83
Fig. 6.1 The hyperboloid model and the Klein model for I
6.2 Proposition (The Metric for the Klein Model). Let hK be the home ото rphism
from B" to H" that sends the point χ = {x\, ..., x„) e B" to
f(*l
,*«, l)e
лД-1М12'
H". (Note that this image point is the intersection о/И" with the line through 0 and
(x, l)el"xl' = E"-1.; The ball B", equipped with the pull-back by hK of the
hyperbolic metric ofW is the Klein model for hyperbolic η-space. The distance
d(x, y) between two points x, у in this model is given by the formula:
l - (χ I y)
cosh d(x, y)
yi - ||*||2vT^"
where (χ \ y) is the Euclidean scalar product and \\ ■ \\ is the Euclidean norm.
Given two distinct points x, у е B", the unique hyperbolic geodesic line
containing χ and у is the intersection ofB" with the affine line in W through χ and y.
Let jCoo and y^ be the two intersection points of this line with the boundary S"~' of
B", arranged so that x^,, x, y, y^, occur in order on the line through χ and y. The
hyperbolic distance d(x, y) between χ and у is given by
d{x, У) = ~ log(Jc, у, Хм, Уоо) = - log — -
2 2 \ ||лг — лгоо ||. ||y — >Όο ||
Proof. The first formula for d(x, y) is a straightforward computation. Arguing as we
did in (6.1), one can use the projective invariance of the cross ratio and the fact that
Isom(H2) acts transitively on pairs a fixed distance apart, to reduce the verification of
the second formula to the case χ = (0,..., 0) and у = (^f-r, 0,..., 0), where r —
d(x, y). In this case Хм = (—1, 0,..., 0) and y^, = (1,0,..., 0), and (x, y, iM, y^)
is e
2r
a
A great advantage of the Klein model is that the geodesic lines are very easy to
describe: they are the non-empty intersection of Bn with Euclidean lines. However,
the angle between two geodesic segments in the Klein model that issue from the

84 Chapter 1.6 More on the Geometry of M"
same point is not equal to the Euclidean angle observed by simply regarding these
segments in В" С Е". Thus considerations of angles in the Klein model do not
conform to Euclidean intuition.
For instance, orthogonality can be understood as follows. We can consider the
projective space P" as the union of its affine part E" and the hyperplane at infinity. A
hyperplane in the Klein model is a non-empty intersection of B" with a hyperplane
Η of P". The pole of this hyperplane is, by definition, the point of χ e Ρ" χ Β" such
that the lines which issue from χ and are tangent to the boundary S"_1 of B" cut S"~'
along Η П §"-'.
Fig. 6.2 The pole of a hyperplane with respect to the sphere S"
6.3 Exercise. Show that the hyperbolic lines orthogonal to the hyperplane defined
by Η are the non-empty intersections of B" with the lines through the pole of H.
For η — 2, construct the common perpendicular to two non-intersecting geodesic
lines in the Klein model.
The Mobius Group
6.4 Definitions.
(1) Let E" = E" U {oo} be the one point compactification of E". There is a natural
homeomorphism from S" с E"+1 to E", given by stenographic projection pN from
the north pole N = (0,..., 0, 1) e §": identify E" with the hyperplane xn+l = 0
in E"+1 via^the map (x\,..., xn) м> (х\,..., xn, 0); the stereographic projection
pN : S" -»■ E" maps the north pole N to oo and maps each χ e S" χ {Ν} to the point
of intersection of the hyperplane E" with the line through N and x.

The Mobius Group 85
(2) A circle in Ё" is either a Euclidean circle in E" or the union of a straight line
in E" and {oo}. More generally, a k-sphere in E" is either a ^-dimensional sphere in
E" or the union of a ^-dimensional affine subspace of E" and {00}.
(3) The cross ratio of four distinct points x, y, v, w situated on a circle of E" is
the real number (x, y, v, w) defined by the formula
\\x-w\\.\\y-v\\
(χ, у, ν, w)
\\х-ь\\.\\у-\»\\'
where || · || is the Euclidean norm; by convention, ^ = Oand ~ = l,and ||jc-oo|| =
00 for all χ eE".
(4) An inversion (or reflection) Is of E" with respect to an (n — l)-sphere S is
defined as follows. If S is a sphere contained in E" with centre a and radius r > 0,
then /5 exchanges a and 00 and is defined otherwise by
r2
\\x-af
In other words, /5 sends each point χ φ a to a point on the geodesic ray from a that
passes through x; this point is characterized by the fact that the product of its distance
from a with the distance from χ to a is equal to r2. In particular, the points of S are
precisely those left fixed by /5. If S is the union of {00} and a hyperplane Η с Е",
then Is fixes 00 and its restriction to E" is the Euclidean reflection m-
We leave the reader to check the following properties of inversions.
6.5 Proposition. Let Is be an inversion ofE" with respect to a sphere S. Then,
(1) Is maps spheres ofE" to spheres ofE";
(2) Is preserves the cross ratio;
(3) /5 preserves the Euclidean angles between intersecting spheres;
(4) the stereographic projection рн from S" onto E" is the restriction to S" of the
inversion o/E"+1 with respect to the sphere whose center is the north pole N
and whose radius is \/2.
From this proposition it follows that the stereographic projection/j/y maps spheres
to spheres and preserves cross ratios and angles. We define an inversion ofS" with
respect to an (n — Y)-sphereS С S" to be the restriction to S" of the inversion of E"+1
with respect to the unique и-sphere that is orthogonal to S" and contains S. Note that
stereographic projection p^ conjugates such an inversion of S" to the inversion ofE"
with respect to the (η ~ l)-sphere/?w(5).
6.6 Definition. The Mobius group Mob(rc) (resp. Mob(S")) is the group of
transformations ofE" (resp. of S") generated by inversions in (n — l)-spheres.
Exercise. Show that the subgroup of the Mobius group Mob(rc) fixing the point 00 is
the group of similarities of E" (extended by the identity on {00}). For instance any

Chapter 1.6 More on the Geometry of M"
Fig. 6.3 The stereographic projection viewed as the restriction of an inversion
homothety fixing the origin is the product of two inversions with respect to spheres
centred at the origin.
The Poincare Ball Model for W
As with the Klein model, in the Poincare ball model the points of hyperbolic space are
represented by the points of the unit ball B" in Ε", but the geometry imposed upon the
ball is quite different. In particular, we shall see that the geodesic lines in the Poincare
model are the intersection of B" with those Euclidean lines and circles which meet
the boundary of B" orthogonally. A great advantage of the Poincare model is that it
is conformally correct in the sense that the angle between two geodesic paths issuing
from a point is equal to the Euclidean angle between these paths.
6.7 The Poincare Ball Model. We define the Poincare metric on B" by pulling back
the metric from H" с Ε"'1 via the homeomorphism hP : B" —»■ H" given by
hp(x) = ( 2x , !±Ш еГх1= Ε"·',
Vi-и2 i-iwiv
where || jc|| is the Euclidean norm of x. This map associates to χ the point of intersection
of Ш" with the line through the points (0, -1) and (x, 1) in W χ Κ = Ε"''.
More explicitly, the hyperbolic distance d(x, y) between x, у е В" in the Poincare
model is given by the formula
(1 + |W|2)(1 + W2) - 4(x | y) , , „ \\x-y\\2
cosh d(x, y) — —-τ- = 1+2
(1 - IMI2)(1 - IMI2) (1 - M2)(l - IMI2)'
where (x | y) is the Euclidean scalar product.

The Poincare Ball Model for H" 87
6.8 Proposition. In the Poincare ball model for hyperbolic η-space, the unique
geodesic segment joining distinct points x,y e B" is the arc which they bound on
the unique Euclidean circle (or line) that passes through them and intersects the
boundary of B" orthogonally. If Χχ, and y^, are the points of intersection of this
circle with S"_1, arranged so that jcoq, x, у, у χ, occur in order on the circle, then the
hyperbolic distance between χ and у is given by the formula
d(x, y) ~ log (X, у, Хоо, Уж).
Proof. We shall deduce the stated properties of the Poincare model from those of
the Klein model by examining the following explicit description of the isometry
h := (hK)~lhp : B" -> B" between the two models (see figure 6.4). Briefly, this
isometry is obtained as the composition of two projections, the first stereographic,
the second orthogonal.
Fig. 6.4 The map h := (%) ' hp from the Poincare ball to the Klein model
Consider Bn as the intersection of the unit ball Bn+l in E"+I=l"xl with the
hyperplane Κ" χ {0}. Letps : B" -» S" с W χ R be the stereographic projection
from the south pole of S", sending B" onto the upper hemisphere of S"; namely,
ps : χ н> (, +^||2, |~H2). In other words, ps is the restriction to B" of the inversion
of E"+1 in the sphere with centre (0, ... ,0,-1) and radius \/2. The isometry h =
{hK)~lhP : χ м> 1+^,|2 is obtained by post-composing the stereographic projection
ps with the orthogonal projection from the upper hemisphere of S"CR"x {0} onto
fi"cl"x {0}.

88 Chapter 1.6 More on the Geometry of M"
We claim that the isometry h sends the set of arcs of those circles and lines in B"
that are orthogonal to the boundary §""' onto the set of straight line segments in B".
Indeed ps sends the former set of arcs onto the set of arcs in the upper hemisphere
of S" that are contained in circles orthogonal to the equatorial sphere §"~'; and the
orthogonal projection from the upper hemisphere of S" onto B" then sends the set
of such circular arcs onto the set of straight line segments in B". Thus, since the
geodesic segment joining two distinct points in the Klein model is a straight line
segment, the geodesic arc joining two distinct points x,yeBn in the Poincare model
is the arc bounded by χ and у on the unique Euclidean circle (or line) that passes
through them and meets §"_1, the boundary of β", orthogonally. Let x^ and yoo be
the points of intersection of this circle with S"~', arranged so that x^, x, y, y^ occur
in order on the circle.
Because ps is the restriction of an inversion of E"+1 the cross ratios (x, у, х^, у^)
and (ps(*),Ps(y),*oo, Уоо) are equal (6.6(4)). The points ps(x),ps(y), *co> Уса lie ОП
a semicircle orthogonal to Κ" χ {0}, so it is easy to calculate the cross ratio of their
projections onto W χ {0}:
6.9 Exercise. Let p, q e E" be distinct points on a semicircular arc with endpoints
a and b. Let p' and q' denote the orthogonal projections of ρ and q onto the line
segment [a, b]. Prove that (p, q, a, b) = (p', q', a, b)112.
Hence (h(x), h(y), Xoo, yoo) ~ (ps(x), Psb>), *οο,}Όο)1/2 = (x, У, *oo, >Όο)1/2·
Therefore, log(x, y, *«,, joo) is equal to j log (h(x), h(y), Xoo, yoo), which is the
hyperbolic distance in the Klein model between h(x) and h(y), by (6.2). Since h is an
isometry, this establishes the desired formula for d(x, y) in the Poincare model. G
In Chapter 2 we described how reflections through hyperplanes generate Isom(H").
In the Klein model for hyperbolic space (based on B"), the hyperplanes are simply
the intersections of B" with Euclidean hyperplanes. The next proposition describes
the hyperplanes in the Poincare ball model of hyperbolic space, and the subsequent
corollary records what this description tells us about the structure of the isometry
group of the model.
6.10 Proposition. Let hP : B" -»■ H" be the isometry from the Poincare ball to W
defined in (6.7J. This map sends the set of intersections ofB" with (n — 1)-dimensional
spheres KE" orthogonal to S"_1 bijectively onto the set of hyperplanes ΗζΙ".
Moreover, hp conjugates inversion in S to reflection through hp(S), i.e., hplshp~l —
rhp(.S)-
We proved in (2.18) that reflections through hyperplanes generate IsomQHI"),
hence:
6.11 Corollary. The group ofisometries of the Poincare ball model for W is
generated by the restrictions to B" of inversions ofE" in spheres orthogonal to S"_1. Thus
Isom(H") is naturally isomorphic to the Mobius group Mob(S"-1).

The Poincare Ball Model for Шп 89
Proof of 6.10. One way to prove this proposition is to calculate an explicit formula
for hplrfjhP, the conjugate of a hyperplane reflection. We indicate two other, more
instructive, proofs.
Let S be an (n — l)-sphere orthogonal to S"~l, and consider the restriction to
B" of the inversion /5. According to (6.5), /5 sends the set of circles orthogonal to
S"~' to itself and preserves cross ratios. Hence, by the the preceding proposition, /5
is an isometry of the Poincare ball. Its fixed point set is S Π Β". But, according to
(2.18), reflections in hyperplanes are the only isometries of W whose fixed point sets
separate H" into two connected components; and hyperplanes can be characterized
as those subsets of H" which arise as such fixed point sets. Since the fixed point set
of the isometry hPIshpl has this separation property, and its fixed point set is hP(S),
we deduce that hPIshpl is the reflection in the hyperplane hP(S).
It remains to be shown that every hyperplane in Ш" arises as hP{S) for some S.
Since every hyperplane is the bisector of some pair of points in И", it suffices to
show that the set of points in the Poincare model that are equidistant from a fixed
pair of points x0 and y0 form an (n — l)-sphere orthogonal to S"_1. According to the
first formula in (6.7), the set of such points is determined by the equation:
ll*o-z||2 = *||j>o-z||,
where к — (I — ||*||2)/(1 — 1Ы12)· It is an ancient observation of Apollonius that this
is the equation of a sphere (if к φ 1) or a Euclidean hyperplane (if к = 1). Moreover,
since we know that the geodesic lines in the Poincare model are the intersection with
B" of (Euclidean) lines and circles orthogonal to S"_1, and since we also know that
the connected components of the complement of a hyperplane in Ш" are geodesically
convex, the above bisecting sphere/hyperplane must meet S"-1 orthogonally, as was
to be proved. G
A different approach to proving the proposition is to first factor hP as hKh, where
h — (hK)~rhP is the map from the Poincare to the Klein model considered in (6.8).
As in that proof, we write h — π ops, where/55 is a stereographic projection sending
B" onto the upper hemisphere of the unit sphere S"+ с W χ R and π is orthogonal
projection onto Β" = Β" χ {0}. The map ps sends the set of intersections of B"
with (n — l)-spheres orthogonal to S"_1 onto the set of half-spheres in S^_ that are
orthogonal to the equator §""'. The orthogonal projection π sends the set of such
half-spheres onto the set of intersections of B" with affine hyperplanes of E". The
latter is precisely the set of hyperbolic hyperplanes in the Klein model. To finish
the proof, one notes that if S is an (n — l)-sphere orthogonal to §""', then hPIshpl
is an isometry that fixes the hyperplane hP(S); it is not the identity, so it must be a
reflection through this hyperplane.

90 Chapter 1.6 More on the Geometry of Mn
The Poincare Half-Space Model for W
6.12 Definitions. Let H" denote the upper half-space {{x\,..., xn) e E^ | xnj>
0} с Ε". The Cay ley transform is the restriction to B" of the inversion /5 : E" —»■ E",
where S is the sphere with centre (0,..., 0, — 1) and radius л/2.
The Cayley transform gives a homeomorphism from the Poincare ball model for
H" to H". We define the Poincare half-space model for W to be the half-space H"
endowed with the unique metric with respect to which this homeomorphism is an
isometry.
The boundary dH" of the half-space model H" is the set of points {(x\ ,...,*„)
e E" I xn = 0} U {oo} с Е". One can extend the Cayley transform by considering
the restriction of the above inversion /5 : E" —»■ E" to the closure of β"; the resulting
map sends §"_1, the boundary of β", bijectively onto ЗЯ".
Applying the considerations of (6.5) to the Cayley transform, one sees
immediately that:
6.13 Proposition. Propositions 6.8 and 6.10 and Corollary 6.11 remain valid if one
replaces the Poincare ball model B" by the half-space model H" and §""' by dH".
In Chapter II.8 we shall describe a natural compactification ШГ of H" that depends
only on the metric (and not on any specific model). The subspace formed by the
ideal points of this compactification is homeomorphic to §"_1, and there are natural
bijections from it to the sphere bounding the ball model in E", and to dH". We shall
see that isometries of W extend uniquely to homeomorphisms of И ; the induced
action of Isom(#") on ЗИ" is faithful and transitive.
We have seen that the generators of the isometry group of the Poincare ball
model are restrictions to B" of inversions of E". Thus they extend to continuous
homeomorphisms of the closed ball B" U S"_1. Similarly, there is a natural extension
of isometries of the half-space model to homeomorphisms of ЗЯ". With respect to
these actions, the above bijections from ЗИ" to §""' and dH" are equivariant.
The study of individual isometries of H" is well-documented and we shall not
repeat it here. However, it is worth recording some important examples: for the action
of Isom(H") on the Poincare ball model, the subgroup fixing 0 e В" с Е" consists of
the restrictions to B" of the usual linear action of 0(n) on E"; for the action of Isom(H")
on the Poincare half-space model #", the subgroup fixing the point {00} e ЭЯ"
is generated by the restrictions to #" of Euclidean translations, homotheties and
reflections of E" preserving H" — it is isomorphic to Isom(E"_1) χ: Κ, where the
left hand factor consists of those isometries of H" that preserve subsets of the form
Μ"-1 χ {a} and the action of the right hand factor is conjugation by Euclidean
homotheties а м> е'а.

IsometriesofH2 91
Since the actions of Isom(H") on H" and ЗИ" are both transitive, these examples
serve to describe the isotropy subgroups of arbitrary points in И , up to conjugation
in Isom(H").
Isometries of H2
We conclude our considerations of the standard models for hyperbolic space with a
classical description of Isom(H2). This requires that we introduce a complex
coordinate in the Poincare model: identify E2 with the complex plane C, the unit ball B2
with the disc {z e С \ \z\ < Ц and the upper half-plane H2 with {z e С | Зг > 0},
where Зг is the imaginary part of z. Given two distinct points^ and w in B" or H'\
let с be the unique circle in the Riemann sphere С U {00} = E2 that passes through
z and w and meets the boundary of the model orthogonally. Let Zoo and Woo be the
two points at which the circle с cuts the boundary of the model, labelled so that
Zoo, Z, w, Woo occur in order on c. The hyperbolic distance between ζ and w is
d(z, W) ~ log (Z, W, Zoo, Woo),
where (z, w,Zoo, Woo) = (ζ-οαΓ-^ω°°) 1S rea^anc* &eaieT man one.
In (6.12) we related B" to H" by the Cayley transform; in the present context
this is given by the formula ζ ь-> ^zf- (We warn the reader that the term Cayley
transform is also commonly used to describe the orientation preserving isometry
H2 -+ B2 defined by ζ м> g.)
6.14 Proposition. The group of isometries of the Poincare half-space model for H2
is precisely the group of maps H2 —> H2 obtained by letting the group GL(2, Ж) act
thus:
if ad — be > 0 then the action of I ,1
if ad — be < 0 then the action of[ , I is
az + b
гн> ;
cz + d
с d
az + b
гн> — -.
cz + d
The kernel of this action consists of all scalar multiples of the identity matrix. Thus
Isom(H2) = PGL(2, R).
The group of isometries of the ball model for Ш2 is precisely the group of maps
of the form:
pz + q PZ + q
гн> , ζ ι-» ——-,
qz + p qz + p
where p,q e С and \p\2 — \q\2 — 1.

92 Chapter 1.6 More on the Geometry of M"
Proof. A direct calculation shows that the given maps are indeed isometries of the
half-space model H2, and the map GL(2, K) -»■ Isom(#2) described by the given
formulae is a homomorphism with kernel the group of scalar multiples of the identity.
To see that the given action is transitive on H2, it suffices to consider ζ м> az with
a > 0 and ζί-* z + b.
The Cayley transform conjugates the given maps of H2 to those of B2, hence the
latter group of maps acts transitively on B2. This group also contains 0(2) (in the
form of the maps ζ м> pz and ζ м> pz, with \p\ = 1), and since this is the stabilizer
of 0 in Isom(B2), it follows that the given maps are the whole of Isom(H2). G
М% as a Riemannian Manifold
In Chapter 2 we defined the metric spaces MnK. The main purpose of this section is
to exhibit MnK as the length space associated to a Riemannian manifold. Initially, we
defined the set MnK to be a certain submanifold of a Euclidean space. We shall give an
explicit description of a Riemannian metric on this submanifold. We shall then prove
that M" is the associated length space. We do so by combining some facts about
Isom(M^) with an explicit calculation of the Riemannian length of certain curves.
We shall also give explicit formulae for the Riemannian metrics on the Poincare
models of hyperbolic space.
We begin by recalling once more that a Riemannian metric on a differentiable
manifold X is an assignment to each tangent space TXX of a positive definite bilinear
form (i.e., a scalar product) such that these forms vary continuously with x. By
relaxing the requirement that these bilinear forms must be positive definite one
obtains the notion of a pseudo-riemannian metric.
The Riemannian Metric on M\
As we pointed out in Chapter 3, E" has a natural Riemannian metric, the Euclidean
Riemannian metric ds2 = Σ"=ί dx2 (see Chapter 3 for notation). More explicitly,
one identifies the tangent space TXE" of E'! at χ with the set of pairs (χ, υ), where υ
is a vector in E"; then one endows TxWl with the unique vector space structure and
scalar product so that the projection (χ, υ) ы> υ is a linear isometry of TXE" onto E'!.
S" is the differentiable submanifold of E'!+1 consisting of vectors of Euclidean
norm one. As such it inherits an induced Riemannian metric: the tangent space TxSn
to S'! at χ e S" is the vector subspace of TxEn+l consisting of pairs (χ, υ) with
(x | v) = 0 (Euclidean scalar product), and the restriction to this subspace of the
given scalar product on TxE"+i defines a Riemannian metric on §". In other words,
for every χ eSn, the map (χ, υ) ы> υ gives a bijection from TXS" to the «-dimensional
subspace x1 С E"+1 orthogonal to x, and the Riemannian metric on S" is the unique
assignment of scalar products that makes all of these maps isometries.

Μ" as a Riemannian Manifold 93
In an entirely analogous way, one can obtain a Riemannian metric on W by
regarding it as a differentiable submanifold of E"·'. The latter has a natural pseudo-
riemannian metric ds2 = Q2"=1 ^A~) ~~ dx\+v *^е desired Riemannian metric on Ш"
is obtained by restricting the given bilinear form on TXE"·' to TXW for each χ e W
(these restrictions are positive definite). More explicitly: the tangent space TXW to
W at χ e Ш" is the vector subspace of TxEn·' consisting of pairs (χ, υ) such that
υ e Xх (that is, (x | υ)=0); for every χ e H", the map (χ, υ) ы> υ gives a bijection
from ГЛИ" to the и-dimensional subspace Xх С Е"1 orthogonal to x; the desired
Riemannian metric on H" is the unique assignment of scalar products that makes all
of these maps isometries. (The restriction of (· | ·) to r1 is positive definite for all
χ e W.)
The natural Riemannian metric on MnK is obtained from the preceding special
cases by scaling. There are two equivalent ways of doing this. If к > 0, then one
can either identify M" with S" and multiply each of the scalar products defining
the Riemannian metric on the latter by a factor of l//c, or else one can view M" as
the Riemannian submanifold {x e E"+1 | (x | x) = \/к} с E"+1 and give it the
induced Riemannian metric. Similarly, if к < 0, then one can either identify MnK with
W and multiply each of the scalar products defining the Riemannian metric on the
latter by a factor of 1/— к, or else one can view M" as the Riemannian submanifold
{x e E'!+1 | (x | x) = \/к } с Ε"·1 and give it the induced (pseudo-)Riemannian
metric.
We should be absolutely clear about the logical structure of what we are doing at
this point. We have introduced a Riemannian metric on the manifold M"; this allows
us to speak, for example, of the Riemannian length of curves inM" and of Riemannian
isometries from M" to itself or other manifolds. But a priori the Riemannian length
of a curve may be different from its length in the metric space M^, and we may not
assume that a Riemannian isometry from M" to itself (i.e., a map whose derivative
preserves the Riemannian metric) is an isometry of the metric space MnK, nor vice
versa.
6.15 Lemma. Given к e M, suppose that M1' is equipped with the Riemannian
metric described above. Let d denote the metric on M" defined in (2.10).
(1) The action of lsom(M", d) on MnK is by Riemannian isometries.
(2) For all χ e MnK and r > 0 (with r < DK = π/y/ic if к > 0), if κ ψ 0 then
the induced Riemannian metric on {у е Μ" \ d(x, у) = г] с М" makes it
isometric, as a Riemannian manifold, to a sphere of radius —)= \ sin(rv/if)| in
E". (If к < 0 then smir-s/κ) = sin(i>V—~ic) = i sinh(rV—к)■)
Proof. Because the metric and Riemannian metric on M" were defined by compatible
scaling processes, it is enough to consider the cases M" = E", S", Ш". It is obvious
that the group D(n) of displacements of E'\ the orthogonal group 0(n + 1) acting
as usual on S" с E'!+1, and the group 0(n, 1)+ acting linearly on W с Е"·1, all
preserve the Riemannian metric. Thus (1) is a consequence of (2.21).

94 Chapter 1.6 More on the Geometry of M"
Having proved (1), it suffices to verify assertion (2) for a particular choice of x,
because Isom(M") acts transitively. In the case of §", we let χ = (1, 0, ... , 0). By
definition, {y e S'! | d§„(x,y) = r} = {y e E"+1 | ||>>|| = 1, cosr = (у \ x)} =
{iy\. · · · . Уп+ύ e E'!+1 | y\ = cos r, Σί> ι yf = sin2 rb which is a Euclidean sphere
of radius sin r. The Riemannian metric which this set inherits as a submanifold of
S'! is, by definition, the same as that which it inherits from the ambient manifold E'!.
This proves (2) in the case of §".
The case of W с Ε"·1 is analogous. Let χ = (0,... , 0, 1) e ШГ cP1. Then,
[у е ШГ | d№(x,y) = r] = {y e E"·1 | (y \ y) = -1, coshr - -{y | *>} =
{Oi, - - - , y„+i) e E"'1 | %+i = cosh r, Σ"=ι yf = sinh2 /·}. This is a sphere of
radius sinh r in the affine subspace Ε" χ {cosh г] с Ε"·', and the projection of this
subspace onto its first factor gives a (Riemannian) isometry to E". G
6.16 The Exponential Map. We have constructed a Riemannian metric on M",
which gives a norm on each tangent space TXM". Each non-zero vector υ e TXM"
can be written uniquely as ru, where r is a positive number and и is a vector of
norm one. We call (r, u) the polar coordinates of v. Given a point χ e Г*М", the
exponential map at χ is the map exp* from TXM" to M" that sends the origin to χ
and sends the vector with polar coordinates (r, u) to the endpoint of the geodesic
segment issuing from χ with initial vector и and length r (as defined in Chapter 2).
If к < 0 then the domain of exp* is the whole of TXM"; if к > 0 then ехрл is only
defined on the open ball of radius DK = π/^/κ.
More explicitly, for the case к = — 1, if one identifies unit vectors in TXW to
unit vectors in E'!'', as described above, then
expx(ru) = (cosh r)x + (sinh r)u.
Similarly, for the case к = 1, if one identifies unit vectors in TxSn to unit vectors in
E"+', as described above, then
expx(ru) = (cos f)x + (sin r)u.
In (2.11) we proved that for each pair of points x,y e M" (assuming that d(x,y) <
DK), there is a unique geodesic joining χ to у in M". Hence expx is a bijection —
onto M" if к < 0, and onto B(x, π/^/κ) if к > 0 — and the explicit expression
above shows that exp* is a diffeomorphism. In particular, one can use expx to pull the
Riemannian metric on M" back to TXM", where the availability of polar coordinates
facilitates an elegant and instructive description of it, (6.17(2)). This process is rather
like using planar charts to describe the surface of the Earth.
Notation. The metric arising from the given scalar product on TXM" makes it isometric
to Euclidean space, and hence there is an associated Riemannian metric on TXM".
Let du2 denote the induced Riemannian metric on the unit sphere in TXM".

Μ" as a Riemannian Manifold 95
6.17 Proposition.
(1) For every к е K, ί/ге metric on MnK that was defined in (2.10) is equal to the
length metric associated the Riemannian metric given above.
(2) For every χ e MnK, the Riemannian metric on TxMnK that is obtained by pulling
back the given Riemannian metric on MnK via the exponential map expx, is given
in polar coordinates (r, u) by the formula
ds2 =dr1 sinh2(V-K' r) du2 if к <0;
к
ds2 =dr2 Η— sin2(Vif r) du2 if к > 0;
к
ds2 = dr2 + r2 du2 ific=0.
Proof. We first prove (2). The case к = 0 is clear, because the exponential map is an
isometry and the Euclidean Riemannian metric is given in polar coordinates by the
formula ds2 = dr2 + ^du2. The given formulae for к ф0 follow from 6.15(2) and
the fact that the smooth curve г м> expx(ru) is orthogonal to spheres centred at x.
We shall deduce (1) from (2) in the case к = —1; the other cases are entirely
similar. Given x, у е Ш" with d(x, y) = r, there is a unique unit vector u0 e ТхШп
such that у = expx(ruo). Consider a piecewise differentiable curve с : [0, 1] —► И"
joining χ to y. By writing с in the form c(t) = expx(r(t)u(t)) we obtain a formula for
its Riemannian length:
hie) = l л/Ш\2 + (sinhr(t))2\u(t)\2 dt > J \r(t)\ dt>\\ r(t)dt
Jo Jo I ./o
One obtains equality in this expression if and only if u(t) = 0 and r(t) > 0 for all
t e [0, 1]. These conditions are satisfied by the path, t м> expx(truo) (and by no
other, up to reparameterization). Thus the Riemannian distance between χ and у is
r = d(x, y). D
6.18 Proposition. The Riemannian metrics for the Poincare ball and half-space
models of hyperbolic space are:
(1) for the ball model В" с Е",
Ads2
ds1
JE
(1 - Μ2)2
(2) and for the half-space model {x e Ε" | x„ > 0},
ds2=d4,
where \\x\\ is the Euclidean norm of χ and ds\ = 5Z"=1 dxj is the Euclidean
Riemannian metric.

96 Chapter 1.6 More on the Geometry of M"
Proof. The formula for the ball model can be calculated directly as the pull-back
of the Riemannian metric on W by the isometry hp : B" —> W с Ε"·' defined
prior to (6.8). Alternatively, one can verify that the isometries of the Poincare ball
preserve the given Riemannian metric; the action is transitive, so in order to prove the
proposition it then suffices to calculate the Riemannian length of curves emanating
from О € В"; this can be done using polar coordinates on Bn, as in the preceding
proposition.
The Riemannian metric on the half-space model H" can be deduced from that on
Bn by using Cayley transform B" -> H". D
6.79 Exercises
(1) Prove that the action of Isom(M") on the Riemannian manifold MnK induces
a simply transitive action on the associated bundle of «-frames, i.e. the space of
(n + l)-tuples (x, u\,... , u„), where χ e M" and u\,... ,u„ e TXM" are mutually
orthogonal unit vectors, (cf. 2.15.)
(2) Using the Poincare ball model, construct in the hyperbolic plane for each
η > 4 a regular n-gon with vertex right angles, and more generally regular rc-gons
with vertex angles equal to a < (η — 2)π/η.
Given numbers alt... ,a„ e (0,π) with 5Z"=1 oct < (n — 2)π, construct in H2
и-gons with vertex angles αϊ, ..., a„.
(3) The map {x\, ..., xn) м> (χι,... ,xn-u 1) reduces thelength of any rectifiable
path that is contained in the subspace of the upper half-space model consisting of
points with x„ < 1.
(4) Fix I > 0. Let S С Ε2 and S' С И2 be circles of length I with centres с and d
respectively. Letp, q e Sdsiap', q' e У be points such that Zc(p, q) = Zc>(p',q') > 0
and suppose that the tangent lines to S at ρ and q (resp. to S' at// and q') intersect at
r (resp. d). Show that dw(d,p') > dE(r,p) and d^(pf, q') < rfE(p, q).
(Hint: Consider the upper half-space model of hyperbolic space. When endowed
with the induced path metric, the horosphere x-$ — 1 is isometric to E2; regard S as
lying on this horosphere. Identify H2 with the unique 2-plane in И3 that contains S and
take ρ = ρ' and q = q'; note that the hyperbolic segment [p, d] is the intersection of
this 2-plane with the vertical 2-plane containing the Euclidean segment {p, r]. Argue
that r is the image of d under the projection map described in (3).)

Chapter 1.7 M^-Polyhedral Complexes
The simplest examples of geodesic metric spaces that are not manifolds are provided
by metric graphs, which we introduced in (1.9). In this section we shall study their
higher dimensional analogues, M^-polyhedral complexes. Roughly speaking, an
M^-polyhedral complex is a space that one gets by taking the disjoint union of a
family of convex polyhedra from MnK and gluing them along isometric faces (see
(7.37)). The complex is endowed with the quotient metric (5.19). The main result in
this chapter is the following theorem from Bridson's thesis [Bri91] (see (7.19)).
Theorem. If an MK-polyhedral complex has only finitely many isometry types of
cells, then it is a complete geodesic metric space.
Metric Simplicial Complexes
We begin with an informal description of an M^-simplicial complex. There are
technical reasons why simplicial complexes are easier to work with than general polyhedral
complexes, so since any polyhedral complex can be made simplicial by subdivision
(7.49), it makes sense to begin with the simplicial case.
There are two equivalent ways to view a metric simplicial complex. The first is
to imagine building the complex from a disjoint union of geodesic simplices in MK
by gluing simplices together along isometric faces. (One restricts the gluing so that
each simplex injects into the quotient.) The quotient pseudometric (7.4) is called
the intrinsic pseudometric of the complex. This is the approach to MK -simplicial
complexes that we shall emphasize. An alternative approach is described in the
appendix to this chapter. This second approach is more appropriate in situations
where one wishes to use geometry to study a problem that has been encoded in the
combinatorics of an abstract simplicial complex.
Whichever approach one takes, the issues that one faces are the same. One hopes
to obtain a complete geodesic metric space, but as we have already seen in the 1-
dimensional case (1.9), in order to do so one must add additional hypotheses. Under
very mild hypotheses one can prove that the intrinsic pseudometric is a metric and
the complex is a length space (7.10). Under the same hypotheses we prove that each
point in the complex has a neighbourhood that is isometric to a neighbourhood of
the vertex in the к -cone over the space of directions at that point (7.16).

98 Chapter 1.7 MK-Polyhedral Complexes
If the complex under consideration is locally compact, then one can deduce the
existence of geodesies under mild hypotheses using the Hopf-Rinow Theorem (3.7).
However we do not wish to assume that the complexes under consideration are locally
compact. (This is particularly important for the applications to complexes of groups
in Part III.) Instead, following [Bri91], we consider complexes that have only finitely
many isometry types ofsimplices. (Note that whenever one has a cocompact group
action on a complex, a natural thing to do is to metrize the simplices of the quotient
and extend equivariantly; this gives a complex with only finitely many isometry types
of cells.)
7.1 Geodesic Simplices in MnK. Let η < m be positive integers. An η-plane10 in M™
is, by definition, a subspace isometric to MnK. We say that (n + 1) points in M™ are in
general position if they are not contained in any (n — l)-plane.
A geodesic n-simplex S С Μ™ is the convex hull of (n + 1) points in general
position; the points are called the vertices of S. If к > 0, then the set of vertices of S
is required to lie in an open hemisphere, i.e. in an open ball of radius DK/2.
A face Τ С S is the convex hull of a non-empty subset of the vertices of S; if
Τ φ S then Τ is called a proper face. Note that a face of a geodesic и-simplex is a
geodesic и'-simplex for some n' < n. The interior of S is the set of points that do not
lie in any proper face.
7.2 Definition of an M^-Simplicial Complex. Let (5λ : λ e Λ) be a family of
geodesic simplices Sx С Μ"'-. Let X = UXeA(S\ χ {λ}) denote their disjoint union,
let ~ be an equivalence relation on X and let К = X/~. Let ρ : X —> КЪе the natural
projection and define ρλ : Sx —»■ К by ρχ(χ) :=■ p(x, λ).
К is called an MK-simplicial complex if
(1) for every λ e Λ, the map ρχ is injective, and
(2) if px(Sx) Dpx'(Sx') φ 0 then there is an isometry hXtx· from a face Τχ С Sx onto
a face Τχ С Sx· such thdXp{x, λ) = p{x!, λ') if and only if χ1 = 1ΐχχ{χ).
The set of isometry classes of the faces of the geodesic simplices Sk is denoted
Shapes^).
Terminology. Mq (resp. M\ , M_i) simplicial complexes are often called piecewise
Euclidean (resp. spherical, hyperbolic) complexes. M^-simplicial complexes are also
called metric simplicial complexes, or simplicial complexes of piecewise-constant
curvature.
7.3 Definitions (Simplices, Stars and Segments). Let К be as in (7.2). A subset
S С К is called an m-simplex if it is the image under some ρχ of an m-dimensional
face of Sx. If Γ is a subset of S and p~[' (Γ) is a face of p~[' (5), then Τ is called a. face
of S. The interior of S is the image under ρχ of the interior of p^1 (S). The support of
a point χ e К is the unique simplex supp(jc) that contains χ in its interior.
10 Any such subspace is an intersection of hyperplanes.

Metric Simplicial Complexes 99
We define a metric ds on S by ds(px(x),px(y)) = dSx(x,y). Condition 7.2(2)
ensures that this is independent of λ. Observe that if Γ is a face of S then dj is the
restriction of ds. A bijection K\ -> Κχ between MK-simplicial complexes is called
a simplicial isometry1' if it maps each simplex of K\ isometrically onto a simplex of
K2.
Let* e K. The (closed) star oix, denoted St(x), is the union of the simplices of
К that contain x. The open star oix, denoted st(x), is the union of the interiors of the
simplices of К that contain x.
The image under/?λ of a geodesic segment [χχ, ух] С 7\ с 5λ is called a segment
in the simplex Τ = ρχ(Τχ) and is denoted [x, y], wherex = Ρχ(χχ) and у = px(yx). The
length of [x, y] is defined to be dT(x, y). (Condition 7.2(2) ensures that the notation
[x, y] and the definition of length are unambiguous.)
К comes equipped with the quotient pseudometric described in (5.19) — the
definition was in terms of chains. In the present setting, we can use the extra structure
inherent in the definition of К to restrict the sort of chains that have to be considered.
The following combinatorial device is useful in this regard.
7.4 Definition (m-Strings and the Intrinsic Pseudometric). An m—string in К from χ
toy is a. sequence Σ = (xq, x\ ,..., xm) of points of К such that χ = xq, у = xm, and
for each i = 0,..., m — 1, there exists a simplex S(i) containing x, and jc,+ i . We call
m the size of Σ, and define the length of Σ to be
/=o
Every m-string determines a path in K, given by the concatenation of the segments
[xi,Xi+i]. We denote this path Ρ(Σ). When the integer m is not important, we may
refer to Σ simply as a string.
The intrinsic pseudometric on К is defined by:
d(x, y) := ϊηί{Ζ(Σ) | Σ a string from χ to у }.
If there is no string from χ to y, then d(x, y) := oo.
We now have two pseudometrics on K, the quotient pseudometric (defined in
terms of m-chains (5.19)), and the intrinsic pseudometric (defined in terms of m-
strings).
7.5 Lemma. For any MK-simplicial complex K, the intrinsic pseudometric and the
quotient pseudometric coincide. More precisely, there is an m—chain joining χ to у
in К if and only if there is an m-string of the same length joining χ to y.
1' sometimes called an isometric isomorphism

100 Chapter 1.7 MK-Polyhedral Complexes
Proof. We have ρ : JJ Sx -> K. Every m-chain С = (χχ, у χ,..., xm, ym) projects
to an m-string of the same length {p(x\),p(xi), ■. ■ ,р(хт),р(ут))- Conversely,
to each m-string Σ = (χ0,χχ,... ,xm) we can associate an m-chain Cs =
(jci, у\,...,xm, ym) such that for each i there exists λ e Л with jc, = (x^, λ), % —
(y[-, λ), and pxi^y'il = [xi,Xi+i]. It is clear that Cs projects to Σ and that
Ζ(Σ) = l(Cs). D
Until further notice, К will denote a fixed MK~simplicial complex
and d will denote the intrinsic pseudometric on K.
7.6 Remark. If S is a simplex in К then ds(x, y) > d(x, y) for all x, у е S, but one
does not get equality in general. For instance, consider the 1-complex that has three
vertices and three 1-simplices of lengths 1, 1 and 3 connecting the vertices pairwise.
If S is the simplex of length 3, then the restriction of d to S does not coincide with
ds.
If Shapes(iT) is finite, then one can always subdivide so as to arrange that d = ds
for all simplices S.
The following simple example shows that the intrinsic pseudometric on К will
not be a metric in general.
7.7 Example. Consider the metric graph that has two vertices and countably many
edges e„ connecting these vertices, where e„ has length \/n. Subdivide the edges
to make it simplicial. The distance between the two original vertices in the intrinsic
pseudometric is zero.
The pathology in this example derives from the fact that there are points at which
the following quantity is zero.
7.8 Definition. Let χ e К and let
ε(χ) := inf {ε(χ, S) \ S С К a simplex containing x},
where ε (χ, S) := inf {ds(x, T) \ Τ a face of S and χ φ Τ}. (If S = {χ} then we define
ε(χ, S) to be oo.)
7.9 Lemma Fix χ e K. If у е К is such that d(x, y) < ε(χ), then any simplex S
which contains у also contains x, and d(x, y) = d$(x, y).
Proof. It is enough to show that if Σ = (jco, ..., xm) is an m-string of length /(Σ) <
ε(χ) from χ = x0 to у = xm, with m > 2, then Σ' = (jc0, jc2, ..., xm) is an (m - 1)-
string with Ζ(Σ') < Ζ(Σ).
From the definition of an m-string, we know that there is a simplex S( 1) containing
both xx and X2, and since Ζ(Σ) < ε(χ), the point x0 = χ also belongs to 5(1). As

Metric Simplicial Complexes 101
^ί(ΐ)(*ο> xi) ^ 6?ί(ΐ)(*ο> χύ + ds(i)(*i> *г)> We deduce that Σ' is an (m — l)-string of
length < Ζ(Σ). D
7.10 Corollary. If ε(χ) > 0 for every χ e K, then the intrinsic pseudometric is a
metric and (K, d) is a length space.
Proof. The lemma shows that d(x, y) > 0 if χ φ у. It is then obvious from the
definition that d is a length metric. D
The following simple example shows that we must impose extra hypotheses in
order to ensure that d is a complete length metric.
7.11 Example. Let L be the quotient of the disjoint union (JN([0, 1/2"] x{«})bythe
equivalence relation that identifies the terminal point of the и-th interval to the initial
point of the (n + l)-th interval. Then ε(χ) > 0 for every χ e L, and L is isometric to
[0, 2).
In this example there are infinitely many simplices in a ball of finite radius. If
one has only finitely many simplices in any ball then (K, d) will be a proper length
space, hence complete and geodesic (3.7). (For an alternative proof see Ballmann
[Ba90].)
The following result is the first of a number in which the set of isometry classes
Shapes(iT) plays a prominent role. In order to work with this set we need a concrete
realization of it.
7.12 Definition (Model Simplices). We represent each of the isometry classes in
Shapes(iT) by a fixed geodesic simplex S С М" that is isometric to the simplices
in that class. 5 is called a model simplex. If S is an и-simplex in К then (S, ds) is
isometric to a unique model simplex 5. We associate to S an isometry/5 : S -> S
called the characteristic map of S.
Note thatd(fs(x),fs(y)) < d^(x,y) for allx,y eS (compare with 7.6).
7.13 Theorem. IfShapes(K) is finite, then (K, d) is α complete length space.
Proof. First we prove that d is a length metric. According to (7.9), it suffices to show
that ε(χ) > 0 for every χ e K. Let S e Shapes(if) and let χ be a point in the interior
of S. Define ε(χ) > 0 to be the minimum of the finite set of numbers d(j(y), T), where
Γ is a face of some S e Shapes(iT) and j ranges through the isometries of S onto
those faces of S whose interiors are disjoint from T. It follows immediately from
(7.9) that if χ e К is such thatfs(x) = x, then ε(χ) > ε(χ). In particular ε(χ) > 0.
In order to prove that К is complete, we must show that every Cauchy sequence
(jc„) in К contains a convergent subsequence. Let Sn be a simplex of К containing
xn. Because Shapes(iT) is finite, we can pass to a subsequence such that S„ = So for
every η e N. Letx„ =f<f (x„). By passing to a further subsequence, we may assume
that (xn) converges in S0 to a point x.

102 Chapter 1.7 MK-Polyhedral Complexes
Let Τ — supp(x). Fix εο > 0 such that εο < ε(χ) and d(x,j(x)) > 2εο for every
j e Isom(r) with χ φ j(x). Note that if 5 — ? = S0 then either fs(x) = f?(x) or else
d(fs(x)Js'(x)) > so-
We fix an integer N such that d(xn, x) < εο/З and d(xn, x^) < Sq/3 for all η > N.
Then, d(fSn(x),fsN(x)) < d(fs„(x), x„) + d(x„, xN) + d(xN,fSfl(x))· But d(fSr(x), x„) =
d(fs„(x)ifs„(Xn)) is no greater than d(x,xn) (see the last sentence of (7.12)), and
similarly d(fsN(x),XN) < d(x,xN). Therefore d(fs„(x),fsN(x)) < ε0 if η > Ν, and
hence fSn(x) =fSfi(x). So if we let χ —fSfi(x), then for all η > N we have d(xn, x) —
d(fs„(xn)>fs„(x)) — d(x„,x), which goes to zero as η -> oo. D
In (7.19) we shall see that if Shapes(if) is finite then К is actually a complete
geodesic space. In the appendix (7A.4) we consider a weaker hypothesis which is
enough to ensure that К is a complete length space but not enough to ensure the
existence of geodesies.
Geometric Links and Cone Neighbourhoods
In this section we study the local structure of M^-simplicial complexes. We shall show
that if ε (χ) > 0 (in the notation of (7.8)) then the geometry of К in a neighbourhood of
χ is entirely determined by the infinitesimal geometry at* (see 7.16). The infinitesimal
geometry at χ e К is encoded in the space of directions of geodesies issuing from
that point, which is called the geometric link Lk(x, K).
Let us first reflect on how to describe the neighbourhood of a point in a geodesic
simplex in terms of the tangent space at that point.
7.14 The Link of a Point in a Geodesic Simplex. Let χ be a point of a geodesic
m-simplex S in M", where m > 1. The geometric link Lk(x, S) of χ in S is the set of
unit vectors at χ that point into S, i.e. the set of initial vectors of geodesic segments
joining χ to the points of S. The scalar product on the tangent space TXM" induces
a length metric on the set of unit vectors making it isometric to §""'. Thus we may
identify Lk(jc, S) with a subset of §"~'. If χ is a vertex of S then this subset is a
geodesic simplex; if χ lies in the interior of S then Lk(x, S) is isometric to §""'.
An equivalent way to view Lk(jc, S) is as the set of ~ equivalence classes of
geodesic segments [x, y] in S, where у e S \ {x\ and [x, y\] ~ [x, уг\ if [x, yi] с
[jc, уг]. The distance between the classes of [x, y] and [jc, /] is the angle Z^(y, /).
Motivated by this, in what follows we shall write Zs(u, u') to denote the distance
between points u, u' e Lk(x, S).
If ε > 0 is less that the distance from χ to the union of those faces of S that
do not contain x, then the ε-neighbourhood of χ in S is naturally isometric to the
ε-neighbourhood of the vertex in the /c-cone CK(Lk(x, S)): the isometry sends tu e
CK(Lk(x, 5)) to the point a distance t from χ along the geodesic segment in the
direction u. (See 5.8)

Geometric Links and Cone Neighbourhoods 103
The geometric link of a point in an M^-complex is assembled from the links of
the points in the model simplices by the natural identifications.
7.15 The Link Lk(jc, K) of a Point in K. Fix χ e K. Given y, / e st{x) % {x},
we say that the segments [x, y] and [x, y'] define the same direction at χ if one of
them is contained in the other. (Note that every equivalence class is contained in a
simplex.) The geometric link of χ in К (usually just called the link) is the set Lk(jc, K)
of directions at x. If χ lies in the simplex S, then the link of χ in S, denoted Lk(jc, S)
is the subset of Lk(jc, K) consisting of those directions which are contained in S.
Suppose S = Px(Sx) (notation of (7.2)). The distance (angle) between two
directions и, и' е Lk(jc, S) is defined to be Zsx(mx, u'x), where их, и'х е ΙΜχχ, Sx) are
the directions of the unique geodesic segments in Sx that project onto geodesic
segments in the classes и and u' respectively. Note that ρχ induces a canonical isometry
Lk(jtx, Sx) -> Lk(jc, S) and that if Γ is a face of S then Lk(jc, T) is a subspace of
Lk(jc, S).
We construct a pseudometric on Lk(jc, K) in strict analogy with (7.4). An m-
string in Lk(jc, K) joining и to υ is a sequence Σ = (и0, ·.., um) of points in Lk(jc, K)
such that щ = u,um = v, and for each i there is a simplex S(i) such that щ and щ+\
are contained in Lk(jc, S(i)). The length of Σ is defined to be
ΐφ) := Σ Zs(i)(M,·, ui+i),
and the intrinsic pseudometric on Lk(jc, K) is the function:
d(u, ν) := ΐηί{Ζ(Σ) | Σ a string joining и to υ }.
If there is no string connecting и and υ, then d(u, υ) :— сю.
The intrinsic pseudometric on Lk(jc, K) coincides with the quotient metric
associated to the projection Цл Lk(jtx, 5λ) -> Lk(jc, K) induced hyp : Цл 5λ -> К where,
by definition, Lk(jcx, Sx) = 0 if χ φ p(Sx) and otherwise χ = ρχ(χχ). Note that if
χ e К is a vertex, then Lk(jc, if) is an Mi-simplicial complex whose model simplices
Shapes(Lk(jc, K)) form a subset of {Lk(]c, S) \ χ a vertex of 5 e Shapes(if)}.
7.16 Theorem. Let К be an MK-simplicial complex, and let χ e K. If the number
ε(χ) defined in (7.8) is strictly positive, then B(x, ε(χ)/2) is naturally isometric to the
open ball of radius ε(χ)/2 about the cone point in СДЬк^, К)).
This is the first of a number of results in which we shall need the following device
for examining m-strings.
7.17 Definition (The Development of an m-String). We fix a base point χ in M^, a
base ray с : [0, DK] -> Щ with c(0) = x, and an orientation of M\. (The point of
the orientation is that it provides a uniform way of ordering pairs of directions that
are not antipodal.)

104 Chapter 1.7 MK-Polyhedral Complexes
Let χ be a point of К and let Σ = (jc0, ..., xm) be an m-string in st{x) that avoids
χ (in the sense that χ does lie on any of the segments [*,-, jc;+i])· Let S(i) be a simplex
containing [xt, x-l+{\ and jc. Let Σ = (Ιο, · · ·, 3cm) be the sequence of points in M2K
such that Xq lies in the image of c, and for each i we have d(x, xi) — ds^(x, xd and
rf(jc,-_i,3c;) = dSft(xi-.i,x;), and if 3c,3c;-,3c;+i are not aligned, then the initial vectors
[3c, 3c,] and [3c, 3c,+ i] occur in the order of the given orientation of M^. The sequence
Σ is uniquely determined by these conditions and is called the development of the
m-string Σ in M2.
An important point to note is that the sum of the distances d(3cj_i, 3c,) is equal to
/(Σ).
Fig. 7.1 The development of Σ
The Proof of Theorem 7.16. We identify B{x, ε{χ)) with the ball of radius ε(χ) about
the cone point in CK (Lk(jc, K)) by writing tu to denote the point a distance t along a
segment issuing from jc in the direction и e Lk(jc, K). Let dc denote the pseudometric
on CK (Lk(jc, K)) (now transferred to B(x, ε(χ))). What we must prove is that for any
y, y' € B(x, ε(χ)/2) we have d(y, y1) — dc(y, y'). (Lemma 7.9 ensures that the above
identification is well-defined, i.e. if d(x, y) < ε(χ) then there is a unique segment
joining χ to y.)
Let у — tu and / = fV. Lemma 7.9 tells us that t — d(x, y) — dc(x, y) and
therefore we restrict attention to the case t, f > 0. Note that d(y, /) < t + f and
dc(y, y')<t+ f.
Claim 1: Ifd(u, u') < π then d(y, /) < dc(y, /) < t + f.
By definition, d(u, u') is the supremum of the lengths of the strings in Lk(jc, K)
connecting и to u'. Thus there exists an m-string Σ = (ио, ..., um) with и = и0 and
u' — um such that d(u, u') < /(Σ) < π. Let S(i) be a simplex such that и,, m,-+i €
Lk(jc, 5©)^
From Σ we shall construct a corresponding m-string Σ joining у to У by first
constructing its development. Tc this end, we fix a basepoint Зс е M% and choose
3c^,3c^ e Ml such that d(x,xf) = t,d(x,x%) = f7 and zf (3c|,lJ) = Ζ(Σ).
Then we choose points 3ti,... ,3cm_i on the geodesic segment Γ^,χ^] such that

The Existence of Geodesies 105
Z-4*/.*i+i) = dhkix<s0))(ui, Uj+ι). Let i,- = d(x,xf) and let jc, = р,щ. Note that
[x-„ χ-,+ι] С 5© and dS([)(xh xi+i) — d(xh x1+\).
The m-string Σ = (xq, ..., xm) joins у = jco to y' = xm in 5(jc, e(jc)), and
therefore d(y,y') < Ζ(Σ) = Σ/4(*/.*ί+ι) = ^0Σ,ϊ^) < t + f. But ^(у,У) =
infg d(x^,xm). Thus Claim 1 is proved.
If d(y, /) = t + f, then d(u, u') > π by Claim 1, and therefore <fc(y, y') — t + f.
In the light of this observation, the theorem follows from Claim 1 and:
Claim 2: Ifd(y, y') < t + f7, ί/геи й?(и, и') < π and dc(y, у') < d(y, у').
Let Σ = (xq, ..., xm) be an m-string joining у = xq to y' = xm in B(x, ε), and
suppose Ζ(Σ) < t + f. The triangle inequality tells us that χ is not contained in the
path determined by Σ. Thus jc, can be written uniquely as χι — t\Ui, and we have an
m-string Σ = (и0, ..., um) joining и to и' in Lk(jc, K).
Let Σ — (xq, ... ,xm) be the development of Σ in M^. If Ζ(Σ) were greater
than π, then for some i with 0 < i < m there would exist I- e \χι,χι+\\ such
that Z^\x0,x\) = тг; let i< = rf(3c,Зф. Then, for Σ' := (x0,...,xh%) and Σ" :=
(x„ xi+i,..., xm) we would have Ζ(Σ) = Ζ(Σ') + Ζ(Σ") > (ί0 + φ + (tm - φ = t + f,
which is a contradiction. Thus π > Ζ(Σ) = ZiK)(xo> *m) > й?(и0, ит) = d(u, и').
Since ύ?(χ0,3cm) and dc(y, y') are obtained by applying the law of cosines to the MK-
triangles with vertex angles Z- (xq, xm) and d(u, u') respectively (see 5.6), we deduce
that Ζ(Σ) > d(x0, xm) > dc(y,Xy'). Therefore d(y, y') = inf/(Σ) > dc(y, y'). D
The following map was implicit in the arguments of the preceding proof.
7.18 Definition (Radial Projection). The map St(x) \ {jc} -> Lk(jc, K) that associates
to each point у the direction of [jc, y] is called radial projection from jc. Thus the m-
string Σ considered in the preceding proof was the radial projection of the m-string
Σ. If jc is a vertex then this map takes open simplices to open simplices and sends
segments in ii(jc) to segments in Lk(jc, K).
The Existence of Geodesies
In this section we shall prove the theorem of Bridson [Bri91] mentioned in the
introduction.
7.19 Theorem. Let К be a connected ΜK-simplicial complex. 7/rShapes(iT) is finite,
then К is a complete geodesic space.
The outline of the proof of this theorem is as follows. We shall define a useful
subset of the set of m-strings called taut strings. This class is large enough to ensure
that the infimum in the definition of the intrinsic pseudometric on К can be taken
over taut strings rather than all strings (7.24). On the other hand, taut strings are
rather efficient in the sense that their size is bounded above by a linear function of

106 Chapter 1.7 MK-Polyhedral Complexes
their length (7.30); this will be proved by an induction on the dimension of K. Thus
the proof of Theorem 7.19 is reduced to showing that if one fixes m then there is
a shortest m-string among all those with specified endpoints. This is proved by a
simple "quasi-compactness" argument (7.27).
The proof of Theorem 7.19 will also provide us with a characterization of the
geodesies in M^-simplicial complexes that will be useful in Chapter II. 11 when we
come to construct polyhedral complexes using the gluing techniques introduced in
(5.19).
Taut Strings
Before defining what it means for an m-string to be taut, we must make some
observations about small subcomplexes of K. Suppose that S and S' are closed simplices in
К which have non-empty intersection, and consider L — SUS' equipped with its
intrinsic metric, i.e. the metric such that d(x, y) = ds(x, y) if jc, у € S, d(x, y) = d$> (x, y)
if x, у e S', and d(x, y) — infzeSns'№(*, z) + ds>(z, y)) if χ e S and у е S'. Then
L is a geodesic metric space, and the minimal m-string associated to any geodesic
segment has size at most 2 (cf. 5.24).
7.20 Definition of a Taut String. An m-string Σ = (x0, x\,... , xm) in К is taut if
it satisfies the following two conditions for i = 1, ... , m — 1 :
(1) there is no simplex containing {х,~.\, jc,-, jc,+i};
(2) if x-,-.\,Xi e S(i) and x-„Xi+\ e S(i + 1) then the concatenation of the line
segments [jc;_i, jc,] and [jc,-, jc,+i] is a geodesic segment in L = S(i) U S(i + 1).
Notice that if a string is taut then only its first and last entries can lie in the interior
of a top dimensional simplex of K.
7.21 Lemma. For a fixed integer m, if Σ is an m-string from χ to у in K, and if Σ is
of minimal length among all m-strings joining these points, then Ρ(Σ) (the path in
К determined by Σ) is the path determined by some taut n—string with η < m.
Proof. Let Σ = (χο,χι, ... ,xm). Suppose that for some i there exists a simplex
S containing x-,~i, x, and jc,+i, and let Σ' denote the (m — l)-string obtained from
Σ by deleting the entry x-t. The triangle inequality for d$ gives ds(xi~i,Xi+i) <
ds(xi~i,Xi) + ds(xi,Xi+\), with equality if and only if jc, lies on the line segment
[jci_bjci+i]. Thus Ζ(Σ') < /(Σ). But /(Σ) is minimal, so in fact Ζ(Σ') = /(Σ), and
hence jc, must lie on the line segment [jc,_ ι, x-l+\ ]. This implies that Σ' determines the
same path as Σ. We can repeat this procedure until no simplex of К contains three
successive entries of the resulting string — the first condition for tautness.
We now show that Σ satisfies the second condition for tautness. Let S and S' be any
simplices containing {x,_i, jc,} and {jc,, x,+i} respectively. Every geodesic segment in
the complex L — S U S' can be expressed as the concatenation of one or two line
segments. So if the concatenation of the line segments [x;-\,xi\ and [jc,, jc,+i] were

The Existence of Geodesies 107
not a geodesic segment in L, then we could replace jc,- by some xj e S Π S' to obtain
an m-string from jco to xm which would be strictly shorter than Σ, contradicting the
minimality of /(Σ). D
7.22 Example. Figure 7.2 illustrates the fact that a taut string is not necessarily a
local geodesic. Неге К is a planar 2-complex with three 2-simplices, and the string
(a, b, c) is taut but not a local geodesic.
Fig. 7.2 A taut string which is not a local geodesic
7.23 Lemma. Let χ be a vertex of К and let Σ = (jco, ..., xm) be a taut string in
st{x), the open star ofx. Suppose thatxi φ χ for each i — 0, ..., m. If к > 0, assume
in addition that for each i, ds(i)(x,xt) < DK/2, where S(i) is a simplex containing
[*,·,*,·+1].
Then, the image Σο/Σ under radial projection (as defined in (7.18)) is a taut
string in Lk(jc, K) of length < π.
Proof. The radial projection of Σ into Lk(jc, K) is Σ = (и0, · ■ ■, u,„), where щ is the
direction determined by the segment [jc, x,]. Let Σ = (xo, ■ · ■ , 3cm) be the
development of Σ in Ml with respect to the base point χ e M\ (see 7.17); if к > 0, we
have d(x,Xi) < DK/2. We claim that the concatenation of the geodesic segments
[3co,ϊι],..., \xm-\,Xm\ is the geodesic segment [x0,xm]. As Ζ(Σ) = Z^(x0,xm), it
will follow that Ζ(Σ) < π.
Since every local geodesic in B(x, DK/2) is a geodesic, it is enough to show that
the concatenation of [x/_i, 3c,·] and [3c,, 3c,+i] is pc,-_ j, 3c/+)]. If this were not the case,
then arbitrarily close to 3c; on the ray issuing from χ and passing through 3c,- there
would be apointjej such that ύ?(χ,_ι, 3c-) + d(3tj-,3c;+i) < й?(Зс,_1,Зс,) + u?(3c,, 3c/+i). Let
S(i) and S(i + 1) be simplices containing [jc,_i , jc,] and [jc,, jc,+i] respectively. Let y,
be a point of St(x) \ st(x) such that jc, e [jc, y,·]; note that y,- e S(i) П S(i + 1). As
Xj is in the open star of jc, we have jc, φ у,-, and if 3fJ- is chosen close enough to 3c;,
there is a point x!; e [jc, y;] such that d{x, x!;) = d(x, 3cJ). As jcJ e S(i) Π S(i + 1) and
d(jc;_i,xj) = u?(3c,_ι,Зф and d(xi,xfj+l) = d(xj,x'j+l), we would have d(xj_ι, jc,) +
й?(Зс„ jc/+i) > d(xj_i, jcJ) + ^(jc,, jcJ+1), contrary to the hypothesis that Σ is taut.
It remains to show that Σ is taut. The first condition for tautness is obvious. To
establish the second condition, for each i we consider the map from S(i) U S(i + 1)

108 Chapter 1.7 MK -Polyhedral Complexes
(with its intrinsic metric) to CK (Lk(jc, S(i) U S(i + 1))), as in (7.16). The argument of
the previous paragraph shows that the image under this map of the concatenation of
[jc,_ι, χ;] and [xj, xi+i] is a geodesic in CK (Lk(jc, S(i)US(i +1))). Therefore, by (5.10),
the concatenation of [m,-_i, и,·] and [и,·, И/+1] is a geodesic in pS(i) U pS{i +1). Π
The Main Argument
In this subsection we present the proof of Theorem 7.19, as outlined above. In the
first step (7.24), we prove that the distance between two points is the infimum of the
length of taut strings joining them. In the second step (7.28), we prove that for every
I > 0, there is an integer N depending only on the finite set Shapes(X) such that the
size of taut strings of length < I is bounded by N.
7.24 Proposition. Let К be an MK-simplicial complex with Shapes(K) finite. Then
for any two points x,y e K, we have
d(x, y) = inf{/(E): Σ a taut string fromxto y].
The following lemma is taken from the thesis of Gabor Moussong [Mou88] who
used m-strings to prove the existence of geodesies in locally compact complexes.
7.25 Lemma. IfL is a finite MK-simplicial complex, and if the points χ and у can
be joined by an m-string in L, where m is a fixed integer, then there is a shortest
m-string from χ toy in L
Proof. Let X С Lm+l denote the set of m-strings from χ to у in L. We show that X
is closed and hence compact. The length function I on m-strings is continuous on X,
and therefore attains a minimum if X is closed.
Notice that for any w, ζ € L, the set st(w)D st(z) is empty if and only if there is no
closed simplex of L containing both w and z. Thus, if ζ = (го, Z\, ■ ■ ■ , гт) is not an
m-string, then for some i between 0 and m — 1 we have st(zd Π st(Zi+i) = 0. Hence ζ
has a neighbourhood disjoint from X, namely (L x... Lxst(Zi)xst(Zi+i)xLx.. .xL).
Thus X is closed. D
7.26 Model Complexes (Quasi-Compactness). Given К with Shapes(if) finite, and a
positive integer m, one can build a finite set of model subcomplexes, that is, connected
complexes obtained by taking at most m (not necessarily distinct) simplices from
Shapes(if) and identifying faces by isometries. Any subcomplex K0 of К which can
be expressed as the union of at most m closed simplices must be simplicially isometric
to one of these models, i.e. there is a bijection from Ко to the model which respects
the simplicial structure and restricts to an isometry on each simplex S equipped with
its intrinsic metric d$. The existence of this finite set of models allows us to pass

The Main Argument 109
from the case of finite complexes to the case of interest, complexes with Shapes(iT)
finite.
Notice that Ко with the induced metric from К is not in general isometric to the
model with its intrinsic metric. However, since the length of a string is defined in
terms of the local metrics ds, a given m-string in Kq and the corresponding m-string
in the model have the same length. This is the key to the following lemma.
7.27 Lemma. If К is an MK-simplicial complex with Shapes(K) finite, and two points
χ and у can be joined by an m—string in K, where m is a fixed integer, then there is a
shortest m-string from χ toy in K.
Proof. For any fixed pair of elements χ and у there are only finitely many bipointed
models, (K1; x1, /), for (Kq; x, y) as Kq runs over all subcomplexes of К which contain
both χ and у and can be expressed as the union of at most m closed simplices. Thus,
any m-string from χ to у in К corresponds to an m-string of the same length from
x1 to / in one of the finitely many models under consideration, and vice versa. The
present lemma now follows by application of (7.25) to each of the models. Π
Proof of 7.24. Apply (7.21) and (7.27). D
We are now in a position to prove the required lower bound on the length of taut
m-strings. The idea is to localize the problem to the star of a single vertex and then
use (7.23) to reduce to a lower-dimensional complex. The proof which we present
here is a refinement of the original proof [Bri91], and appears in some unpublished
seminar notes of R. L. Bishop.
7.28 Theorem. Let К be an MK-simplicial complex with Shapes(if) finite. Then, for
every I > 0 there exists an integer N > 0, depending only on Shapes(iT), such that
for every taut m-string in К of length at most I we have m < N.
Proof. We proceed by induction on the dimension of K. In order to clarify the
induction process, we articulate an intermediate conclusion and use the following
scheme of proof:
Ф„ : The statement of the theorem is true for complexes of dimension < n.
Ф„ : Suppose that К is of dimension n. Then there exists a constant R, depending
only on the finite set Shapes(iT), with the following property: if χ is a vertex of
К and Σ = (xq, ..., xm) is a taut m-string contained in st(x), and if [x, jc,] has
length less than DK /2 for each i, then m < R.
Assertions Φι and Φι are trivial. We shall prove: Φ„_ι => Φ„ => Φ„ for all
η > 2.

110 Chapter 1.7 MK-Polyhedral Complexes
Фи-1 => Φ» :
Let Κ, Σ and χ be as in statement Φ„.
If the path determined by Σ were to pass through x, then χ would have to be
an entry in Σ. The first condition for tautness would then imply that Σ has size
at most 2. Thus we may take R > 3 and restrict our attention to the case where
Σ = (xq, x\ , ... , xm) with m > 2 and the path Ρ(Σ) determined by Σ does not pass
through x. We can then radially project Σ into Lk(x, K), as in (7.18).
According to (7.23), the image of Σ under this projection is a taut m-string Σ in
Lk(x, K) of length smaller than π ■
Now, by applying condition Φ„_ι to the spherical complex Lk(x, K), we obtain
a constant N(x), depending only on Shapes(Lk(x, K)), such that any taut string in
Lk(x, K) of length at most π has size at most N(x).
There are only finitely many possibilities for Shapes(Lk(x, K)) as χ ranges over
the vertices of K, and the set of these possibilities depends only on Shapes(if). So
setting R equal to the minimum of the constants N(x) as χ ranges over the set of
vertices of К finishes the proof of the present implication.
Ф„ => Ф„ ■■
If Ф„ were false, then in some и-dimensional metric simplicial complex К with
only finitely many isometry types of simplices there would be taut chains of bounded
length and arbitrarily large size, and hence taut strings of arbitrarily small length and
arbitrarily large size. But because Shapes(if) is finite, there is a constant η such
that any set of diameter at most η is contained in the open star of some vertex
χ e К intersected with B(x, DK/2) (one simply chooses η small enough so that each
S e Shapes(if) is the union of the subsets S \ Vn(S χ st(x)), where χ runs over the
vertices of S and Vn denotes the ??-neighbourhood). Thus we obtain a contradiction
to*„. □
The Proof of Theorem 7.19. Given x, у е К, we can apply the preceding result to
obtain abound on the size of taut strings joining χ to y. Then, by (7.28), one need only
take the infimum in the definition of d(x, y) over m-strings with m fixed. According
to (7.26) there is a shortest such m-string joining χ to y. □
We amplify two aspects of our proof of the existence of geodesies.
7.29 Corollary. If К is an MK-simplicial complex with Shapes(K) finite, then every
local geodesic of finite length in К is the concatenation ofafinite number of segments
(each contained in a simplex).
7.30 Corollary. If К is an MK-simplicial complex with Shapes(K) finite, then there
exists a constant a > 0 such that every taut m-string in К has length at least am — 1.
Proof. It is clear from the definition that a substring of a taut string is itself taut. And
according to (7.28) there exists an integer N such that if m > N then the length of every

Cubical Complexes 111
taut m-string in К is at least 1. Let α = 1/Ν and let μ be the integer part of am. Given
a taut m-string Σ = (x0, ... , xm), we can view it as the concatenation of the μ + 1
substrings Σι = Oo,... , xN), Σ2 = (xN,... , xOT), ■ ■ ■ , Σμ+1 = (χμΝ,... , xm).
Each of the first μ of these strings has length at least 1. Hence the length of Σ is at
least μ = [am] > am — 1. D
This technical observation has the following consequence relating the metric
structure of К to its combinatorial structure. The 1-skeleton of K, denoted #(1), is
the graph whose vertices are the vertices of К and whose edges are the 1-simplices
ofK.
7.31 Proposition. Let К be an MK-simplicial complex with Shapes(if) finite. Let
K\ denote the 1-skeleton ofK, considered as a metric graph with all edge lengths
1; let d\ be the associated metric. Then the natural injection j : K\' ^-» К is a
quasi-isometry12.
Proof. Because Shapes(iT) is finite, every point of К is within a bounded distance of
the image of/ And if we set к equal to the length of the longest edge of any simplex
in Shapes(iT), then clearly d(j(u),j(v)) < kd\{u, v) for all vertices u, ν e #(1).
Let the constant a be as in (7.30). We fix a taut m-string Σ = (xq, ... , xm)
determining a path in К that is a geodesic joining the vertices j(u) and j(v). Let
yQ = j(u) and, inductively for i — 1, ... , m — 1, define y, to be a vertex of К that lies
in St(y^ ι) Π St(xi+i). Let ym = j(v). This gives a sequence of vertices connecting j(u)
toj(v), and since each successive pair of these vertices cobounds an edge in if(1\ we
deduce that d\(u, v) < m. But from (7.28) we have d(j(u),j(v)) > am — 1. Hence
adi(u, υ) - 1 < d(j(u),j(v)) < kd{{u, υ). D
Cubical Complexes
We turn our attention briefly to cubical complexes. These are more rigid objects than
M^-simplicial complexes and in many ways they are easier to work with. Moreover,
there exist many interesting examples (see II.5).
The unit и-cube /" is the и-fold product [0, 1]"; it is isometric to a cube in E"
with edges of length one. By convention, 1° is a point.
The faces of the 1-cube [0, 1] are the subsets {0}, {1} and [0, 1], the first two
have dimension 0 and the last has dimension 1. A face of/" is a subset S of/" which
is a product Si χ ■ ■ ■ χ Sn of faces of [0, 1]; the dimension of S is the sum of the
dimensions of the 5,·. (Each ^-dimensional face is isometric to /*.)
We define a cubical complex, by mimicking the definition of an M^-simplicial
complex, using unit cubes instead of geodesic simplices.
12 See 1.8.14 for the definition of quasi-isometry

112 Chapter 1.7 MK-Po]yhedral Complexes
7.32 Definition of a Cubical Complex. A cubical complex К is the quotient of a
disjoint union of cubes X = JJA I"x by an equivalence relation ~. The restrictions
Px : I"k -> К of the natural projection ρ : X -> К = X/~ are required to satisfy:
(1) for every λ e Л the mappx is injective;
(2) if ρλ(Ιηί) Γ)ρχ,(Γ>-') φ 0 then there is an isometry hx_x, from a face Tx С I">
onto a face 7V С 7"λ' such \haipx(x) = px>{x!) if and only if χ1 = /ϊλ,χ'(χ).
7.33 Basic Structure. Let К be as in (7.32). A subset С С К is called an
Tridimensional cube if it is the image under some ρλ of an m-dimensional face of /"A.
The notions of segments, m-strings, and so on, are defined as in the simplicial case, as
is the pseudometric on K. In contrast to the simplicial case, the intrinsic pseudometric
on a cubical complex is always a metric because the number ε(χ) defined in (7.8) is
positive for every χ e K. If К is finite dimensional then d is also complete (a simplified
version of the argument given in (7.9) applies). In fact, if К is finite dimensional,
then it is a complete geodesic space (see (7.43)).
If К is not finite dimensional then it might not be complete. For instance, let Η be
an infinite dimensional Hubert space, fix an orthogonal basis, and consider the cubes
spanned by the finite subsets of the basis elements; these form a cubical complex
that is not complete.
MK -Polyhedral Complexes
We began this chapter by studying simplicial complexes rather than more general
polyhedral complexes, partly because simplicial complexes are easier to work with
and partly because we did not wish to obscure the main ideas by the troublesome
technicalities which must accompany a precise definition of polyhedra in general.
Now we shall consider the general case.
Intuitively speaking, an MK -polyhedral space is a cell complex whose cells are
endowed with local metrics making each isometric to the convex hull of a finite
number of points in M"; these local metrics agree on the intersection of cells. (A
precise definition is given in (7.37).) One can measure the length of paths using these
local metrics, and the intrinsic pseudometric on the complex is obtained by setting the
distance between two points equal to the infimum of the lengths of paths connecting
them. As in the simplicial case, we are interested in the question of whether this
infimum is attained. We also wish to establish that polyhedral complexes have the
sort of local cone structure described in (7.16).
In order to establish the existence of geodesies, we shall describe a canonical
process of subdivision that transforms any MK -polyhedral complex К into an isometric
MK -simplicial complex called the second barycentric subdivision ofK.
Recall that a subset of a geodesic space is said to be convex if every geodesic
segment whose endpoints lie in the subset is entirely contained in the subset. The
convex hull of a subset Ρ of a geodesic space is the intersection of all convex sets
containing P.

MK-Polyhedral Complexes 113
7.34 Convex MK-Polyhedral Cells. Fix к е I. By definition, a convex MK-
polyhedral cell С С Μ" is the convex hull of a finite set of points Ρ С М"; if
к > 0, then Ρ (hence С) is required to lie in an open ball of radius DK/2. The
dimension of С is the dimension of the smallest13 m-plane containing it. The interior of С
is the interior of С as a subset of this m-plane.
Let Я be a hyperplane in M". If С lies in one of the closed half-spaces bounded
ЬуЯ, and if#DC φ 0,thenF = ЯП С is called a/ace of C; if F φ С then it is called
a. proper face. The dimension of a face F is the dimension of the smallest m-plane
containing it. The interior of F is the interior of F in this plane. The O-dimensional
faces of С are called its vertices. The support oix e C, denoted supp(X), is the unique
face containing χ in its interior.
The following discussion is parallel to (7.14) and (7.15). Given χ e C, the
directions at χ pointing into С form a space Lk(jc, C). The distance (angle) between
two directions u, ύ e Lk(jc, C) is denoted Zc(«, «')■ If С has dimension m, then
Lk(jc, C) can be identified with a subset of Sm~'; if χ is an interior point then Lk(x, C)
is the whole of Sm~'; if χ is a vertex of С then Lk(jc, C) is a convex polyhedral cell
in§m-'.
7.35 Exercise. Let С be a convex M^-polyhedral cell of dimension m, let χ e С and
suppose that supp(x) has dimension k > 0. Prove that Lk(x, C) is isometric to the
spherical join (5.13) of S*~' and a convex Mi-polyhedral cell of dimension m — k — 1.
7.36 Proposition. Let С be a convex MK-polyhedral cell. Then:
(1) Each face of С is a convex ΜK—polyhedral cell.
(2) The intersection of any two faces of С is again a face.
(3) С has only finitely many faces.
(4) С is the convex hull of its set of vertices.
(5) If С is the convex hull of a set P, then there is a unique minimal subset of Ρ
with convex hull C, and this is the vertex set of C.
(6) Iff : С -> Ci is an isometry from С to a convex MK-polyhedral cell C\, and if
F is a face ofC, thenf(F) is a face ofC\.
Proof. Suppose that С is the convex hull of the finite set P. Let F be a face of С and
suppose that F = С П Я where Я is a hyperplane. Because the open half-spaces
defined by Я are convex, F = Η Π С is the convex hull of Η Π P. This proves (1).
The content of (2) is that the intersection of two closed half-spaces X\ and X2,
bounded by hyperplanes Hi and Яг say, is contained in a half-space bounded by a
hyperplane that intersects X\ ПХ2 in H\ ПЯг- Part (3) follows immediately from (1),
and (4) and (5) are proved by induction on the dimension of faces.
Part (6) is a consequence of the fact that any isometry between subsets of M"
is the restriction of an element of Isom(M") (see (2.20)). Any such global isometry
13 i.e. the intersection of all hyperplanes containing С

114 Chapter 1.7 MK-Polyhedral Complexes
obviously takes the intersections of С with hyperplanes to intersections of C\ with
hyperplanes. D
For more details, see Eggleston's book on convexity [Eg77]. (Note that because
convex polyhedral cells in M" correspond to convex polyhedral cones in R"+1, one
can convert most questions about such cells into questions concerning the convex
geometry of Euclidean space.) Further references include [Berg77], [Bro88] and
[Grii67].
7.37 Definition of an MK-Polyhedral Complex. Let (Cx : λ e Λ) be a family of
convex Μ*-polyhedral cells and let X = {JXsA(Cx χ {λ}) denote their disjoint union.
Let ~ be an equivalence relation on X and let К = X/~. Let ρ : X -> К be the
natural projection and defineρχ : Cx —> К Ъурх(х) := p(x, λ).
К is called an MK—polyhedral complex if:
(1) for every λ e A, the restriction of ρχ to the interior of each face of Οχ is
injective;
(2) for all λι, "kj. e Λ and x\ e Cxn x% e ϋχ2, if pxt(xO = Рх2(хг) then there
is an isometry h : supp(xi) -> suppfe) such thatpx^iy) = Px2(h(y)) for all
у е suppOi).
The set of isometry classes of the faces of the cells Cx is denoted Shapes(if).
The auxiliary definitions and terminology needed to describe the geometry of
Μ*-polyhedral complexes are essentially the same as in the simplicial case:
7.38 Cells, Stars, Links and the Intrinsic Pseudometric. Let К be as in (7.37). A
subset С С К is called an η-cell if it is the image px(F) of some и-dimensional face
Fd Cx; the interior of С is the image under ρχ of the interior of F.
The intrinsic pseudometric on К is the quotient pseudometric d associated to
the projection ρ : JJX Cx -> K. As in (7.5) one shows that it is equivalent to define
d(x, y) to be the infimum of the lengths of piecewise geodesic paths in К joining χ to
y. a piecewise geodesic path is a map с : [a, b] -> К such that there is subdivision
a = t0 < t\ < · ■ ■ < tk = b and geodesic paths c, : [ί,·_ ι, i,] -> Cx; such that for
each t e [ί,·_ι, t;] we have c(t) = px;(ci(t)); the length 1(c) of с is 1(c) := XJL, 1(a).
(7.37(2) ensures that 1(c) is independent of the choice of subdivision.)
Fix χ € K. The open star of x, denoted st(x), is the union of the interiors of the
cells that contain x. The geometric link Lk(x, K) of χ in К is the space of directions at
χ endowed with the quotient metric associated to the projection JJA Lk(*x, Cx) ->
Lk(jc, K) induced by Цл Cx -> К (compare with (7.15)). As in the simplicial case,
one can give a more explicit description of the metric on Lk(jc, K), except that now one
has to use piecewise geodesic paths in place of m-strings (as with the pseudometric).
As in (7.8), for each χ e К one defines ε(χ) to be the distance from χ to the closure
of st(x) minus st(x), where distance is measured in the local metrics dc(px(x), Pxiy)) =
dCi(x, y) on the cells С С К. By following the proof of (7.16) we obtain:

Barycentric Subdivision 115
7.39 Theorem. Let К be an Μ «-polyhedral complex, and let χ e K. Ifs(x) > 0 then
B(x, ε(χ)/2) is naturally isometric to the open ball of radius ε(χ)/2 about the cone
point in CK (Lk(jc, K)).
7.40 Examples. We use the notation of (7.37).
(1) К is a metric graph if and only if all of its cells have dimension at most 1.
(2) К is an Μκ-simplicial complex if and only if each of the cells Cx is a geodesic
simplex, each of the maps ρχ is injective, and the intersection of any two cells in К
is empty or a single face.
(3) К is a cubical complex if and only if each of the cells Cx is isometric to a
cube Γλ, each of the maps ρ χ is injective, and the intersection of any two cells in К
is empty or a single face.
(4) There are many interesting polyhedral complexes all of whose cells are cubes,
but they do not all satisfy the conditions of (3). We shall use the term cubed complex
to describe this larger class of complexes, except that in the 2-dimensional case we
shall use the term squared complex.
A simple example of a cubed complex that is not a cubical complex is the n-
dimensional torus obtained by forming the quotient of/" by the equivalence relation
that associates to each point in the (n — l)-dimensional faces its orthogonal projection
on the opposite face.
7.41 Exercises
(1) Describe all of the isometrically distinct squared complexes that arise as the
quotient of a single square I1 by an equivalence relation.
(2) Describe all of the isometrically distinct polyhedral complexes that one can
obtain as the quotient of a single equilateral triangle and describe the links of the
vertices in each case.
One of the complexes that you should consider is called the dunce hat, which can
be described as follows: if the vertices of the triangle are xq, xx , x2, then the dunce hat
is the space that you get by identifying [x0, xi ] with [x\, x2] by the isometry mapping
jeo to X2, and by identifying [jco, χι] with [x2, xq] by the isometry mapping x0 to x2.
The dunce hat has one vertex, the link of which has two 0-cells, with a loop based at
each and a third edge connecting them.
Show that the dunce hat is simply-connected. Then prove that it is contractible
(this is harder).
Barycentric Subdivision
We shall now explain how to subdivide an MK -polyhedral complex so as to obtain
an isometric M^-simplicial complex. The approach is rather obvious: one subdivides
each of the cells of K, or equivalently JJA Cx (in the notation of (7.37)), into geodesic

116 Chapter 1.7 MK-Polyhedral Complexes
simplices. However there are some technical difficulties: the process of subdivision
must be sufficiently canonical to ensure that the subdivisions of all cells agree on
faces of intersection, and one has to arrange for the conditions of (7.40(2)) to hold.
7.42 The Barycentre of a Convex Polyhedral Cell. Let С С М" be a convex
polyhedral cell. We shall define a point be in the interior of С that is fixed by all
isometries of C.
First we define a barycentre for finite sets in M". In the case к = 0, we identify E"
with the hyperplane in E"+1 formed by the points ν = (x0,.. .,x„) with Σ"=0 χ,■ — ί.
Given k such points «i,..., vk, we define their barycentre to be \(Σί=ι vj).
In the case к = 1, we identify S" with the unit sphere about 0 e E"+1. We define
the barycentre of a finite set of points {i>i, ..., υ*} contained in an open hemisphere
of s» to be x;t, VII Σ^=ι 4fll-
In the case к = — 1, we identify W with the upper sheet of the hyperboloid in
E"1. The barycentre of a finite subset {i>i,..., υ*} in Ш" is the unique point on the
hyperboloid that lies on the half-line that issues from 0 and passes through Y^=l Ц.
The barycentre of a finite subset V с Μ" (assuming that V is contained in an
open ball of radius DK/2) is defined by rescaling the metric on E'!, S" or Ш".
The barycentre of a convex polyhedral cell С С М" is defined to be the barycentre
of its set of vertices; it is denoted be-
7.43 Lemma. Let С be a convex ΜK -polyhedral cell. The barycentre of С lies in the
interior of С and is fixed by any isometry of С
Proof. It is clear from the definition that be lies in the convex hull of the vertex set
of С and not in the convex hull of any proper subset of it. Thus be is in the interior
of С (by 7.36(5)). The invariance of be under isometries follows from the invariance
of the vertex set (7.36(6)). D
7.44 Barycentric Subdivision of a Convex Cell. Let С С Мк be a convex polyhedral
cell. The, first barycentric subdivision ofC, denoted C", is the M^-simplicial complex
defined as follows. There is one geodesic simplex in C" corresponding to each strictly
ascending sequence of faces Fo С F\ ■ ■ ■ С Fk of C; the simplex is the convex hull
of the barycentres of the F,. Note that the intersection in С of two such simplices
is again such a simplex. The natural map from the disjoint union of these geodesic
simplices to С imposes on С the structure of anM^-simplicial complex — this is C".
Notice that if xx ,x2 € С belong to the same simplex of С and F\ and Fj. are the
minimal faces of С containing these points, then F\ С Fj. or Fj. С F\.
As an immediate consequence of (7.36(6)) we have:
7.45 Lemma. Any isometry С ι -> Сг between convex Μ'κ -polyhedral cells defines
a simplicial isometry C[ —> C'2.
7.46 Lemma. LetK = JJ Cx/~ be an ΜK-polyhedral complex, andletpx : C\ -> К
be the natural projection. Then:

Barycentric Subdivision 117
(1) The restriction ofpx to each simplex of the barycentric subdivision C'k is injec-
tive.
(2) Lei λ], λ2 e Λ, let S\ and S2 be simplices ofCx and C^ and suppose that pxt
andpx, are injective. If{x e Si | pxt (x) e px2(S2)} is not empty, then it is a face
ofSx. '
Proof. (1) If ρχ(x) = px(y), then there is an isometry h : supp(x) -> supp(y) such that
у = h(x). Therefore the faces supp(jc) and supp(y) have the same dimension. If χ and
у are in the same simplex of C'x, then by the last sentence of (7.44), supp(x) = supp(y)
and h is the identity.
(2) It follows from the second condition in the definition of a polyhedral complex
that the set described in (2) is a union of faces. It therefore suffices to show that it is
convex.
Letxuyi e Ckl andx2,y2 e Cx2 be such that/jx,(xi) = px2(x2) and/jXlCy,) =
Рхг(уг)- Then supp(xt) and supp(jC2) (resp. supp(yi) and supp^)) have the same
dimension. According to the last sentence of (7.44), if x\,y\ belong to the same
simplex S\ of the barycentric subdivision of Cx, then we may assume that supp(xt) С
suPPCyi X ш which case supp(jc2) С supp^)- Condition (2) in the definition of a
polyhedral complex yields an isometry h : supp(yi) -> supp(y2) such that h{y\) = y2
and px2(h(x)) = pxt(x) for each χ e supp(yi). As p\2 is injective, we also have
h(x\) = x2, hence/?λ,([^ι,}Ί]) = рьА'ьй!), showing that the set in (2) is convex.
D
7.47 Barycentric Subdivision of MK-Polyhedral Complexes. Let К be an MK-
polyhedral complex. Let/? : JJA Cx -> К be as in (7.37). For each cell Cx, we index
the simplices of the barycentric subdivision C'x by a set /λ; so C'x is the MK -simplicial
complex associated to \\, 5, -> Cx.LetA' = Τ]Λ/λ. By composing the natural maps
Ц/л Si -> Cx and ρ : Цл Cx -*■ К we get a projectionp' : Ц,еЛ, 5, -> К. Let К' be
the quotient of Ц,еЛ/ 5, by the equivalence relation [x ~ у iff p'(x) = p'iy)]-
K' is called the^rsi barycentric subdivision ofK. There is a natural identification
ofsets£-» K'.
7.48 Lemma. With the notation of (7.47):
(1) K' is an MK-polyhedral complex.
(2) The restriction ofp' to each simplex 5, is an injection.
(3) The natural map К -> К' is an isometry.
Proof. For (1) We must check the conditions of (7.37): the first is obviously satisfied
and the second follows from (7.45). Part (2) is a restatement of (7.46(1)), and (3) is
an easy consequence of the definitions. D
In general K' will not be an MK -simplicial complex because although the cells
of K' are geodesic simplices (in their local metrics), the intersection of two cells is in
general a union of faces rather than a single face. To remedy this, we take the second

118 Chapter 1.7 MK-Polyhedral Complexes
barycentric subdivision ofK, which is by definition the first barycentric subdivision
of if.
7.49 Proposition. If К is an Μκ -polyhedral complex, then the second barycentric
subdivision K" of К is an MK-simplicial complex and the natural map К -> К" is
an isometry.
Proof. This follows immediately from (7.46(2)) and (7.48). D
7.50 Theorem. Let К be an MK-polyhedral complex. //Shapes(iT) is finite, then К
is a complete geodesic metric space.
Proof. Shapes(if") is the set of isometry classes of the simplices in the
second barycentric subdivisions of the model cells in Shapes(iT). In particular, since
Shapes(if) is finite, Shapes(if') is finite, and the theorem follows from (7.49) and
(7.19). D
We end this section with an example to indicate how Theorem 7.50 together
with our earlier results on the nature of geodesies can be combined to construct new
examples of complete geodesic spaces.
7.51 Example. Let K\ and Кг be 2-dimensional M0-polyhedral (i.e. piecewise
Euclidean) complexes with Shapes(ifi) and Shapes^) finite. For i = 1,2, let
c, : Sj -> Ki be a closed local geodesic, where 5, is a circle of length 1(a). By
scaling the metrics on K\ and Кг we can arrange that l(c\) = 1(сг) = 2π. Consider
the metric space obtained by gluing a cylinder S1 χ [0, 1] to the disjoint union of K\
and Кг by attaching the ends of the cylinder to the К by the maps c,.
According to (7.29), each a is the concatenation of a finite number of segments
(in the sense of (7.3)). It follows that we may subdivide the cells of K\ and Кг into
smaller polyhedra so that the curves c, run in the 1-skeleton. We can then triangulate
the cylinder S1 χ [0, 1] so that its attaching maps c, map cells isometrically to
cells. The resulting quotient complex K\ U (S1 χ [0, 1]) U Хг/~ is then a Euclidean
polyhedral complex.
We shall see many more specific examples of this construction in Chapter II. 11.
7.52 Exercise. Show that the subdivision in (7.51) is possible even if K\ and Кг are
not 2-dimensional.
More on the Geometry of Geodesies
In this section we gather a number of technical results concerning geodesies in MK-
polyhedral complexes К with Shapes(if) finite. First we note that although we have
shown that for every χ e К there is a unique geodesic joining χ to у whenever

More on the Geometry of Geodesies 119
у e B(x, ε(χ)), it does not follow that К is locally uniquely geodesic. Indeed we shall
now show (7.55) that requiring К to be locally uniquely geodesic is a surprisingly
stringent condition.
7.53 Definition. The injectivity radius of a geodesic space X is the supremum of the
set of non-negative numbers r such that any two points in X a distance < r apart are
joined by a unique geodesic. We denote this number injrad(X).
An easy compactness argument shows that if a compact geodesic space X is
locally uniquely geodesic then injrad{X) > 0. We shall show that M^-polyhedral
complexes with Shapes(if) finite share this property. For this we need the following
lemma.
7.54 Lemma. If К is an MK-simplicial complex with Shapes(X) finite, then there
is a constant εο > 0 such that for every χ e К there exists у е К with B(x, εο) с
B(y, s(y)/4), where s(y) is as defined in (7.8).
Proof. Let K^ denote the и-skeleton of К (that is, the union of the closed simplices
of dimension at most и). We shall construct a constant εο so that, for all n, if χ e if(n)
lies in the δεο-neighbourhoodof ίΓ(π_1) then there exists у e ίΓ(π_1) with5(jc, 4εο) с
B(y, e(y)/4).
Let Σ denote the disjoint union of the model simplices S e Shapes(if). Consider
the map η : Σ -> (0, oo) defined oris eS by:
7j(i) = min{u?(5, F) | F a face of S and s φ F}.
Note that while η is not a continuous map, its restriction to the interior of each face
is continuous.
Let ηο be 1/8 of the minimum value attained by η on Σ(0), the 0-skeleton of Σ.
Then, inductively for и < D = dim(K), we define r\n to be 1/8 of the minimum value
attained by η on the compact set (on which η is continuous) obtained by deleting the
open η„_χ-neighbourhood of Σ(π_1) from Σ(π). Notice that 8η„ < ηη-\- Let εο = ηο-
Arguing by induction on n we prove that if χ e K<-n^K(-"~1^ then either 8η„ < ε (χ)
or else there exists у e Kin~l) with B{x, 4ηη) с B(y, e(y)/4). The case n = 0
follows from Lemma 7.9. Consider χ e K^"\ If χ lies in K^ minus the open η„-\-
neighbourhoodofίΓ("~1)thenε(x) > 8η„. On the other hand, if thereexistsz e #(π_1)
such that d(x, z) < ηη-\, then B(x, 8ηη) с B(x, ηη-\) с Β(ζ, 2tj„_i). But then, by
induction, either ε(ζ) > 8tj„_i and hence 5(x, 2η„) С β(ζ, ε(ζ)/4), or else there
exists у e K^"-2) such that 5(jc, 4?jn) С B(z, 4τ7„_ι) С В(у, е(у)/4). D
Recall that a metric space is said to be r-uniquely geodesic if every pair of points
a distance less than r apart can be joined by exactly one geodesic segment.
7.55 Proposition. Let К be an Μκ-polyhedral complex with ShapesiK) finite. The
following conditions are equivalent:

120 Chapter 1.7 MK -Polyhedral Complexes
(1) К is locally uniquely geodesic;
(2) К has positive injectivity radius;
(3) Lk(x, K) is π-uniquely geodesic for every χ e K-
Proof. Subdividing if necessary, we may assume that К is simplicial.
That (2) implies (1) is trivial. If (1) holds then, by (7.39), for every χ & К л
neighbourhood of the cone point in CK(Lk(x, K)) is uniquely geodesic. By (5.11),
this implies that Lk(jc, K) is 7r-uniquely geodesic, thus (3) holds.
A second application of (5.11) shows that if Lk(x, K) is π-uniquely geodesic
then the whole of CK(Lk(x, P)) is uniquely geodesic. So by (7.39), B{x, ε(χ)/4) is
uniquely geodesic. Hence, by the preceding lemma, there is a constant εο such that
for every χ e X the ball B(x, ε0) is contained in a uniquely geodesic subspace of K.
In particular, injrad(K) > εο· Thus (3) implies (2). D
For future reference we note a result related to (7.55).
7.56 Lemma. Let К be an Μκ -polyhedral complex with Shapes(K) finite. If the
points χ and у lie in the same open cell ofK, then for sufficiently small ε > 0 there
exists an isometryfrom B(x, ε) to B(y, ε) that restricts to an isometry from B(x, ε) Π С
to B(y, ε) Π С for every closed cell С containing x.
Proof. If we take ε < j πήη{ε(χ), s(y)}, then the metrics induced on B(x, ε) and
B(y, ε) are the length metrics given by measuring the length of paths in these balls
using the given local metrics ав on the individual cells В that contain χ and y. Thus
it is enough to exhibit a bijection from B(x, ε) to B(y, ε) that restricts to an isometry
from B{x, ε) Π Β to B(y, ε) Π Β, where these intersections are endowed with the
restriction of the local metric ав-
This observation allows us to argue in the models S e Shapes(if) and hence in
M". More specifically, it is enough to show that given two points x1 and y' in M" (with
d(x/,y') < π/^/κ if к > 0), there is a canonical isometry of MK that interchanges
χ and у and has the property that it sends every hyperplane containing χ and у to
itself. (We need this isometry to be canonical so as to ensure that the local isometries
induced on each B(x, ε) Π Β agree on common faces.) The reflection of M" in the
hyperplane bisector H(x!', y') of x1 and / has the desired property. D
Arguments Using Finite Models
Proposition 7.55 is an example of how, in many respects, complexes with Shapes(if)
finite behave as if they were compact (or cocompact) spaces. We shall now present
two more examples of this phenomenon. Our main interest in the first result lies
with the corollary, but in the next chapter we shall have need of the stronger, more
technical statement of the proposition.
7.57 Proposition. Let χ and у be points in an MK-polyhedral complex K. Suppose
that Shapes(if) is finite. Suppose that there is a unique geodesic segment joining χ

More on the Geometry of Geodesies 121
to у in К and let с : [0, 1] —> К be a linear parameterization of this segment. Let
c„ : [0, 1] —*■ Κ, η = 1,2,... be linearly reparameterized geodesies in K, and
suppose that the sequences of points c„(0) and c„(l) converge to χ and у respectively.
Then, c„ -> с uniformly.
7.58 Corollary. Let К be an MK-polyhedral complex with Shapes(if) finite. If К is
uniquely geodesic, then geodesies in К vary continuously with their endpoints.
Proof of 7.57. Subdividing, we may assume that К is an M^-simplicial complex
К with Shapes(if) finite. Suppose x„ -> χ and y„ -> y. Let c„ : [0, 1] -> К be
a linearly reparameterized geodesic joining x„ to y„ and let с : [0, 1] -> Ρ be
the linearly reparameterized geodesic joining χ to y. There is a uniform bound on
d(x, xn) + l(c„) + d(yn, y), so by (7.30) there exists an integer N such that image of
each of the paths cn lies in a connected subcomplex K„ С К that can be expressed
as the union of at most N closed cells, and which contains geodesic segments from
χ to xn and from у to y„. We fix a definite choice of K„ for each positive integer n.
Because Shapes(if) is finite, there are only finitely many equivalence classes of
such complexes K„ modulo the relation: K„ ~ Km if there is a map K„ -> K,„ that
fixes x, у and is a simplicial isometry (i.e., a homeomorphism whose restriction to
each closed simplex in Kn is a local isometry onto a closed simplex in Km). Such
a map is an isometry between K„ and Km equipped with their intrinsic metrics (not
their induced metrics from K). We shall call ~ equivalence classes models, denoted
μ.
This notion of a model carries a little more information than that considered in
(7.26) and (7.27). A model μ in the present sense consists not only of a (model)
compact M^-complex Ζ,μ with two distinguished points χβ and γμ, it also comes
equipped with a specified family of simplicial isometries φη : Ζ,μ -> K„ с К, one
for each Kn e μ, such that φη(χβ) = χ and φ„(γβ) = у. An important point to note
is that since the maps ф„ are simplicial isometries, they preserve the length of paths.
Thus, when considered as maps to К (as they will be henceforth) they do not increase
distances.
Because there are only finitely many models, after deleting a finite number of
terms if necessary, we may decompose (c„) into finitely many infinite subsequences
so that Κι ~ Kj for all c, and q in any subsequence. We focus our attention on one
such subsequence, and to simplify the notation we write μ for the corresponding
model and c„ for the paths comprising the subsequence.
Let cn = ф~1с„ : [0, 1] -> Lfl. Because фп is length-preserving, c„ is a linearly
reparameterized geodesic of length l(c„). Let x„ = c„(0) and let yn = c„(l). Notice
that by our initial specification of Kn, there exists a geodesic segment of length d{x, xn)
connecting χ to xn in Ζ,μ, and a geodesic segment of length d(y, y„) connecting у to
yn. Hence d(x, y) < d(x, xn) + l(cn) + d(y, yn), which approaches d(x, y) as η —> oo.
Since φ„ does not increase distance, we deduce that d(x, y) = d(x, y). It follows that
since с is the unique reparameterized geodesic [0, 1] -> К joining χ to y, and since
φη is injective, there is a unique linearly reparameterized geodesic с : [0, 1] —> Ζ.μ
joining 3c to у in Ζ.μ; furthermore, с = фпос for all n.

122 Chapter 1.7 MK-Polyhedral Complexes
We can now appeal to (3.11) to see that the sequence of paths cn : [0, 1] -> Ζ,μ
converges uniformly to c. But d{c{t), cn{t)) = d((pnc(t), (p„cn(t)) < d(c(t), c„(t)).
Hence c„ -> с uniformly. D
Given an MK-polyhedral complex K, model cells S, S; e Shapes(iT) and points
χ e S and у e S, we consider the sets of points in К that lie above χ and y:
Χκ = {x e К \ 3 isometry/ from a face of 5 to a closed cell in К with/(3c) = x},
Yk = {y <= К | 3 isometry g from a face of S to a closed cell in К with g(y) = y}.
7.59 Proposition. Let К be an MK-polyhedral complex. If Shapes(if) is finite then
for all S,S e Shapes(X) and allχ e S and у e S, the set of numbers {d(x, y) \ χ e
Хк, у e Υ κ) is discrete.
Proof The number of points of Χχ or Yf> that lie in any given closed cell of К
is bounded by the maximum of the orders of the isometry groups of the model
cells S e Shapes(if). Hence, for each finite subcomplex L С Р the set of numbers
{dL(x, y) | χ e Χκ Π L, у е Υκ Π L} is finite, where L denotes the intrinsic metric on
L (not the induced metric from K).
Let Σν = [J{di{x, y) \ χ 6 Χκ Π L, у e F^ Π L}, where the union is taken only
over those subcomplexes L С К which can be expressed as the union of at most N
closed cells. For each integer N there are only finitely many such subcomplexes up
to simplicial isometry, so the final sentence of the previous paragraph implies that
ΣΝ is finite. But according to (7.28), for every I > 0 there exists an integer N such
that every geodesic in К of length I > 0 is contained in a subcomplex which is the
union of at most N cells. In particular, {d(x, y) | χ e Хк, у e Υκ} Π [0, ΐ) С ΣΝ. Ο
Alternative Hypotheses
We close this chapter with a couple of exercises and examples to illustrate both the
usefulness and deficiencies of alternatives to the hypothesis that Shapes(if) is finite.
Further examples and exercises are contained in the appendix.
Let us first consider how one might weaken the hypothesis that Shapes(if) is finite
so as to accommodate infinite dimensional complexes. An obvious hypothesis to try
is that Shapes(if) contains only finitely many model simplices in each dimension. In
this generality we have:
7.60 Exercise. Let К be a metric simplicial complex. If Shapes(X) contains only
finitely many model simplices in each dimension then the intrinsic pseudometric on
К is actually a metric.
On the other hand, we do not necessarily obtain geodesic metric spaces in this
generality.

Abstract Simplicial Complexes 123
7.61 Example. Let Δη denote the standard Euclidean simplex of dimension и (i.e. the
convex hull of the standard basis vectors in W+l. For every positive integer η we have
Δ" с Δ"+1 via the inclusion (xu ■ ■ ■ , xn+x) н> (xu ■ ■ ■ ,xn+l, 0)of K"+1 intoK"+2.
Let Δ°° be the union of Δ" for η in Ν \ {0} with the obvious piecewise Euclidean
structure. We consider the complex К obtained by joining two copies Δ~, Δ~ of
Δ°° along the subcomplex opposite the vertex Ρ = (1, 0,...) (that is, we join them
along An± П {(*,, · · · , *,,+,) e R'!+1 | xx = 0} for every и in Ν χ {0}). We claim
that there is no geodesic joining P+ and P_ in K. Indeed, the distance from Ρ to the
opposite face of Δ" decreases to the limit 1 as η -> oo, and the distance between P+
and P_ in К is 2, which is strictly less than the length of any path joining P+ to P_
inK.
On the other hand, it is easy to verify the following:
7.62 Exercise. Let К be a metric simplicial complex. If for every vertex χ e К each
ball of finite radius about К (in the intrinsic pseudometric) contains only cells which
are modelled on a finite subset of Shapes(X), then (K, d) is a complete geodesic
metric space.
Appendix: Metrizing Abstract Simplicial Complexes
In this appendix we describe an alternative approach to metric simplicial complexes.
This approach is often appropriate in situations where one wishes to use geometry to
study a problem that has been encoded in the combinatorics of an abstract simplicial
complex.
We shall recall the definition of an abstract simplicial complex, define the affine
realization of such a complex, and discuss related notions such as barycentric
coordinates. We shall then recast the definition of a metric simplicial complex in this
language.
In (7A. 13) we generalize (7.13) by giving a weaker criterion for the intrinsic
metric on a metric simplicial complex to be complete. We shall also define join
and cone constructions for simplicial complexes and explain how they are related to
the analogous metric constructions given in Chapter 5. Finally, we give a criterion
(7 A. 15) for spaces to have the homotopy type of a finite simplicial complex.
Abstract Simplicial Complexes
7A.1 Definitions. An abstract simplicial complex К consists of a non-empty set V
(its set of vertices) together with a collection S of non-empty finite subsets of V, such
that {υ} e S for all υ e V, and if S e S then every non-empty subset Τ of S is also
in S. The elements of S are called the simplices of K. A simplex S e S is called an

124 Chapter 1.7 MK-Polyhedral Complexes
η-simplex if S has cardinality η + 1 and η is called the dimension of S. The elements
of S are called its vertices and the non-empty subsets of S are called its faces. The
dimension dim К of К is the supremum of the dimensions of its simplices.
A simplicial complex Ко is called a subcomplex of К if its set of vertices Vo is a
subset of V and each simplex of Ко is also a simplex of K. To each и-simplex S of К
is associated a subcomplex 5 of К of dimension η whose set of simplices is the set
of faces of S.
Consider the equivalence relation on V generated by: и ~ υ if {и, υ} e 5. The
complex if is said to be connected if V consists of a single ~ equivalence class.
Let K\ and Кг be be abstract simplicial complexes with vertex sets V\ and V?
respectively. A simplicial map from if] to #2 is a map/ : Vj —> V2 that sends each
simplex of K\ to a simplex of #2·
7A.2 Definition of Cones, Joins and Links. Let K\ and Кг be abstract simplicial
complexes whose vertex sets V\, Уг are disjoint. The simplicial join of K\ and Кг,
denoted K\ * K2, is a simplicial complex with vertex set V\ U Уг; a subset of Vi U V2
is a simplex of K\ * #2 if and only if it is a simplex of K\, a simplex of ЛГ2, or the
union of a simplex of K\ and a simplex of ЛГ2. Note that
dim(i4 * #2) = dimiT, + dim #2 + 1.
For example, the join of a/j-simplex and a ^-simplex is a (p + q + l)-simplex.
If К is a simplicial complex with vertex set V, then the simplicial cone over K,
denoted C(K), is the simplicial join of К and a complex with only one vertex i>o. The
vertex υ0 is called the cone vertex of C(K) and dim C(K) = dim К + 1.
Given a simplex 5 in an abstract simplicial complex K, the link of 5 in K, denoted
lk(S, K), is the subcomplex of К consisting of those simplices Τ such that Τ Π S = 0
and Γ U 5 is a simplex of if.
Note that the subcomplex of К whose simplices are the faces of simplices
containing S is isomorphic to the join S * lk(S, K).
7A.3 Affine Realization. Let К be an abstract simplicial complex with vertex set
V. Let W denote the real vector space with basis V. The affine realization \S\ of a
simplex S of К is the affine simplex in W that has vertex set S; in other words, \S\
consists of those vectors χ = ^2vsSxv ν with xv e [0, 1] and ^2vsSxv = 1. We give
\S\ the topology and affine structure induced from the finite dimensional subspace of
W spanned by the vertices of S; this subspace is isomorphic to M"+1, where η is the
dimension of S. Given two points χ and у in \S\, the affine segment with extremities
χ and у is denoted [jc, y].
We define the affine realization (or simply the realization) of К to be the subset
\K\ of W which is the union of the affine realizations of the simplices of K. (It
is convenient to abuse terminology to the extent of writing S and К to mean the
realization of S and K, and we shall do so frequently.) The realization of S is called
a closed simplex in K. The coordinates xv of a point χ of К С W are called its
barycentric coordinates.

Abstract Simplicial Complexes 125
The interior of a simplex S is the set of points in its affine realization whose
coordinates χυ, υ e S, lie in (0, 1). Given a point χ e K, the open star st(x) of χ
is defined to be the union of {x} and the interior of those closed simplices S which
contain x; the closed star St(x) of χ is the union of the closed simplices which contain
x. (If υ is a vertex, then its open star st(v) is the set of points χ of К with barycentric
coordinates χυ > 0.)
Let K\ and Кг be abstract simplicial complexes with vertex sets V\ and V2 and let
W\ and W2 be the real vector spaces with bases V\ and V2 respectively. If/ : V\ —> V2
gives a simplicial map from K\ to K2, then the linear map W\ —> W2 sending υ e Vj
to /(υ) e V2 maps \K\\ to IX2I and its restriction to each \S\ is an affine map onto
1/(5)I. This map, which we denote \f\ : \Ki\ -> \K2\ will be called the (affine)
realization of the simplicial map/.
7A.4 The Barycentric Subdivision. Let К be an abstract simplicial complex with
vertex set V. The barycentric subdivision K' of К is the abstract simplicial
complex whose vertices are the simplices 5, of K, and whose и-simplices are the sets
{So, ■ ■ ■, Sn} where S0 с S\ С ■ · ■ С Sn.
There is a natural bijection from the affine realization of K' to the affine realization
of K: this map is affine on each simplex of K' and sends the vertex of K' corresponding
to the и-simplex S of К to the barycentre of the affine realization of S (namely the
point with barycentric coordinates xv = 0 if υ φ S and xv = \/(n + 1) if ν e S) and
is affine on each simplex of K'.
7A.5 Topology on the Affine Realization. For the moment we do not consider any
topology on the affine realization of K, but mention in passing that a useful topology
for the purposes of homotopy theory is the weak topology, which is characterized by
the property that a subspace of К is closed if and only if its intersection with each
closed simplex is closed.
An alternative topology on К can be obtained by introducing on W the scalar
product such that the basis V is orthonormal and considering the associated metric.
The restriction of this metric to К is called the metric of barycentric coordinates.
If К is not locally finite, then the topology associated to this metric is not the weak
topology. Indeed if a vertex υ lies in infinitely many simplices, then any set containing
exactly one point (φ υ) in the interior of each simplex of St(v) is closed in the weak
topology, but in general it is not closed with respect to the metric of barycentric
coordinates because ν might be an accumulation point of this set.
On the other hand, an important theorem of Dowker [Dow52] shows that the
identity map of К is a homotopy equivalence from К equipped with the weak topology
to К equipped with the topology induced by the metric of barycentric coordinates.
7A.6 Barycentric Coordinates for Geodesic Simplices in M™. For a geodesic n-
simplex S С Μ™ with vertices vq, ..., vn one has natural barycentric coordinates,
which are defined as follows (compare with 7.42).
к = 0: Identify Em with the hyperplane in Em+1 formed by the points у =
Суо, · · · ,Уп) with ^2Т=оУ' = 1· АПУ point x e S С Em can be written uniquely

126 Chapter 1.7 MK-Polyhedral Complexes
as χ = 5Ζ"=0*υ,υ; with xVl e [0, 1]; the reals numbers xv. are the barycentric
coordinates of x.
к > 0: Identify M™ with the sphere of radius 1 /л/к centred at the origin in Em+1.
The и-simplex Sq in Em+1 spanned by the vertices щ, . ■ ■, v„ does not contain 0 and
therefore is mapped bijectively onto the simplex S by the radial projection from 0
to M™. The barycentric coordinates of points χ e S are defined to be the barycentric
coordinates of the corresponding points in So.
к < 0: Identify M™ with the upper sheet of the hyperboloid Y^LX xf — x^+l =
Ι/λ/κ in E"1'1 and proceed as in the case к > 0.
Barycentric coordinates are preserved by isometries of M™. If S is an «-simplex
of an M^.-simplicial complex, then the barycentric coordinates of a point χ e S are
defined to be the barycentric coordinates of the unique point χ e S that is mapped to
χ by the characteristic map/5 : S —> S (see 7.12). The barycentre of a simplex S is
the point whose barycentric coordinates are all equal.
7A.7 Affine and Projective Maps. Given geodesic и-simplices S с М™ with
vertices vq, ... ,v„ and S' С Μ™ with vertices v'Q,..., i^, there is a unique map from
S to 5' sending υ, to υ,- and preserving the barycentric coordinates. Similarly, given
an и-simplex Τ = {щ,..., un} in an abstract simplicial complex, there is a unique
map from S to the affine realization of Τ sending υ, to щ for all i and preserving the
barycentric coordinates. Such a map will be called an affine map.
A projective map from S to S' (or \T\) is one for which there exist positive real
numbers λ,- such that the point of S with barycentric coordinates (jeo, ..., x„) is sent
to the point with barycentric coordinates
λο^ο λ„χ„
Σί=0 λ-iXi Σί=0 λ-Λ'
The image under a projective map of a geodesic segment in S is a geodesic segment
in S', but in general a linear parameterization of this segment is not mapped to a
linear parameterization of its image.
7A.8 Subdivision. Given a geodesic и-simplex S С Μ™ with vertex set V =
{vq, ...,v„] and a point υ e S \ V, one can subdivide S into smaller и-simplices: the
и-simplices Sy in this subdivision correspond to subsets V С V of η vertices whose
convex hull does not contain υ; Sy is the convex hull of V U {υ}. With respect to the
barycentric coordinates of S and Sy>, the inclusion Sy ^-» S is projective.
Metrizing Simplicial Complexes
Let if be an MK -simplicial complex as defined in (7.2). Associated to К one has an
abstract simplicial complex, whose vertex set V is the set of vertices of K; a subset
S С У is an (abstract) simplex if and only if S is the vertex set of a simplex in K. There
is a canonical bijection from К onto the affine realization of this abstract simplicial

Metrizing Simplicial Complexes 127
complex; this map extends the natural bijection between the sets of vertices and
is affine on each simplex with respect to the barycentric coordinates as defined in
(7A.6).
Conversely, when presented with an abstract simplicial complex K, one can turn
it into a geometric object (an MK-complex) as follows: metrize the simplices of
the realization of К by choosing an affine isomorphism to a geodesic simplex in
M"; choose these isomorphisms so that the induced metrics coincide on faces of
intersection. This construction leads to a well-defined notion of a piecewise linear
(PL) path in К and the intrinsic pseudometric on К is the infimum of the lengths of
PL paths joining them.
7A.9 Alternative Definition of an MK -Simplicial Complex. Let к be a real number.
An MK-simplicial complex consists of the following information:
(1) an abstract simplicial complex K;
(2) a set Shapes(X) of geodesic simplices 5, С М"·;
(3) for every closed simplex S in the realization of K, an affine isomorphism/5 :
S —» S, where S e Shapes(X); if S' is a face of S, thenf^1 ofs, is required to be
an isometry from S onto a face S.
A piecewise linear (PL) path in К is a path that is the concatenation of a finite
number of affine segments. The length of a segment / in a simplex S С К is defined
to be the length of/J"' (/). The intrinsic metric on К is defined by setting d{x, y) equal
to the infimum of the length of PL-paths joining χ to y.
Note that the set of PL paths in К depends only on the affine structure of К and
not on the chosen maps^. It is easy to see that the above definition of the intrinsic
metric d agrees with that given in (7.4) and therefore (K, d) is a length space; it is a
complete geodesic space if Shapes(iT) is finite (7.19). Moreover, the characterization
of geodesies in (7.29) shows that if Shapes(if) is finite then d(x, y) is equal to the
length of the shortest PL path connecting them (i.e. the infimum in the definition of
d is attained).
7A.10 Regular MK -Simplicial Complexes. An MK -simplicial complex К in which
all of the model simplices S e Shapes(if) have the same edge lengths is called
regular. We describe the cases к = 0 and к = 1.
In the notation of (7A.3), let W be the vector space with basis the vertex set of
K, endowed with the metric associated to the scalar product for which this basis is
orthonormal. Each и-simplex S of K, with the induced metric ds, is isometric to a
regular «-simplex in E" with edge lengths \/2. In this way К becomes a piecewise
Euclidean complex. If we wish, we may rescale the metric so that each edge has
length one; this is called the standard Euclidean realization of K.
Consider the unit sphere in W (the set of vectors of norm one) with the induced
length metric. Radial projection from the origin 0 e W onto the unit sphere restricts to
an injection on K, and this projection identifies each и-simplex S of К with a spherical
«-simplex whose edge lengths are all π/2. In this case, К becomes a piecewise
spherical complex in a natural way, called the all-right spherical realization of K.

128 Chapter 1.7 MK-Polyhedral Complexes
If К is of finite dimension и, then in the above examples Shapes(if) = {Δ' \i =
0, 1,..., и}, where Δ' is the Euclidean (resp. spherical) г-simplex in E'+1 (resp. §')
spanned by the vectors of the standard basis of K'+1.
7A. 11 Exercises
(1) Prove that the identity map of any abstract simplicial complex induces a
bi-Lipschitz homeomorphism between any two regular MK -simplicial complexes
associated to К (where к is not fixed).
(2) Let К and d be as in (7 A.5). Prove that if Shapes(if) is finite then the canonical
map from К to the standard Euclidean realization of the underlying simplicial which
is the identity on the set of vertices and affine on each simplex is a homeomorphism.
(3) Let K\ and Кг be abstract simplicial complexes. Show that the all-right
spherical realization of K\ * Кг is the spherical join (as defined in 5.13) of the all-right
realizations of K\ and Кг-
(4) Let К be an MK-simplicial complex and fix a vertex ν e K. In (7.15) we
defined the geometric link Lk(u, K) of υ in K; it is a spherical complex. On the
other hand, the subcomplex St(v) \ st(v) is endowed with an induced MK-simplicial
structure from K. Prove that the map which associates to each χ e St{v) \ st{v)
the initial vector of [υ, χ] e Lk(u, K) is a map that is not affine in general, but is
projective in the barycentric coordinates.
A Criterion for Completeness
7A.12 Definition. Given any geodesic и-simplex S in M™, there is an affine map
05 of S onto the standard и-simplex Δ" in E"+1 spanned by the basis vectors. Note
that φ5 is unique modulo the action of Isom(A"). The distortion of S is the smallest
number λ such that j^d^six), 0sOO) < d(x, y) < λά{φ{χ), ф{у)) for every x, у € S.
The distortion of an MK -simplicial complex К is the supremum of the distortions
of the model simplices S e Shapes(if).
7A.13 Theorem. Let К be a finite dimensional MK -simplicial complex. If the
distortion of К is finite, then К is a complete length space.
Proof. Equip К with its intrinsic pseudometric d. It follows from (7.10) that (K, d)
is a metric space and it is clear that it is a length space. (However it need not be a
geodesic space, 7A.14.)
Let Kq be the affine realization of the simplicial complex associated to К and let
φ : Κ —> Ко be the canonical map which is affine on each simplex. We view Ко as a
regular M0-simplicial complex. For each model и-simplex S e Shapes(if) we have
a bi-Lipschitz map ф$ : S —» Δ". To say that К has finite distortion means that there
is a uniform bound, λ say, on the Lipschitz constants of these maps. It follows that
the length of each PL path in К is at most λ times the length of the same path in Kq,

Spaces with the Homotopy Type of Simplicial Complexes 129
and vice versa. Since the pseudometric on each of the complexes is defined by taking
the infimum of the lengths of such paths, it follows that the map φ : К —> Ко is a
bi-Lipschitz homeomorphism (with Lipschitz constant λ). In particular, since Ко is
complete (K, d) is complete. D
7A.14 Example. Consider the metric graph with two vertices y, ζ and countably
many edges joining them, the и-th of which has length (1 + £). By introducing a
new vertex in the middle of each edge we make this graph simplicial. It has bounded
distortion, but it is not a geodesic space: d(y, z) = 1 whereas every path from у to ζ
has length strictly larger than 1.
Spaces with the Homotopy Type of Simplicial Complexes
We close with a result which shows that many geodesic spaces have the homotopy
type of locally compact complexes. There are much more general results of this type
(see [Вогбб], [Hu(S)65], [WestW]), but the simple result given below indicates the
ideas that we wish to exemplify and is sufficient for the purposes of Part II.
Let X be a topological space. A collection of subsets of a space X is said to be
locally finite if each point* e X has a neighbourhood that meets only finitely many
of the given sets. An abstract simplicial complex is said to be locally finite if each
vertex belongs to only finitely many simplices.
7A.15 Lemma. Let Xbea geodesic space and suppose that there exists ε > 0 such
that X can be covered by a locally finite collection ofopen ballsU = {B(xt, ε) | i e /}
such that B(xj, ε) and B(xit 3ε) are convex and uniquely-geodesic, and suppose that
geodesies in these balls vary continuously with their endpoints. Then X is
homotopy equivalent to the geometric realization of a locally finite simplicial complex К
(namely the nerve of the covering Ы). Moreover, ifX is compact then К is finite.
Proof Consider the abstract simplicial complex К whose vertex set {i>;}i<=/ is indexed
by the elements of/ and which has an r-simplex {i>,0, ..., vir\ for each non-empty
intersection [^=0В(х^, ε) φ 0. This complex is called the nerve of the covering U.
We consider the affine realization of this complex, which we also denote K, with the
weak topology (7A.5).
We shall define maps/ : X —> К and g : К —> X such that/g is homotopic to the
identity of К and gf is homotopic to the identity of X. In the second case it suffices to
prove that for every χ e X there exists i e / such that χ has a neighbourhood V with
V U gf(V) С B(xi, 3ε), for then we obtain a homotopy h, from ho = gf to h\ = idx
by defining h,(x), for each t e [0, 1], to be the point a distance td(x, gf(x) from gf(x)
on the unique geodesic segment [x, gf(x)]', this segment is contained in B(xt, 3ε) and
is assumed to vary continuously with its endpoints.
The desired map / : X —> К is obtained by using a partition of the unity
subordinate to the covering Li. For each i e /, let fi : X —> Ж be the function

130 Chapter 1.7 MK-Polyhedral Complexes
which is identically zero outside of the ball B{xn ε) and which maps χ e B{x, ε,) to
ε — d(x, Xi). Then/(jc) is defined to be the point whose j-th barycentric coordinate is
m
Note that/ maps B{xit ε) to the open star of υ,.
The desired map g : К —> X is defined inductively on the skeleta of K. For each
vertex ν of K, let Sf(v) be the set of points χ e К whose barycentric coordinate xv
is not smaller than any of the other barycentric coordinates of x. (This subset of К
corresponds to the closed star of the vertex υ in the barycentric subdivision K' under
the bijection from K' to К described in 7A.4.) First, for each i e / we map υ; to xt.
Proceeding by induction, we assume that a continuous map g has been constructed
on the r-skeleton K(r) of К and that, for each vertex υ;, this map sends K(r) Π Slf{v{) to
Bfa, ε). We extend g across each (r + l)-simplex S = {vio,..., v-h+l} by mapping the
barycentre bs of S to a point in the intersection (Ύ=0 B{xir ε), and then mapping the
affine segment [£>5, y] affinely onto the geodesic segment [g(bs), g(y)] for each point
у in the boundary of S. This extension of g is well-defined and continuous because
for each у there is a vertex υ, of S such that both g(bs) and g(y) lie in Bfa, ε), where
geodesies are unique and vary continuously with their endpoints. By construction,
if υ is a vertex of a simplex S of K, then g maps S into the ball B(xt, 3ε), and hence
g maps St(vi) to B(jc,·, 3ε).
It follows that gf maps B(xi, ε) to В(дс,-, 3ε) and the argument at the end of the
second paragraph shows that gf is homotopic to the identity.
To see that/g is homotopic to the identity, note that if S is a simplex of К with
vertices {«,·„, ..., vir}, then g(S) С 1_|=ο^(*<)> ε), and hence/g(5) is contained in the
union of the open stars in if of the vertices of S. This completes the proof, because
if L is the realization of any locally finite simplicial complex, then by a standard
argument, any map F : L —> L that sends each simplex S into the union of the open
stars of the vertices of 5 is homotopic to the identity. (One proves this by proceeding
one simplex at a time using the obvious "straight line" homotopies, [Spa66, 3.3.11].)
D

Chapter 1.8 Group Actions and Quasi-Isometries
In this chapter we study group actions on metric and topological spaces. Following
some general remarks, we shall describe how to construct a group presentation for
an arbitrary group Γ acting by homeomorphisms on a simply connected topological
space X. If X is a simply connected length space and Γ is acting properly and
cocompactly by isometries, then this construction gives a finite presentation for Γ. In
order to obtain a more satisfactory description of the relationship between a length
space and any group which acts properly and cocompactly by isometries on it, one
should regard the group itself as a metric object; in the second part of this chapter we
shall explore this idea. The key notion in this regard is quasi-is ometry, an equivalence
relation among metric spaces that equates spaces which look the same on the large
scale (8.14).
Group Actions on Metric Spaces
First we need some vocabulary.
8.1 Terminology for group actions. An action of a group Γ on a topological space
X is a homomorphism Φ : Γ -> Homeo(X), where Homeo(X) is the group of self-
homeomorphisms of X. We shall usually suppress all mention of Φ and write γ.χ for
the image of χ e X under Ф(у), and γ.Υ for the image of a subset Υ c. X. We shall
write Γ.Υ to denote (Jy<=r Y-Y-
An action Φ is said to be. faithful (or effective) if кегФ = {1}. An action is said
to be free if, for every χ e X and every γ e Γ \ {1}, one has γ.χ ψ χ. And it is said
to be cocompact if there exists a compact set К с X such that X = Г.К. Given an
action of a group Г on a space X, we write Гх for the stabilizer (isotropy subgroup)
of χ e X, that is Γχ = {γ e Γ | γ.χ = χ].
If X is a metric space, then one says that Γ is acting by isometries on X if the
image of Φ is contained in the subgroup Isom(X) С Homeo(X).
8.2 Definition of a Proper Action. Let Г be a group acting by isometries on a
metric space X. The action is said to be proper (alternatively, "Г acts properly on
X") if for each χ e X there exists a number r > 0 such that the set {γ e Γ |
y.B(x, г) П B(x, r) φ 0} is finite.

132 Chapter 1.8 Group Actions and Quasi-Isometries
8.3 Remarks
(1) It is more usual to define an action of a group Γ on a topological space X to be
proper if for every compact subset К С X the set of elements {γ e Γ | γ.ΚΠΚ φ 0}
is finite. The definition that we have adopted is equivalent to the standard definition
in the case of actions by groups of isometries on proper metric spaces, but in general
it is more restrictive. In fact our definition implies that every compact subset К С X
has an open neighbourhood U such that {γ e Γ | y.U Π U φ 0} is finite.
To see this, cover К with finitely many open balls B(xh r,), where each x, is in
К and each of the sets S(i) = {γ e Γ | B(xh 2r,) Π y.B(xh 2r,) φ 0} is finite. Let
U be the union of the balls 5(дс,·, r,). If there were infinitely many distinct elements
y„ e Г such that yn.U HU φ 0, then for some fixed indices k and i\ there would be
infinitely many y„ e Г such that yn.B(xio, r,0) Π B{xi{, r„) φ 0. But then we would
have infinitely many elements y~lym € S(i0), which is a contradiction.
(2) Let X be a length space. If one endows the universal covering of X with the
induced length metric, then the action of π\Χ by deck transformations on X is a
proper action (see 3.22(1)). (This statement would not remain true if one were to
replace "3 r > 0" by "V r > 0" in the definition of properness.)
(3) In (8.2) we do not assume that the group Γ acts faithfully on X. However, if
the action of Γ on X is proper then every isotropy subgroup Γχ will be finite.
8.4 Exercises
(1) Let X be a length space. If there is a group that acts properly and cocompactly
by isometries on X, then X is complete and locally compact, so by the Hopf-Rinow
Theorem it is a proper geodesic space.
(2) Show that the action of a "group Г on a locally compact space X is proper in
the sense of (8.3(1)) if and only if:
(i) Г\Х with the quotient topology is Hausdorff,
(ii) for every χ e X, the isotropy subgroup I\ is finite, and
(iii) there is a r^-invariant neighbourhood t/of χ with I\ = {γ e Γ | y.UCiU φ 0}.
(3) If a group Γ acts properly by isometries on a metric space X, then the pseudo-
metric on Γ\Χ that the construction of (5.19) associates to the equivalence relation
χ ~ γ.χ is the metric described in (8.5(2)) and the metric topology is the quotient
topology.
We gather some basic facts about proper actions on metric spaces.
8.5 Proposition. Suppose that the group Γ acts properly by isometries on the metric
space X. Then:
(1) For each χ e X, there exists ε > 0 such that if γ.Β(χ, ε) Π Β(χ, ε) φ 0 then
Υ еГ,.
(2) The distance between orbits in X defines a metric on the space Y\X of Υ-orbits.

Group Actions on Metric Spaces 133
(3) If the action is proper and free, then the natural projection ρ : X —> Γ\Χ is a
covering map and a local isometry.
(4) If a subspace Υ ofX is invariant under the action of a subgroup ЯСГ, then
the action of Η on Υ is proper.
(5) If the action of Τ is cocompact then there are only finitely many conjugacy
classes ofisotropy subgroups in Γ.
Proof. Let χ e X. By properness, there exists r > 0 such that the orbit Γ.χ of χ meets
the ball B(x, r) in only a finite number of points. To prove (1), choose ε small enough
to ensure that B{x, 2ε) Π Γ.χ = {χ}.
The distance d(T.x, Г.у) = d(x, Г.у) between orbits is obviously a pseudometric,
and it follows from (1) that d(T.x, Г.у) = d(x, Г.у) is strictly positive, thus (2) is
true. (3) follows easily from (1), and (4) is immediate from the definitions. To prove
(5), we fix a compact set К whose translates by Г cover X, and cover К with finitely
many balls В(ъ, г,·) such that each of the sets {γ e Γ | y.5(jc,, r,) Π B(xh η) φ 0} is
finite; let Σ be the union of these sets. For each χ e X there exists γ e Γ such that
χ e γ.Κ. Since γ~ιΓχγ = Γγ-ιχ, we have γ~ιΓχγ С Σ. D
In the following proof we need some elementary facts about lifting maps to
covering spaces. Such material is covered in any introductory course on algebraic
topology, e.g. [Mass91].
8.6 Proposition. Let X be a length space that is connected and simply connected,
and suppose that the group Г с Isom(X) acts freely and properly on X. Consider
T\X with the metric described in (8.5(2)): d(T.x, Г.у) := infyer d(x, y.y). LetN(T)
be the normalizer of Г in Isom(X).
Then, there is a natural isomorphism Ν(Γ)/Γ = Ьот(Г\Х), induced by the map
ν ы> v, where ν e Λ'(Γ) and ν e Isom(r\X) is the map Y.x ы> r.(v(jc)).
Proof. By definition, Ν(Γ) = {ν e Isom(X) | νγν~ι e Γ Vy e Γ}. So if ν e N(T)
then ν.Γ = Γ.ν, and hence ν.(γ.χ) e Γ.(ν.χ) for every χ e X and γ e Γ. Therefore
the map ν : Τ.χ м> T.(v.x) is well-defined, ν is obviously an isometry. If ν is the
identity map, then for any choice of basepoint xq e X, we have v.xq = γ.χο, for
some γ e Γ. But this implies that ν = γ, because both ν and γ are liftings to X
of the identity map on Γ\Χ and these lifts coincide at one point xo, so they must
coincide everywhere, because X is connected. Thus ν м> ν defines an injective
homomorphism Ν(Γ)/Γ -> Ьот(Г\Х).
It remains to prove that this map is surjective. Let JL be an isometry of Г\Х, and
let ρ : X —> Γ\Χ be the natural projection (which is a covering map, by (8.5(3)).
As X is connected and simply connected, the composition JL ο ρ lifts to a continuous
bijection μ : X -> X projecting to JL. This map μ is a local isometry, because ρ is
locally an isometry. It follows that μ is an isometry, because it preserves the lengths
of the curves in X, and X is assumed to be a length space. Moreover, for each γ e Γ,
the isometry μ~*γμ projects to the identity of Γ\Χ, hence it is an element of Г, since
X is connected. D

134 Chapter 1.8 Group Actions and Quasi-Isometries
The Group of Isometries of a Compact Metric Space
Let У be a compact metric space. One defines a metric on the group Isom(F) by
d(a, a') := sup d(a.y, a'.y).
8.7 Proposition. The metric on Isom(F), defined above, is invariant by left and
right translations. With the topology induced by this metric, Isom(F) is a compact
topological group.
Proof. The invariance of the metric by left and right translations is obvious, as is the
fact that а ы> α-1 preserves distances from the identity in Isom(F). The composition
(α, β) м> αβ is continuous because ά{αβ, α'β') < ά{αβ, αβ') + ά{αβ', α'β') =
ά{β, β') + d(a, α'), hence Isom(F) is a topological group.
To see that Isom(F) is compact, one can apply Arzela-Ascoli theorem (3.10):
the set of isometries of Υ is equicontinuous, so every sequence has a subsequence
converging uniformly to a map from Υ to Y; the limit map is obviously an isometry.
D
Presenting Groups of Homeomorphisms
In this paragraph we construct a presentation for an arbitrary group Γ acting by
homeomorphisms on a simply connected topological space X. If X is a simply
connected length space and Γ is acting properly and cocompactly by isometries, then
the construction which we shall describe gives a finite presentation for Γ. Our proof
relies on a simple observation concerning the topology of group presentations (8.9).
8.8 Free Groups and Presentations. We write F(A) to denote the free group on a
set A. The elements of F(A) are equivalence classes of words over the alphabet u
A±l: a word is a finite sequence a^ ... an where a, e A±l; one may insert or delete
a subword of the form aa~l, and two words are said to be equivalent if one can pass
from one to the other by a finite sequence of such deletions and insertions. A word
αϊ... a„ is said to be reduced if each α, φ aj}^. There is a unique reduced word in
each equivalence class and therefore we may regard the elements of F(A) as reduced
words.
The group operation on F(A) is given by concatenation of words — the empty
word is the identity.
Let G be a group and let S С G be a subset. We write ((S)) to denote the smallest
normal subgroup of G that contains S (i.e. the normal closure of S).
A presentation for a group Г consists of a set A, an epimorphism π : F(A) -» Γ,
and a subset 1Z с F(A) such that ((71)) — кеглч One normally suppresses mention
14 Given a set A, the elements of A~l are by definition the symbols a~l where α € A- The
notation A±] is used to denote the disjoint union of A and A~].

Presenting Groups of Homeomorphisms 135
of π and writes G — (А | TZ). The presentation is called finite if both of the sets Λ
and TZ are finite, and Γ is said to be finitely presentable (or finitely presented) if it
admits such a presentation.
The Cayley graph С^(Г) of a group Г with a specified generating set Л was
defined in (1.9). Directed edges in СА(Г) are labelled by the generators and their
inverses, and hence there is a 1-1 correspondence between words in the alphabet
A±l and edge paths issuing from each vertex of С^(Г). An edge path will be a loop
if and only if the word labelling it is equal to the identity in Г. The action of Г on
itself by left multiplication extends to a free action on С^(Г): the action of yo sends
the edge labelled a emanating from a vertex у to the edge labelled a emanating from
the vertex yoy.
Basic definitions concerning 2-complexes (adapted to the needs of this chapter)
are given in the appendix.
8.9 Lemma. Let Υ be a group with generating set A and let TZ be a subset of the
kernel of the natural map F(A) —> Γ. Consider the 2-complex that one obtains
by attaching 2-cells to all of the edge-loops in the Cayley graph СА(Г) that are
labelled by reduced words r e TZ. This 2-complex is simply-connected if and only if
((ft»=ker(F(.A)-»r)-
Proof. The Cayley graph CA(F(A)) of F(A) is a tree. СА(Г) is the quotient of this
tree by the (free) action of N := ker(F(.4) ~» Г). Thus there is a natural identification
7Г1(С^(Г), I) — Ν, and a word in the generators A±l defines an element of N if and
only if it is the label on an edge-loop in СА(Г) that begins and ends at the identity
vertex.
Let и e F(A) be a reduced word and consider the vertex of Сд(Г) that is reached
from the vertex 1 by following an edge path labelled и. If we attach the boundary of a
2-cell to СА(Г) by means of the edge loop labelled r that begins at this vertex, then by
the Seifert-van Kampen theorem [Mass91], the fundamental group of the resulting
2-complex will be the quotient of N by {{u~lru)). More generally, if for every reduced
word r e TZ and every vertex ν of СА(Г), we attach a 2-cell along the loop labelled
r that begins at v, then the fundamental group of the resulting 2-complex will be
the quotient of N by the normal closure of TZ. In particular, this 2-complex will be
simply-connected if and only if N is equal to the normal closure of TZ. D
In the following proposition we do not assume that Г is finitely generated, nor
do we make any assumption concerning the discreteness of the action.
8.10 Theorem. Let X be a topological space, let Г be a group acting on X by
homeomorphisms, and let U С X be an open subset such that X = Г. U.
(1) IfX is connected, then the set S = {у е Г | y.U П U φ 0} generates Γ.
(2) Let As be a set of symbols as indexed by S. IfX and U are both path-connected
andX is simply connected, then Γ = (As I TZ), where
TZ — {aSlaS2a~^ | s,- e S; U ilsi.U ils^.U φ 0; s\s2 = S3 in Γ}.

136 Chapter 1.8 Group Actions and Quasi-Isometries
Proof. First we prove (1). Let Я С Г be the subgroup of Γ generated by S, let
V = H.U and let V' = (Γ χ H).U. If V(l V φ 0, then there exist heH,ti e Г \ Я
such that h~lh'.U Π U φ 0 and hence h' e HS с Я, contrary to assumption. Thus
the open sets V and V" are disjoint. V is non-empty and X is connected, therefore
V" = 0 and Я = Г.
We now prove (2). Let К be the combinatorial 2-complex obtained from the
Cayley graph СдДГ) by attaching a 2-cell to each of the edge-loops labelled by the
words r € TZ. According to the previous lemma, it is enough to show that К is simply
connected.
Fig. 8.1 The relations in Theorem 8.10
Fix xo € U. For each s e S we choose a point jcs e (7 П s.t/ and then (using the
fact that U is path-connected) we choose a path from xo to xs in (7, and a path from
xs to i.^o in s.U; call the concatenation of these paths cs. Letp : С^(Г) —> X be the
Γ-equivariant map that sends 1 to xq and sends the edge labelled as emanating from
1 to the path cs. Because X is assumed to be simply connected, we can extend this
to a continuous Γ-equivariant map ρ : К -> X.
Let D be the standard 2-disc. To complete the proof of the proposition, we must
show that every continuous map I : 3D —> Сд5(Г) can be continuously extended to
a map Ό —> if. It is enough to consider locally injective edge loops. Because X is
simply-connected, the map pi : 3D —> X extends to a map 0 : D —> X. Because
D is compact and U is open, there is a finite triangulation Τ of D with the property
that for every vertex υ of Τ there exists γυ еГ such that φ maps all of the triangles
incident at ν into yv.U.
Let ί be a triangle of Τ with vertices vi,V2,vi. We have
φ(ή^γυι.υηγυ2.υηΥυ,.υ
= γυΓ(υηγ-ιγυ2.υηγ-ιγη.υ)
= Υο2·{γ^Υνι.υηυηγ-ιΥυιΜ)φ0.

Presenting Groups of Homeomorphisms 137
It follows that si := γυίιΥυ2> s2 '■= y„2'>V and ί3 := γυιγ^ are elements of S,
and aSlaS2a~l e 1Z. We can therefore extend the map υ м> у„ to a map 0 from the
1-skeleton of Τ to Сд5(Г) by sending the edge connecting v, and υ,·+ι to the edge
labelled s, emanating from γυ. (indices mod 3). If и, е 3D, then we choose the γυ.
so that the restriction of φ to 3D is a reparameterization of I.
Finally, since φ sends the boundary of each triangle in Τ to a circuit in С^5(Г)
labelled by an element of 1Z, and since each such circuit is the boundary of a 2-cell
in K, we may extend φ to a continuous map D —> if. D
8.11 Corollary. Λ grow/? is finitely presented if and only if it acts properly and
cocompactly by isometries on a simply-connected geodesic space.
Proof. Given a group Γ acting properly and cocompactly by isometries on a simply-
connected geodesic space X, one chooses a compact set С С XsuchthatT.C = Xand
an open ball B{xq, R) containing C, then one applies the theorem with U = B(x0, R).
Because X is proper (8.4) and the action of Г is proper, the set of generators S
described in (8.10) is finite and hence so is the set 1Z.
Conversely, given a finite presentation (Л \ Щ of a group, one can metrize
the simply-connected 2-complex constructed in the proof of (8.9) as a piecewise
Euclidean complex in which all of the edges have length 1 and all of the 2-cells
are regular polygons. The natural action of Г on its Cayley graph extends in an
obvious way to an action of Г by isometries on this complex; the action is proper
and cocompact (cf. Appendix). D
8.12 Remarks
(1) Theorem 8.10 is due to Murray Macbeath [Mac64]. See page 31 of [Ser77]
for an alternative treatment and additional references.
(2) In 8.10(2) it is necessary to require that X is simply-connected. To see this,
let Г be the group of order four generated by a rotation φ of the circle §' through an
angle 7Г/2 and let U be an open arc of length я on the circle. Then the set S in (8.10)
is {φ, φ~1} and the set TZ is empty, thus (S | TZ) is not a presentation of Γ.
To obtain further examples we consider groups Γ that are finitely generated but
not finitely presentable. There are uncountably many such groups [Rot95]; some
examples are given in section (ΙΙΙ.Γ.5). Let Λ be a finite generating set for Γ and
consider the action of Γ on the Cayley graph С^(Г). If we take U to be the open ball
ofradius 1 about the identity then S = ^4.±1, in the notation of (8.10), and one cannot
present Г with finitely many relations on these generators.
(3) An action of a group Г on a metric space X is said to be cobounded if there
exists a bounded set В с X such that Г.В = X. (8.11) does not remain true if
one replaces the hypothesis 'cocompactly' by 'coboundedly'. For example, consider
1г{Щ and let 5, denote the sequence whose only non-zero term is a 1 in the г-th place.
Define τ,· : £2(Z) -► 12{Щ by σ м> σ +5,. The subgroup Τ С Isom(£2{Щ) generated
by {5, | г" е N} is not finitely generated, but its action is proper and cobounded.

138 Chapter 1.8 Group Actions and Quasi-Isometries
8.13 Exercise. Consider the l2 metric on the space T~[N §' defined in (5.5(2)). Let Τ
be as above. Prove that Т\12{Щ is isometric to the subspace of Y[N §' consisting of
sequences a finite distance from the constant sequence 0.
Quasi-Isometries
Our next goal is to explain the comments that we made at the beginning of this chapter
concerning the relationship between length spaces and the geometry of groups which
act properly and cocompactly by isometries on them.
One of the main themes of this book is the large scale geometry of metric spaces.
In this context one needs a language that will lend precision to observations such as the
following: if one places a dot at each integer point along a line in the Euclidean plane,
then the line and the set of dots become indistinguishable when viewed from afar,
whereas the line and the plane remain visibly distinct. One makes this observation
precise by saying that the set of dots is quasi-isometric to the line whereas the line
is not quasi-isometric to the plane.
8.14 Definition of Quasi-Isometry. Let (Xi,di) and (X2, d2) be metric spaces. A
(not necessarily continuous) map/ : X\ —> X2 is called a (λ, ε)-quasi-isometric
embedding if there exist constants λ > 1 and ε > 0 such that for all x,y &X\
~dx{x,y) - ε < d2(f(x),f(y)) < Xdx{x,y) + ε.
λ.
If, in addition, there exists a constant С > 0 such that every point of X2 lies in the
C-neighbourhood of the image of/, then/ is called a (λ, 8)-quasi-isometry. When
such a map exists, X[ and X2 are said to be quasi-isometric.
8.15 Remark. Some authors refer to the maps defined above as coarse quasi-
isometries. In general the term "coarse" is used to describe properties that are
insensitive to all finite perturbations — when studying such properties there is little
reason to care about whether maps are continuous or not.
8.16 Exercises
(1) Show that if there exists a (λ, e)-quasi-isometry/ : X\ -> X2 then there exists
a (λ', e')-quasi-isometry/' : X2 -> X\ (for some λ' and ε') and a constant к > 0 such
that d(ff'(x/), x1) < к and d(f'f(x), x) < к for all x1 e X2 and all χ e Χι. Such a map
/' is called a quasi-inverse for/.
(2) Prove that the composition of a (λ, e)-quasi-isometric embedding and
a (λ', e')-quasi-isometric embedding is a (μ, i>)-quasi-isometric embedding with
μ = λλ' and ν = λ'ε + ε'. Deduce that the composition of any two quasi-isometries
is a quasi-isometry.
(3) Let X a be metric space. We consider two maps/, g : X —> X to be equivalent,
and write/ ~ g, if sup^ d(f(x), g{x)) is finite. Let [Я denote the ~ equivalence class

Quasi-Isometries 139
of/. The quasi-isometry group ofX, denoted QI{X), is the set of ~ equivalence
classes of quasi-isometries X —> X. Show that composition of maps induces a group
structure on X, and that any quasi-isometry φ : X -» X' induces an isomorphism
φ, : QL{X) -> QL{X').
(4) Give an example of an unbounded metric space X for which the natural map
Isom(X) —> QJ(X) is an isomorphism; give an example where Isom(X) is trivial but
QJ(X) is infinite; and give an example where QJ(X) is trivial but Isom(X) is infinite;
(In the first two cases it suffices to consider discrete subspaces of Μ — but there are
more interesting examples.)
8.17 Examples
(1) A metric space is quasi-isometric to a one-point space if and only it has finite
diameter. More generally, the inclusion Υ <^-> X of a subset Υ of a metric space X
is a quasi-isometry if and only if Υ is quasi-dense in X, i.e. there exists a constant
С > 0 such that every point of X lies in the C—neighbourhood of some point of Y.
For example, the natural inclusion Ζ <^-> К is a quasi-isometry.
(2) Every finitely generated group is a metric space, well-defined up to quasi-
isometry.15 Given a group Г with generating set Л, the first step in realizing the
geometry of the group is to give Г the word metric associated to A: this is the metric
obtained by defining йд(уь γι) to be the shortest word in the pre-image of у~'у2
under the natural projection F(A) ~» Γ. The action of Γ on itself by left multiplication
gives an embedding Γ —> Isom(r, dj). (The action of γο € Γ by right multiplication
У ι-> yyo is an isometry only if γο lies in the centre of Γ.)
The word metrics associated to different finite generating sets A and A' of Γ are
Lipschitz equivalent, i.e. there exists λ > 0 such that ^rf^(/b Yi) < ^.А'(Уь Уг) <
Хйд(уь уг) for all γι, γι е Г. One sees this by expressing the elements of A as
words in the generators A' and vice versa — the constant λ is the length of the longest
word in the dictionary of translation.
If two metrics d and ά on a set X are Lipschitz equivalent then the identity
map id : (X, d) -> (X, d!) is a quasi-isometry. Thus statements such as "the finitely
generated group Γ is quasi-isometric to the metric space Y" or "the finitely generated
groups Γι and Γ2 are quasi-isometric" are unambiguous, and we shall speak like this
often.
(3) The Cayley graph Сд(Г) of a group Г with respect to a generating set A was
defined in (1.10); it is a metric graph with edges of length one. The induced metric
on its vertex set Г is exactly the word metric й?_д described in (2). As a special case
of (1) we see that (Г, dj) is quasi-isometric to Сд(Г) and that the Cayley graphs
associated to different finite generating sets for Г are quasi-isometric.
15 Mikhael Gromov has been extremely successful in promoting the study of groups as
geometric objects [Gro84, 87, 93]. This approach to group theory (which we shall consider at
some length in Part III) lacked prominence for much of this century but was inherent in the
seminal work of Max Dehn [Dehn87]. See [ChaM82] for historical details up to 1980.

140 Chapter 1.8 Group Actions and Quasi-Isometries
One can also think of the inclusion Г <^-> С^(Г) as being given by the natural
action of Г on С^(Г) — it is у м> γ.I. From this point of view, the fact that
Г ^-» С^(Г) is a quasi-isometry provides a simple illustration of (8.19).
Quasi-Isometries Arising from Group Actions
8.18 Lemma. Let (X, d) be a metric space. Let Υ be a group with finite generating
set A and associated word metric dj^. If Υ acts by isometries on X, then for every
choice of basepoint xq e X there exists a constant μ > 0 such that d(y.xo, y'.xo) <
μ d^iy, y')far all γ, γ' e Г.
Proof. Let μ = max{d(xo, a.x0) | a e Λ U A~1}. If ά_&{γ, γ') = η then
γ~ιγ' — а\й2 .. .ап for some α, е .4 U A~l. Let g,- = а^г · · -Щ- By the
triangle inequality, d(y.x0, y'.xq) = d(x0, у~1у'.х0) < d(x0, gi.xo) + d{g\.x0, gi.xo) +
\-d(g„-i-xo, y~V*o)·Andforeachiwehaved,(g,_1.x0,gi-xo) — d(x0, g^gi-xo)
= d(xo, aj.xo) < μ. Π
The following result was discovered by the Russian school in the nineteen fifties
(see [Ef53],[Sv55]). It was rediscovered by John Milnor some years later [Mil68,
Lemma 2].
8.19 Proposition (The Svarc-Milnor Lemma). Let X be a length space. If Υ acts
properly and cocompactly by isometries on X, then Υ is finitely generated and for
any choice of basepoint xq e X, the map γ ы> γ.χο is a quasi-isometry.
z0 = c(0) <4)
Fig. 8.2 The Svarc-Milnor Lemma
Proof. Let С С X be a compact set with Y.C = X. We choose x0 e X and D > 0
such that С С В(х0, D/Ъ) and let A = {y e Γ | γ.Β(χ0, D)(lB(x0, D) φ 0}. Because
X is proper (8.4) and the action of Γ is proper, A is finite.
In (8.10) we showed that A generates Γ. (A second proof is given below.) Let ад
be the word metric on Г associated to A. Lemma 8.18 yields a constant μ such that
d(y.xo, y'.xo) < μ d^iy, y') for all у, у' е Г, so it only remains to bound dj\_(y, γ')
in terms of d(y.xo, y'.xo). Because both metrics are Γ-invariant, it is enough to
compare йд(1, у) and d(xo, y.xo).
+
c(Vi) c(l)

Quasi-Isometries 141
Given у e Γ and a path с : [0, 1 ] —> X of finite length with c(0) = xo and
c(l) = y.jco, we can choose a partition 0 = to < t\ < ... < t„ = 1 of [0, 1] such
that d(c(t;), c(ii+i)) < D/3 for all i. For each i, we choose an element у- e Г such
that d(c(tj), Yi.xo) < D/3; choose γο = 1 and γη — γ. Then, for i = 1, ... , η we
have d(yi.xo, γ;_\.χο) < D and hence α, := /,-ΐ}// € A
У = yo(y0~'yi) · · · {γ^2Υη-ύ{Υ^-ΧΥη) = αχ... α„_ια„.
Because X is a length space, we can choose the curve с considered above to have
length less than d(xo, y.xo) + 1. If we take as coarse a partition 0 = ίο < ?ι < · · · <
ί„ = 1 as is possible with ii(c(i,), c(iI+i)) < D/3, then η < (d(x0, y.jE0)+l)(3/D)+l.
Since у can be expressed as a word of length n, we get йд(1, у) < (d(xo, γ.χο) +
1)(3/D)+1. D
8.20 Exercises
(1) Let φ : Γι —> Гг be a homomorphism between finitely generated groups.
Show that if ^ is a quasi-isometric embedding then ker($) is finite, and that ф is a
quasi-isometry if and only if ker($) and Гг/1т(^) are both finite.
(2) Let T„ denote the connected metric tree in which every vertex has valence η
and every edge has length 1. Prove that if n, m > 3 then T„ is quasi-isometric to Tm.
(Hint: Rather than doing this directly, you can show that every finitely generated
free group occurs as a subgroup of finite index in the free group of rank 2; the
Cayley graph of the free group of rank r is Тгг, so the case where η and m are even
then follows from (1) and the fact that finitely generated groups are quasi-isometric
to their Cayley graphs. In the case where η is odd, consider the Cayley graph of
G2,„ =1*2* Z„; the kernel of the abelianization map G2,„ η· Ζ2 χ Z„ is free and
has finite index.)
8.21 Remark (Commensurability versus Quasi-isometry). Two groups are said to be
commensurable if they contain isomorphic subgroups of finite index.
Commensurable groups are quasi-isometric (8.20(1)), but quasi-isometric groups need not be
commensurable, as we shall now discuss.
First we note that if one uses the Svarc-Milnor lemma to show that finitely
generated groups are quasi-isometric by getting them to act cocompactly by isometries
on the same length space, then in general it can be difficult to decide if the groups are
commensurable. A setting in which one can decide quite easily is that of semi-direct
products of the form Ί? хф Ъ, which were considered in [BriG96].
Each such group acts properly and cocompactly by isometries on one of the
3-dimensional geometries E3, Nil or Sol (see [Sco83] or [Thu97]). These three
possibilities are mutually exclusive and determine the quasi-isometry type of the
group. Ί? Χψ Ζ acts properly and cocompactly by isometries on E3 if ф е GL(2, Z)
has finite order; it acts properly and cocompactly by isometries on Nil if φ has infinite
order and its eigenvalues have absolute value 1; and it acts properly and cocompactly
by isometries on Sol if φ has an eigenvalue λ with |λ| > 1.

142 Chapter 1.8 Group Actions and Quasi-Isometries
In each of the cases E3 and Nil, all of the groups concerned have isomorphic
subgroups of finite index. But in the case of Sol, the groups Ζ2 χφΖ and Ζ2 χφ> Ζ
will have isomorphic subgroups of finite index if and only if the corresponding
eigenvalues λ, λ' have a common power (see [BriG96]). For arbitrary η > 0, the
groups Ζ" χ Ζ have not been classified up to quasi-isometry (although there has
been some progress [BriG96]).
Interesting examples of quasi-isometric groups that are not commensurable also
arise in the study of cocompact arithmetic lattices in Lie groups. In this setting there
are many interesting invariants of commensurability — see [NeRe92] for example.
Further examples of groups that are quasi-isometric but not commensurable are
described in (ΠΙ.Γ.4.23).
Quasi-Geodesics
Let Γ and X be as in the (8.19). One would like to know to what extent the information
encoded in the geometry of geodesies in X is transmitted to Γ by the quasi-isometry
X -> Γ given by (8.19). This leads us to the following definition.
8.22 Definition of Quasi-Geodesics. Α (λ, s)-quasi-geodesic in a metric space X is
a (λ, e)-quasi-isometric embedding с : / —> X, where / is an interval of the real line
(bounded or unbounded) or else the intersection of Ζ with such an interval. More
explicitly,
- \t - t'\ - ε < d(c(t), c(t')) < λ \t - t'\ + ε
λ.
for all t, t' e /. If / = [a, b] then c{a) and c{b) are called the endpoints of c. If
/ = [0, oo) then с is called a quasi-geodesic ray.
Quasi-geodesics will play an important role in Chapter III.H. In particular we
shall see that quasi-geodesics in hyperbolic spaces such as W follow geodesies
closely. In general geodesic spaces this is far from true, as the following example
indicates.
8.23 Exercise. Prove that the map с : [0, oo) —> E2 given in polar coordinates by
t ы> (t, log(l + t)) is a quasi-geodesic ray.
Some Invariants of Quasi-isometry
In the next few sections we shall describe some invariants of quasi-isometry. We
begin with a result that illustrates the fact that quasi-isometries preserve more
algebraic structure than one might at first expect. We shall see further examples of this
phenomenon in Part III.

Some Invariants of Quasi-Isometry 143
8.24 Proposition. Let Y\ and Γ2 be groups with finite generating sets Λι and Λ2. If
Γι and Γ 2 are quasi-isometric and Γ 2 has a finite presentation (Λ2 \ ΊΙ2), then Y\
has a finite presentation {A\ \ 1Z\).
Proof. The strategy of the proof is as follows. We shall build combinatorial 2-
complexes K\ and K2 by attaching a locally-finite collection of 2-cells to the Cayley
graphs of Γι and Γ2; it will be clear that K2 is simply connected and our goal will be
to prove that K\ is simply connected. Because Γ] and Γ2 are quasi-isometric, there
is a quasi-isometry/ : K\ —> K2 with quasi-inverse/' : K2 —> K\. In order to show
that K\ is simply connected, we use/ to map loops in K\ to loops into K2, we choose
a filling (i.e. homotopy disc) in K2 and map it back to K\ using the quasi-inverse
/' : K2 -> #1; a suitable approximation to the resulting (non-continuous) map of a
disc into K\ yields a genuine filling for the original loop in K\.
Let i e {1, 2}. We write C,- to denote the Cayley graph Сд(Г,). Let ρ be the
length of the longest word in TZ2. According to (8.9), the 2-complex K2 obtained by
attaching 2-cells to C2 along all edge-loops of length < ρ is simply connected.
Let/ : Tt —» Γ2 and/' : Γ2 —> Γι be (λ, e)-quasi-isometries and let μ > 0
be such that d(f'f(y), γ) < μ for every γ e Γι. Let m = max{X, ε, μ, ρ} and let
Μ = 3(3m2 + 5m + 1). Let Xi be the complex obtained by attaching a 2-cell to
C\ along each edge loop of length < M, Let I be an edge-loop in C\ that visits the
vertices g\, .. .,gn in that order. We view £ as a map I : 3D —> C)5 where D is a
2-dimensional disc. We will be done if we can show that I has a continuous extension
i:D^K{ (see 8.9).
Let i>i,..., vn be the inverse images of the g,, arranged in cyclic order around
3D, and let φ : 3D —> C2 be a map that sends the edge (subarc) bounded by {υ,, υ;+ι}
onto a geodesic in C2 connecting/(g;) to/(g;+i), where the indices are taken mod n.
Because K2 is simply connected, we can extend φ to a continuous map φ : D —> #2-
We associate to each point χ e D an element ул е Г2 such that either φ(χ) = γχ
or else γχ is a vertex of the edge or open 2-cell in which φ(χ) lies. We specify that
yVi =f(gi). Notice that because φ is continuous, d(yXt, γΧ2) < ρ for all sufficiently
close x\,X2 e D. Notice also that ά{φ{χ), γχ) < 1/2 for every χ e 3D.
We fix a triangulation Τ of D such that й?(уЛ|, уЛ2) < ρ if х\,хг are adjacent
vertices. We require that v\,..., v„ be vertices of T.
Define £|3D = I and £(jc) =f'(yx) for every vertex jc of Τ that lies in the interior
of D. We claim that I sends each pair of adjacent vertices x\, X2 € Τ to elements of
Γι that are a distance at most M/3 apart in Ci. Once this claim is proved, we can
extend I across the edges of Τ be sending each edge to a geodesic in C\, and since
every circuit of length < Μ in C\ bounds a disc in K\ (by construction), we can then
extend I continuously across the faces of T.
We must show that d(l(xi), £(jc2)) < M/3. This is obvious except in the case
where χι is in the interior of D and X2 e 3D, with X2 between υ, and υ;+ι say. In that
case,

144 Chapter 1.8 Group Actions and Quasi-Isometries
d(i(Xl),i(x2))= d(f'(yXl),i(x2))
< d(f'{Yxi)J\Yxi)) + ά(/'(γΧ2),/φ(χ2)) + ά(/'φ(υ;),/φ(χ2))
+ а<Гф(ы), £(«,)) + d(l(Vi), l{x2)).
< (λρ + ε) + (λ/2 + ε) + [каШы), φ(υι+ι)) + ε] + d(f'f(gi), gl) + 1
< (λρ + ε) + (λ/2 + ε) + [λ(λ + ε) + ε] + μ + 1
< Ът1 + 5т + 1 < Μ/3.
D
8.25 Remarks. Let Χ and F be metric spaces. X is said to be a quasi-retract of F if
there exist constants λ, ε, С > 0, and maps/ : Χ -> F, and/' : F -» X such that
difbilfbi)) < Xd{yuyi)+Siorz\\yuy2 e r.andd^O./fe)) < λ^,,χ2)+ε
and d(f'f(xi), x\) < С for all x\,x2 e X. The preceding proof actually shows that
any quasi-retract of a finitely presented group is finitely presented.
Juan Alonso [Alo90] pointed out that by keeping track of the number of faces
in the triangulations that occur in the proof of (8.24), one can deduce that having
a solvable word problem is an invariant of quasi-isometry among finitely presented
groups.
8.26 Exercise. Let Г be a group with a finite generating set Л. Show that if Г is
finitely presentable, and if Г = (Л \ 1Z) is any presentation for Г, then there is a
finite subset W С 11 such that ((W)) = ker(F(.4) -» Г).
The Ends of a Space
Recall that a map / : X —> F between topological spaces is said to be proper if
/~'(C) с X is compact whenever С С F is compact. The following definition is due
to Freudenthal [Fr31].
8.27 Definition of Ends. Let X be a topological space. A ray in X is a map r :
[0, oo) —» X. Let r\, Г2 : [0, oo) —> X be proper rays. r\ and r2 are said to converge
to the same end if for every compact С с X there exists N e N such that r\ [Ν, οο)
and r2[N, oo) are contained in the same path component of Χ χ С. This defines an
equivalence relation on continuous proper rays; the equivalence class of r is denoted
end(f) and the set of equivalence classes is denoted Ends(X). If the cardinality of
Ends(X) is m, then X is said to have m ends.
Convergence of ends is defined by: end(r„) —> end(f) if and only if for every
compact set С С X there exists a sequence of integers Nn such that r„[Nn, oo) and
r[Nn, oo) lie in the same path component of X \ С whenever η is sufficiently large.
We define a topology on Ends(X) by describing its closed sets. A subset В с Ends(X)
is defined to be closed if it satisfies the following condition: if end(r„) e В for all
η e N, then end(r„) —» end(f) implies end(f) e B.

The Ends of a Space 145
Let X be a metric space. By definition, a k-path connecting χ to у is a finite
sequence of points χ = x\,...,x„ = у in X such that d(xj, x,-+i) < fc for i =
1,...,и- 1.
8.28 Lemma. Let X be a proper geodesic space and let к > 0. Let r\ and r2 be
proper rays in X. Let GXo(X) denote the set of geodesic rays issuing from xq e X.
Then:
(1) end(r\) = end(r-i) if and only if for every R > 0 there exists Τ > 0 such that
r\ (t) can be connected to r2(i) by a k-path in X \ B(xq, R) whenever t > T.
(2) The natural map Gx0(X) —> Ends(X) is surjective.
(3) Let r e GXo(X) and let η > 0 be an integer. Let Vn с Gx0(X) be the set of
proper rays r1 such that r(n, oo) and /(n, oo) lie in the same path component
ofX ν B(xo, n). The sets Vn = {endij1) \ V„] form a fundamental system of
neighbourhoods for end(r) in Ends(X).
Proof. Every compact subset of X is contained in an open ball about xq and vice
versa, so one may replace compact sets by open balls B{xQ, R) in the definition of
Ends(X). Part (3) follows immediately from (2) and this observation. Part (1) also
follows from this observation, because if x\,..., xn is a к-path connecting x\ to x„ in
Χ ν B(xq, R + k), then the concatenation of any choice of geodesies [jc,, jc;+i] gives
a continuous path from x\ to x„ in Χ χ B(xq, R).
It remains to prove (2). Let r : [0, oo) —» X be a proper ray. Let c„ : [0, d„] —> X
be a geodesic joining xo to г(и); extend c„ to be constant on [dn, oo). Because X is
proper, the Arzela-Ascoli theorem (3.10) furnishes us with a subsequence of the c„
converging to a geodesic ray с : [0, oo) —> X, and it is clear that end(c) = end(f).
D
8.29 Proposition. IfX\ andXi are proper geodesic spaces, every quasi-isometry
f : X\ —> X2 induces a homeomorphismfr : Ends(Xi) —> Ends(Ay·
77ге ma/? QI(X!) —> Homeo(Ends(X!)) given byf м> /f ii a homomorphism.
Proof. Let r be a geodesic ray in X\ and let/,„(r) be a ray in Хг obtained by
concatenating some choice of geodesic segments [fr(n),fr(n + 1)], η e N. Because/ is a
(λ, e)-quasi-isometry, this is a proper ray. It is clear that endif^r)) is independent of
the choice of the geodesic segments \fr(n),fr(n +1)].
Define/ε : Ends(Xi) —> Ends(X2) by end(f) м> endif^r)) for every geodesic
ray r in Xi. The image under/ of any fc-path in Xi is a {Xk + e)-pafh in X2, so 8.28(1)
ensures that fs is well-defined on equivalence classes, and that it is continuous.
8.28(2) ensures that/£ is defined on the whole of Ends(Xi).
It is clear that if /' : X2 —> Xi and / : Xi —» X2 are quasi-isometries then
f'efs = (f'f)e, and if/' : X2 -> X\ is a quasi-inverse for/, fhen/^/g = (/'/)£ is the
identity map on Ends(Xi). D
In the light of 8.17(2) and the preceding result, the following definition of Ends(r)
is unambiguous.

146 Chapter 1.8 Group Actions and Quasi-Isometries
8.30 Definition (The Ends of a Group). Let Γ be a group and let С be its Cayley
graph with respect to a finite generating set. We define Ends(r) := Ends(C).
8.31 Exercise. Let Tn be a metric tree in which all vertices have valence η > 3 and
all edges have length one. Construct a homeomorphism from Ends(r„) to the Cantor
set.
(Hint: The Cantor set may be thought of as the subspace of [0, 1] consisting
of those numbers whose ternary expansion consists only of zeros and twos. Embed
Γ3 in the plane (topologically), fix a base point, and encode the trajectory of each
geodesic ray as a sequence of zeros and twos describing when the ray turns left and
right; regard this sequence as a ternary expansion. Use (8.20) and (8.29) to pass from
η = 3 to the general case.)
Amalgamated free products and HNN extensions, which appear in part (5) of the
following theorem, will be discussed in more detail in section (ΙΙΙ.Γ.6).
8.32 Theorem. Let Γ be a finitely generated group.
(1) Γ has 0, 1, 2 or infinitely many ends.
(2) Γ has 0 ends if and only if it is finite.
(3) Γ has 2 ends if and only if it contains Ъ as a subgroup of finite index.
(4) Ends(r) is compact. If it is infinite then it is uncountable and each of its points
is an accumulation point.
(5) Г has infinitely many ends if and only if Υ can be expressed as an amalgamated
free product A *c В or HNN extension A*c with С finite, \A/C\ > 3 and
\B/C\ > 2.
This result shows, for example, that if a geodesic space X has three ends, or if
the set of ends is countable, then X does not admit a proper cocompact action by a
group of isometries.
The first four parts of this theorem are due to Hopf [Ho43]. Part (5) is due to
Stallings [St68]; its proof is beyond the scope of the ideas presented here. Part (2) is
trivial. We shall prove parts (1) and (4). We leave (3) as an exercise (8.34).
Proof of 8.32(1) and (4) We fix a finite generating set for Г and work with the
corresponding Cayley graph C. The action of Г on itself by left multiplication extends
to an action by isometries on C, giving a homomorphism Г —> Homeo(Ends(C)) as
in(8.29). Let Η be the kernel of this map. If Ends(C) is finite, then Η has finite index
ίηΓ.
We prove (1) by arguing the contrapositive. Suppose that С has finitely many ends,
and let e0, e\, ej. be three distinct ones. We fix two geodesic rays r\, ri : [0, oo) —» X
with n(0) = r2(0) equal to the identity vertex 1, and end{rt) — е-,. Because Η has
finite index in Γ, there is a constant μ such that every vertex of С lies in the μ-
neighbourhood of H. It follows that there is a proper ray ro : [0, oo) —> С with
end(ro) = eo, d(ro(n), 1) > n, and ro(ri) e Η for every η e N. Let y„ = ro(ri).

The Ends of a Space 147
We fix ρ > 0 such that Г[([р, oo)), r2([p, сю)), and r0([p, oo)) lie in different
path components of С \ 5(1, p). lit, f > 2p, then d(r\(t), r2(f)) > 2p, because any
path joining r\{f) to r2{t') must pass through 5(1, p).
Since Η acts trivially on Ends(C), we have εηά(γ„.η) = end{r{) for i = 1,2. Let
η > Ър. Then y„.r,(0) = y„ lies in a different path component to r,([p, oo)), so уп.Г{
must pass through 5(1, p). Thus yn.ri(t) e 5(1, p) and уп.г2{^) e 5(1, p) for some
t, f > 2p. But y„ is an isometry, so this implies d(r\(t), r2(f)) < 2p, which is a
contradiction.
We now prove (4). Given a sequence of ends e„, we choose geodesic rays rn :
[0, oo) —> С with r„(0) = 1 and end(r„) = e„. The Arzela-Ascoli theorem (or a
simple finiteness argument) shows that a subsequence of these rays converges on
compact subsets to some geodesic ray r; it follows that the corresponding sequence
of ends converges to end(r). Thus Ends(C) is sequentially compact, and since it
satisfies the first axiom of countability (8.28(3), it is compact. It is clear that Ends(C)
is Hausdorff, and a compact Hausdorff space in which every point is an accumulation
point is uncountable (exercise). Thus it suffices to prove that if Ends(C) is infinite
then every e e Ends(C) is an accumulation point.
Given e\, e2 e Ends(C), let D = D(e\, e2) be the maximum integer such that
r\ ([D, oo)) and r2([D, oo)) lie in the same path component of С \ 5(1, D) for some
(hence all) geodesic rays with end{rt) = e, and r,-(0) = 1. Note that en -» e if and
only if D(en, e) -> oo as η -> oo.
Fix e0 e Ends(C) and a geodesic ray rQ with ro(0) = 1 and end(r0) = eo-
Let y„ = ro(ri). We shall construct a sequence of ends e"1 such that em φ eo and
D(em, eo) —> oo as m —» oo. Let e\, e2 be distinct ends, neither of which is eo
and let r, be a geodesic ray with r,(0) = 1 and end(rf) = e, for г = 1,2. Let
Μ = maxD(e;, ej), i,j = 0, 1, 2. If we fix ρ > Μ, then by arguing as in (1) we see
that for large η at most one of the rays γη.Τι can pass through 5(1, p). To each of the
other two rays γη.η we apply the Arzela-Ascoli theorem as in (8.28) to construct a
geodesic ray ή with rj(0) = 1 and ej := end(rj) = εηά{γη.η). Since r0 and ή have
terminal segments in the same path component of С \ 5(1, p), we have D(eo, e'j) > p.
The e': are distinct, so at least one of them is not equal to eo'- rename this end e1.
We now repeat the above argument with e\ replaced by e1 and call the ray
constructed in the argument e2. We replace e1 be e2 and repeat again. At each iteration
of this process the integer Μ is increased. Thus we obtain the desired sequence of
ends em with em φ eo and D(em, eo) —> oo as m —> oo. D
We note a consequence of 8.32(5):
8.33 Corollary. If two finitely generated groups are quasi-isometric and one splits
as an amalgamated free product or HNN extension of the type described in 8.32(5),
then so does the other.
8.34 Exercise. Prove 8.32(3). (Hint: Choose a ball 5 about the identity that separates
the ends of the Cayley graph C, and choose γ e Γ \ 5 whose action on Ends(C) is
trivial. Check that γ has infinite order by considering y".r, where r is a geodesic ray

148 Chapter 18 Group Actions and Quasi-Isometries
r starting at γ that does not pass through B. Finally, argue that the sequences η м> у"
and η ы> γ ~п tend to different ends of C, and that any point of С is within a bounded
distance of (γ).)
Growth and Rigidity
The material that follows is of a different nature to what went before: greater
background is assumed, non-trivial facts are quoted without proof, and some of the
exercises are hard. The material in this section is included for the information of
the reader; our purpose in presenting it is to indicate that there are many situations
in geometry where it is useful to employ techniques involving quasi-isometries of
groups.
There is an extensive literature on the growth of groups. Much of it focuses on
the search for closed formulae which describe the number of vertices in the ball of
radius η about the identity in the Cayley graph of a group (see [GrH97]), but the
coarser aspects of the theory are also interesting.
8.35 Definition. Let Γ be a group with generating set A. Let β^ίη) be the number
of vertices in the closed ball of radius η about 1 e С^(Г). The growth function of Г
with respect to Л is η ы> β^(η).
One says that Γ has polynomial growth of degree < d if there exists a constant k
such that βΑ(η) < knd for all η e N.
8.36 Exercises
(1) Prove that having polynomial growth of degree < d is an invariant of quasi-
isometry. Prove that a group has growth of degree < 1 if and only if it is finite or
contains Ζ as a subgroup of finite index (cf. 8.40).
(2) Show that if Γι and Гг are quasi-isometric then with respect to any finite
generating sets Λι for Г] and Ai for Г2 one has /Ы, (и) ^ βΑ2(η), where ~ is the
equivalence relation on functions N —> M. defined as follows: / «s g if and only if
/ =^ g and g =4 f, where / =^ g if and only if there exists a constant к > 0 such
that/(и) < kg(kn + k) + к for all η e N. (This observation allows one to drop the
subscript A or to write βτ{η) when discussing the ss properties of /Ы(и)·)
(3) Prove that if Г contains a non-abelian free group, then the growth function
of Г satisfies 2" =^ βτ{η). More generally, prove that if Я С Г is finitely generated
then βΗ(ή) =4 βΓ(η).
(4) The growth of Zm with respect to any finite generating set is polynomial of
degree m.
(5) If η > 2, then any group Γ which acts properly and cocompactly by isometries
on W has exponential growth. (Hint: One way to see this is to show that the number
of disjoint balls of radius 1 that one can fit into a ball or radius r in Ш" grows as an

Growth and Rigidity 149
exponential function of r, and this property is inherited by the Cayley graph of Γ.
One can also argue that Γ must have a non-abelian free subgroup.)
8.37 Examples
(1) R. Grigorchuk [Gri83] has constructed finitely generated groups Γ for which
2"°' =^ β(η) =4 2""\ where 0 < a, < a2 < 1.
(2) Recall that the lower central series of a group Γ is (Γ„), where Го = Г
and Γ„ = [Γ, Γ„_ι] is the subgroup generated by и-fold commutators. Γ is said
to be nilpotent if its lower central series terminates in a finite number of steps, i.e.
Гс = {1} for some с е N. All finitely generated nilpotent groups have polynomial
growth — see [Dix60], [ΜΪ168], [Wo64]. Guivarc'h [Gui70, 73], Bass [Bass72] and
others calculated the degree of growth: β(ή) « nd where
с
d=J^(i+l)rkQri/ri+l.
;=o
Gromov proved the following remarkable converse to (8.37(2)) — see [Gro81b]
and [DW84].
8.38 Theorem. Let Г be a finitely generated group. If Υ has polynomial growth then
Γ contains a nilpotent subgroup of finite index.
In order to prove this theorem one must use the hypothesis on growth to construct
a subgroup of finite index Δ С Г for which there is a surjection Δ -» Ζ. Once one
has constructed such a map, there is an obvious induction on the degree of polynomial
growth: the degree of polynomial growth of Δ is the same as that of Γ, and it is not
hard to show that the kernel of any map Δ -» Ζ is finitely generated and the degree
of its polynomial growth is less than that of Γ; by induction the kernel has a nilpotent
subgroup of finite index and hence so does Γ.
In order to construct a map to Z, Gromov proves that a subsequence of the metric
spaces X„ = (Γ, ^d) converges (in the pointed Gromov-Hausdorff sense (5.44))
to a complete, connected, locally compact, finite-dimensional metric space Хю. А
classical theorem of Montgomery and Zippin [MoZ55] implies that the isometry
group of Хм is a Lie group G with finitely many connected components. Using this
theorem, and classical results about the subgroups of Lie groups, Gromov analyses
the action of Г on XM to produce the desired map Δ -» Ζ.
8.39 Corollary. If a finitely generated group Γ is quasi-isometric to a nilpotent group
then Γ contains a nilpotent subgroup of finite index.
A group Γ is called polycyclic if it has a sequence of subgroups {1} = Го С
Г! С ■ · ■ С Г„ = Г such that each Г,- is normal in Γί+1 and Γ,+1/Γ, is a cyclic
group. The number of factors Γ,+ι /Γ, isomorphic to Ζ is called the Hirsch length of

150 Chapter 1.8 Group Actions and Quasi-Isometries
Γ. In [BriG96] Bridson and Gersten proved that the Hirsch length is an invariant of
quasi-isometry for polycyclic groups. 16
Every finitely generated nilpotent group N is polycyclic. Arguing by induction on
the Hirsch length, one can show that the degree of polynomial growth of A' is equal
to the Hirsch length if and only if N does not contain a subgroup of the form Ζ χ φ Ζ
where φ e GL(«, Z) is an element of infinite order, cf. (II.7.16)and(III.r.4.17). And
N contains such a subgroup if and only if it is not virtually abelian. By combining
these facts with Gromov's Theorem (8.38) one gets the following result.
8.40 Theorem. If a finitely generated group is quasi-isometric to Z" then it contains
Z" as a subgroup of finite index.
This theorem is contained in a more general result of P. Pansu concerning quasi-
isometries between nilpotent groups [Pan83]. The proof that we have sketched is
based on that of Bridson and Gersten [BriG96].
8.41 Quasi-Isometric Rigidity of Lattices. In recent years a number of results have
been proved concerning the quasi-isometric rigidity of lattices in semi-simple Lie
groups [Es98], [EsF97], [K1L97], [Schw95]. See [Fa97] for a survey and further
references. There are rigidity results of two types in these works: in the first type
of result one shows that non-uniform lattices in certain Lie groups (e.g. SO(n, 1) in
[Schw95]) are quasi-isometric if and only if they are commensurable; in the second
type of result one shows that if an abstract finitely generated group G is quasi-
isometric to a group in the given class of lattices (e.g. irreducible lattices in higher
rank groups in [K1L97]), then there is a short exact sequence 1 —» F —» G —>· Γ —>· 1,
where F is finite and Γ is a lattice of the given type.
Both types of rigidity results have also been proved for classes of groups other
than lattices (e.g. [FaMo98] and [KaL95].)
Quasi-Isometries of the Model Spaces
We consider which of the model spaces are quasi-isometric.
8.42 Proposition.
(1) M" is quasi-isometric to a point if and only if к > 0.
(2) Ε" is quasi-isometric to E'" if and only if η = т.
(3) H" is quasi-isometric to W if and only if η = т.
(4) Ε" is not quasi-isometric to Mm ifn,m > 2.
16 More generally, Gersten [Ger93a] proved that cohomological dimension is an invariant of
quasi-isometry among groups that have a subgroup of finite index with a compact Eilenberg-
Mac Lane space.

Quasi-Isometries of the Model Spaces 151
Remark. It is clear that W is quasi-isometric to M" for all к < 0.
Proof. In the light of the considerations of growth described in the Exercises (8.36),
the only non-trivial point that remains is to prove that W and Mm are not quasi-
isometric if η φ т. This is explained properly in (III.H.3) in terms of the boundaries
at infinity of the spaces, but it can also be proved using the elementary argument
outlined in the following exercise.
8.43 Exercise (Coarse Filling).
(1) Let k be a positive constant. A map between metric spaces/ : X —> Υ is said
to be ^-continuous if for every χ e X there exists ν > 0 such that d(f(xi ),f{xi)) < k
whenever d(x, jci) < ν and d(x, x2) < v.
Let D"+1 be a closed unit ball in E" and let S" = 3D"+1. Show that the following
property (but not the choice of constants) is an invariant of quasi-isometry of metric
spaces X. Fix η e N.
(Φ„) For all sufficiently large k > 0 there exist constants λ, λ' > 0 such that for
every R > 0 and every χ e X, every ^-continuous map i"^X\ B(x, R) extends to
a λ/t-continuous map D"+' -4Ϊ\ Β(χ, X'R).
(2) Show that W has property Φ„_, for every i > 2, but does not have property
Φ„-ι.
This exercise provides a simple illustration of the fact that one can use coarse
metric analogues of the basic tools of algebraic topology (in this case higher homo-
topy groups) to describe the large scale geometry of a space (in this case dimension).
For more sophisticated applications of this approach, in particular the development of
cohomology theories that are invariant under quasi-isometry — see [Roe96], [Ger95]
and the survey article of Bloch and Weinberger [BlWe97].
8.44 Exercises
(1) Prove that the natural homomorphism GL(«, Ж) -> QJ(E") is injective.
(2) Prove that the image of the natural homomorphism Isom(E") -> QJ(E")
coincides with that of 0(n).
(3) Prove that the natural map Isom(H") -► 02(11") is injective if η > 2.
(4) Prove that the map in (3) is not surjective. (Hint: One can construct quasi-
isometries of W as follows. Given any C1 -diffeomorphism/ of S"_1 = ЗИ", send 0
to itself and for each geodesic ray с : [0, oo) -> W with c(0) = 0, send c(t) to c'(t),
where c'(0) = 0 and c'(oo) =/(c(oo)).)
The conclusion of exercise (3) is valid for other irreducible symmetric spaces
of non-compact type, but the conclusion of (4) is not. (See the references listed in
(8.41).)

152 Chapter 1.8 Group Actions and Quasi-Isometries
Approximation by Metric Graphs
We close our discussion of quasi-isometries with a result that allows one to replace
(perhaps very complicated) length spaces by simple metric graphs in the same quasi-
isometry class. This can be useful when studying quasi-isometry invariants because,
for example, it allows one to use induction on the length of paths (cf. III.H.2).
8.45 Proposition (Approximation by Graphs). There exist universal constants a and
β such that there is an (α, β)-quasi-isometry from any length space to a metric graph
all of whose edges have length one. (Every vertex of the graph lies in the image of
this map.)
Proof. Let (X, dx) be a length space. Consider a subset V С X with the property that
dx{u, v) > 1/3 for all pairs of distinct points u, ν e V. Using Zorn's lemma, we may
assume that V is not properly contained in any subset of X with the same property. In
particular every point of X is a distance at most 1/3 from V. Let Q be the graph with
vertex set V which has an edge joining two vertices u, ν if and only if dx(u, v) < 1.
We consider the metric graph (Q, dg) obtained by setting all edge lengths equal to
1. We claim that the map X —± Q sending χ to a choice of closest point in V is the
desired quasi-isometry.
Since (V, dg) <^-> (Q, dg) and (V, dx) <^-> (X, dx) are obviously quasi-isometries
with universal parameters, it suffices to show that id : (V, dg) —> (V, dx) is a (3, 1)-
quasi-isometry. By construction this map does not increase distances. Conversely,
given u, υ e V and η > 0 we can find a path of length dx(u, ν) + η connecting и to υ
in X. Along the image of this path we choose N equally spaced points /?,, beginning
with и and ending with v, where N is the least integer greater than 3(dx(u, ν) + η) +1.
For each of these points pt we choose щ e V with dx(pit и,·) < 1/3. For each i we
have dx(uj, m,+i) < 1, and hence щ = ui+i or else и, and ui+\ are connected by an
edge in Q. It follows that dg(u, v) < N — 1 < 3(dx(u, ν) + η) + 1 and hence (since
η > 0 is arbitrary) id : (V, dg) -» (V, dx) is a (3, l)-quasi-isometry. D

Appendix: Combinatorial 2-Complexes 153
8.46 Exercises
(1) Calculate the universal constants in the above proposition.
(2) (Quasi-length spaces), fc-paths were defined prior to (8.28). The length of a
fc-path x\,... ,xn is defined to be Σ^fe*;+i)· X said to be coarsely connected if
there is a constant k such that every pair of points in X can be joined by a k-path, and
X is called a quasi-length space if there exist constants k, λ, ε > 0 such that for every
pair of points x, у e X can be connected by a fc-path of length at most λ d(x, y) + ε.
Prove that the class of quasi-length spaces is closed under quasi-isometry and
generalize (8.45) to include such spaces.
Prove that any metric space which admits a cocompact action by a finitely
generated group is coarsely connected, but that it need not be a quasi-length space. (Hint:
Truncate the metric on a nice space by defining d'(x, y) = max.{d(x, y), 1}.)
Appendix: Combinatorial 2-Complexes
Combinatorial complexes are topological objects with a specified combinatorial
structure. They are defined by a recursion on dimension; the definition of an open cell
is defined by a simultaneous recursion. If K\ and Ki are combinatorial complexes,
then a continuous map K\ —» Ki is said to be combinatorial if its restriction to each
open cell of K\ is a homeomorphism onto an open cell of Кг-
A combinatorial complex of dimension 0 is simply a set with the discrete
topology; each point is an open cell. Having defined (n — l)-dimensional combinatorial
complexes and their open cells, one constructs и-dimensional combinatorial
complexes as follows.
Takethe disjoint union of an (n — l)-dimensional combinatorial complex #("_1)
and a family (ex | λ e Λ) of copies of closed и-dimensional discs. Suppose that for
each λ e Λ a homeomorphism is given from дех (a sphere) to an (n — l)-dimensional
combinatorial complex Sx, and that a combinatorial map Sx —> if("_1) is also given;
let φχ : дех —> #("_1) be the composition of these maps. Define К to be the quotient
ofif("_1)UjJA ex by the equivalence relation generated by t ~ φχ(ί)ίοτΆΪΙλ e Aand
all t e дех. Then K, with the quotient topology, is an и-dimensional combinatorial
complex whose open cells are the (images of) open cells in #("_1) and the interiors
of the ex.
8A.1 Attaching 2-Cells. We are interested primarily in 2-dimensional combinatorial
complexes. Let К be such a complex with 2-cells (ел | λ e Λ). Let X denote the 1-
skeleton of K. The attaching maps φχ : дех —> #(1) are combinatorial loops (i.e. edge
loops) in X, and thus one describes К as the 2-complex obtained by attaching the
2-cells ex to X along the loops φχ.
There is an obvious way to endow any combinatorial 2-complex К with a piece-
wise Euclidean structure: one metrizes the 1-skeleton as a metric graph with edges

154 Chapter 1.8 Group Actions and Quasi-Isometries
of length one, and one metrizes each βχ as a regular Euclidean polygon with sides
of length one. We write KE to denote the piecewise Euclidean complex obtained in
this way.
8A.2 The 2-Complex Associated to a Group Presentation. Associated to any
group presentation (A \ 1Z) one has a 2-complex17 К = K(A: 1Z) that is compact
if and only if the presentation is finite. К has one vertex and it has one edge εα
(oriented and labelled a) for each generator a e A; thus edge loops in the 1-skeleton
of К are in 1-1 correspondence with words in the alphabet A±l: the letter a~l
corresponds to traversing the edge εα in the direction opposite to its orientation,
and the word w = a\...an corresponds to the loop that is the concatenation of the
directed edges a\,...,a„\ one says that w labels this loop. The 2-cells er of К are
indexed by the relations r e 1Z; if r = αϊ... a„ then er is attached along the loop
labelled a\... an. The map that sends the homotopy class of εα to a € Γ gives an
isomorphism π\Κ{Α: 1Ζ) = Γ (by the Seifert-van Kampen theorem).
There is a natural Γ-equivariant identification of the Cayley graph Сд(Г) with
the 1-skeleton of the universal cover of K(A: 11): fix a base vertex υ0 € K(A: 1Z),
identify y.u0 with γ, and identify the edge of Сд(Г) labelled a issuing from γ with
the (directed) edge at γ.υ0 in the pre-image of εα.
8A.3 Exercises
(1) Prove that K\ = K(a, b : aba~lb~[) and K2 = K(a, b, с : abc, bac) are tori
but that Kf and Kf are not isometric.
(2) Following Whitehead [Wh36], we define the star graph of a presentation
{A | 1Z) to be the graph with vertices {(α, η) | a e Α, η = ±1} that has one edge
joining (α, η) to (α', η') for each occurrence of the subwords (αηα'~η ) and (α'~η α~η)
among the relators r e 1Z (the terminal and initial letters of a word are to be counted as
being consecutive, thus a"1 ... α~η contributes to the count for the subword α~ηα'η ).
Show that the link of the vertex in К (A: 1Z) is the star graph of (A\1Z).
(3) Using (2), prove that K(a\,..., a„ : a\... a„flj"' ■ ■ ■ <J,7') is a closed surface
(of genus n/ 2) if and only if n is even.
(4) Using (2), prove that if A is finite and w is a reduced word in which a and
a~l both occur exactly once, for every a e A, then К = K(A: w) can be obtained
by gluing a number of closed surfaces together at a point.
8A.4 Homotopy and van Kampen Diagrams. Let К be a combinatorial complex
with basepoint xq e K^°\ We consider edge paths in K; such a path is the
concatenation of a finite number of directed edges ε, : [0, 1] —> #(1) with ε,·(1) = ε,+ι (0),
or else it is the constant path at a vertex. An edge loop is by definition an edge path
17 Sometimes called the Cayley complex.

Appendix: Combinatorial 2-Complexes 155
that begins and ends at the same vertex. Given an edge path с we write с to denote
the same path with reversed orientation (thus, for example, if ε : [0, 1] —> К is a
directed edge then ε : [0, 1] -> К is the inverse ε(0) = ε(1) and ε(1) = ε(0)).
Two edge paths in К are said to be related by an elementary homotopy if one of
the following conditions holds:
(i) (Backtracking) c' is obtained from с by inserting or deleting a subpath of the
form εε, where ε is a directed edge;
(ii) (Pushing across a 2-cell) с and d can be expressed as concatenations с = otufi
and d = au'β, where u~lu' is the attaching loop of some 2-cell of K. (Note that и
or u' may be the empty path.)
Two edge paths are said to be combinatorially equivalent if they are related by
a finite sequence of elementary homotopies. The set of equivalence classes of paths
that begin and end at x0 is denote nf(K, xo)- Concatenation of paths induces a group
structure on nf(K, x0), where the class of the constant path [x0] serves as the identity.
If с is equivalent to the constant path, then we define Area(c) to be the minimal
number of elementary homotopies of type (ii) that one must apply in conjunction
with elementary homotopies of type (i) to reduce с to the constant path. It is a
standard exercise using the Seifert-van Kampen theorem to show that the natural
map π\{Κ, xo) —> π\ (Κ, χ0) is an isomorphism (see [Mass91], for example).
A van Kampen diagram in К is a combinatorial map Δ —» К where Δ is a
connected, simply connected, planar 2-complex. The boundary cycle of Δ, when
oriented, gives an edge loop с in #(1) that is null-homotopic. Δ is called a van
Kampen diagram for c. A simple induction shows that с can be reduced to the
constant loop by applying elementary homotopies of type (i) together with Area(A)
elementary homotopies of type (ii), where Area(A) is the number of 2-cells in Δ.
Thus Area(c) < Area(A). Conversely, if с can be reduced to the constant path [jco] by
elementary homotopies, then arguing by induction of the total number of elementary
homotopies required, one can show that there is a van Kampen diagram Δ for с
with Area(A) = Area(c). This important observation is due to van Kampen [K32b]
(cf. Strebel's appendix to [GhH90]).
If К is the standard 2-complex of a finite presentation К = K(A: TZ), edge paths
с in К are in natural bijection with words in the generators A±[, and two loops are
related by a homotopy of type (ii) if and only if the corresponding words have the
form w = aufi and w' = au'fi, where r = u'u~l e 1Z±[. This means precisely
that there is an equality w' = (ara~' )w in the free group F(A). It follows that if an
edge loop in К is labelled by the word w, then Area(c), as defined above, is the least
integer Nw for which there is an equality of the form
in F(A), where the x, e F(A) are arbitrary and r, e TZ±l. Thus it is natural to define
Area(w) = Nw. The function/(и) = max{Area(w) : \w\ < n, w = 1 in π\Κ] is
called the Dehnfunction of (A | 71). Dehn functions are widely studied in connection

156 Chapter 1.8 Group Actions and Quasi-Isometries
with questions concerning the complexity of the word problem in finitely presented
groups (cf. Section ΙΠ.Γ.5).
The Dehn functions associated to different finite presentations of the same group
(or more generally quasi-isometric groups) are ~ equivalent in the following sense
(see [Ak>90]): given functions g, h : N -> [0, oo), one writes g < h if there is a
constant К > 0 such that g(n) < К h(Kn) + Kn + K, and one writes g ~ /г if g < h
and h < g.
The following picture shows a van Kampen diagram over (a,b,c | [a, b]) for the
word a2bcb~la~lbac~lba~2b~2. The reader might prove as an exercise that no van
Kampen diagram for this word is homeomorphic to a disc.
6A
b/\
a
6 A
b/\
b/\
a
6 л
с
α α
Fig.8.3 A van Kampen diagram over (a, b, с | [a, b])

Part II. САТ(к) Spaces
In Part I we assembled basic facts about geometric notions such as length, angle,
geodesic etc., and presented various constructions of geodesic metric spaces. The
most important of the examples which we considered are the model spaces M"K. The
central role which these spaces play in the scheme of this book was explained in the
introduction: we seek to elucidate the structure of metric spaces by comparing them
to M"K; if favourable comparisons can be drawn then one can deduce much about the
structure of the spaces at hand. We are now in a position to set about this task.
In Part II we shall study the basic properties of spaces whose curvature is bounded
from above by a real number к. Roughly speaking, a space has curvature < к if every
point of the space has a neighbourhood in which geodesic triangles are no fatter than
their comparison triangles in M\. In Chapter 1 we give several precise formulations
of this idea (all due to A.D. Alexandrov) and prove that they are equivalent. In
subsequent chapters we develop the theory of spaces which satisfy these conditions,
concentrating mainly (but not exclusively) on the case of non-positive curvature.
We punctuate our discussion of the general theory with chapters devoted to various
classes of examples.
Unless further qualification is made, in all that follows к will denote an arbitrary
real number. The diameter of Ml will be denoted DK (thus DK is equal to π/л/к if
к > 0, and сю otherwise).

Chapter II.l Definitions and Characterizations of
CAT(k) Spaces
In this chapter we present the various definitions of a CAT(/c) space and prove that
they are equivalent.
The САТ(к) Inequality
Recall that a geodesic segment joining two points ρ and q of a metric space X is
the image of a path of length d(p, q) joining ρ to q. We shall often write [p, q] to
denote a definite choice of geodesic segment, but immediately offset the dangers of
this notation by pointing out that in general such a segment is not determined by
its endpoints, i.e., without further assumptions on X there may be many geodesic
segments joining ρ to q.
By definition, a geodesic triangle Δ in a metric space X consists of three points
p,q,r e X, its vertices, and a choice of three geodesic segments [p, q], [q, r], [r,p]
joining them, its .ш/е?. Such a geodesic triangle will be denoted Δ([ρ, q], [q, r], [r,p])
or (less accurately if X is not uniquely geodesic) Δ(ρ, q, r). If a point χ e X lies in
the union of [p, q], [q, r] and [r, p], then we write χ e Δ.
Recall from (1.2.13) that a triangle Δ = Δ(ρ, q, Ύ) in Мгк is called a comparison
triangle for Δ = Δ([ρ, q], [q, /·],_[>, p\) if d(p, q) = d(p, q), d(q, r) = d(q, r) and
d(p, T) = d(p, r). Such a triangle Δ С М\ always exists if the perimeter d(p, q) +
d(q, r) + d(r, p) of Δ is less than than twice the diameter DK of M^, and it is unique
up to isometry (see 1.2.13). We will write Δ = Δ(ρ, q, r) or Δ(ρ, q, r), according
to whether a specific choice of p, q, ~r is required. A point χ e \д, г] is called a
comparison point for χ e [q, r] if d(q, x) = d(q, x). Comparison points on \p, ~q\
and \p, T\ are defined in the same way. If ρ φ q and ρ φ r, the angle of Δ at ρ is the
Alexandrov angle between the geodesic segments [ρ, q] and [p, r] issuing fromp, as
defined in (1.1.12).
1.1 Definition of a CAT(/c) Space. Let X be a metric space and let к be a real number.
Let Δ be a geodesic triangle in X with perimeter less than 2DK. Let Δ С М\ be a
comparison triangle for Δ. Then, Δ is said to satisfy the CAT(/c) inequality if for
all jc, у e Δ and all comparison points x, у е Δ,
d(x, y) <d(x,y).

The CAT(re) Inequality 159
If к < 0, then X is called a CAT(/c) space (more briefly, "X is CAT(/c)") if X is
a geodesic space all of whose geodesic triangles satisfy the CAT(/c) inequality.
If к > 0, then X is called a CAT(/c) space if X is DK -geodesic and all geodesic
triangles in X of perimeter less than 2DK satisfy the САТ(к) inequality. (In this
definition we admit the possibility that the metric on X may take infinite values. And
we remind the reader that to say X is DK-geodesic means that all pairs of points a
distance less than DK apart are joined by a geodesic.)
Note that in our definition of а С AT(/c) space we do not require that X be complete.
Complete С AT(0) spaces are often called Hadamard spaces.
Fig. 1.1 The CAT(re) inequality
1.2 Definition. A metric space X is said to be of curvature < к if it is locally a
CAT(/c) space, i.e. for every χ e X there exists rx > 0 such that the ball B(x, rx),
endowed with the induced metric, is a CAT(/c) space.
If X is of curvature < 0 then we say that it is non-positively curved.
The definition given above was introduced by A.D. Alexandrov [Ale51]. It
provides a good notion of an upper bound on curvature in an arbitrary metric space.
Classical comparison theorems in differential geometry show that if a Riemannian
manifold is sufficiently smooth (for example if it is C3) then it has curvature < к in
the above sense if and only if all of its sectional curvatures are < к (see the Appendix
to this chapter).
The terminology "CAT(/c)" was coined by M. Gromov [Gro87, p. 119]. The
initials are in honour of E. Cartan, A.D. Alexandrov and V.A. Toponogov, each of
whom considered similar conditions in varying degrees of generality. In Chapter 4
we shall prove that if a complete simply connected length space has curvature < 0
then it is a CAT(O) space.
1.3 Remark. Let X be a geodesic space. If each pair of geodesies c\ : [0, αϊ] —> X
and a : [0, аг] -> X with ci(0) = сг(0) satisfy the inequality d{c\{ta\), ciitai)) <
td(ci(ai), C2O22)) for all t e [0, 1], then one says that the metric on X is convex. It is
easy to see that the metric on a CAT(O) space is convex (cf. 2.2). In general having a
convex metric is a weaker property than being С AT(0) (cf. 1.18). There are, however,
several important classes of spaces in which convexity of the metric is equivalent to
the CAT(O) condition, including Riemannian manifolds (1 A.8) and MK-polyhedral
complexes (5.4).

160 Chapter II. 1 Definitions and Characterizations of CAT(re) Spaces
1.4 Proposition. Let X be a CAT(/c) space.
(1) There is a unique geodesic segment joining each pair of points x, у е X
(provided d(x, y) < DK if к > 0), and this geodesic segment varies continuously
with its endpoints.
(2) Every local geodesic in X of length at most DK is a geodesic.
(3) The balls in X of radius smaller than DK/2 are convex (i.e., any two points
in such a ball are joined by a unique geodesic segment and this segment is
contained in the ball).
(4) The balls in X of radius less than DK are contractible.
(5) (Approximate midpoints are close to midpoints.) For every λ < DK and ε > 0
there exists δ = δ(κ, λ, ε) such that ifm is the midpoint of a geodesic segment
[x, у] С X with d(x, y) < λ and if
max{d(x, m'), d(y, m')} < -d(x, y) + 8,
then d(m, m!) < ε.
Proof. (1) Consider p, q e X with d(p, q) < DK. Let [p, q] and [p, q]' be geodesic
segments joining ρ to q, let r e \p, q] and r1 € [p, q]' be such that d(p, r) = d(p, r1).
Let \p, r] and [r, q] be the two geodesic segments whose concatenation is [p, q].
Any comparison triangle in M\ for Δ([ρ, q]', [p, r], [r, q]) is degenerate and the
comparison points for r and r1 are the same. The CAT(/c) inequality implies that
d(r, r1) = 0, hence r = r'.
If X is proper then it follows immediately from (1.3.12) that [p, q] depends
continuously on ρ and q. We consider the general case. First note that given any positive
number I < DK, there is a constant С = C(l, к) such that if c, c' : [0, 1] -> M% are
two linearly parameterized geodesic segments of length at most I and if c(0) = c'(0),
then d(c(t), c'(t)) < Cd(c(l), c'(l)) for all t e [0,1] (see 3.20). For к < 0 we even
haveu?(c(i), c'(t)) < td(c(l), c'(l)).
Let pn and qn be sequences of points converging to ρ and q respectively. We
assume that d(pn, q„) and d(p, q„) are smaller than I < Dk- Let c, c„, c'n be linear
parameterizations [0, 1] —> X of the geodesic segments \p, q], [p„, qn] and [p, q„]
respectively. Applying theCAT(/c) inequality, we have d(c(t), c„(t)) < d(c(t), c'n(t))+
d(c'n(t), cn(t) < C(d(q, q„) + d(p,pn)), where С = C(l, к) is as above. Therefore c„
converges uniformly to c.
To prove (2), we fix a local geodesic с ". [0, λ] —> X of length λ < DK and
consider S = {t e [0, λ] | c\[o,t] is a geodesic}. This set is obviously closed, and it
contains a neighbourhood of 0 by hypothesis; we must prove that it is open. Suppose
to € S and 0 < to < λ. Because с is a local geodesic, there exists a positive number
ε < λ-t0 such that c|[i0-£,,0+E] is geodesic. Thus c([0, t0 + ε]) forms two sides of the
geodesic triangle Δ = A(c(0), c(tQ), c(to + ε)). The comparison triangle Δ С М\
must be degenerate: if it were not then the CAT(/c) inequality applied to points
χ e [c(0), c(to)] and у e [c(to), c(io + ε)] close to c(t0) would contradict the fact that

Characterizations of CAT(re) Spaces 161
clfo-ε,ΐο-κ]is geodesic. Thus c\
[ο,ίο+ε] ^as length d(c(Q), c(to + ε)), so (ίο, to + ε) с S
and we are done.
Assertion (3) follows from the С АТ(/с) inequality and the fact that balls in Мгк of
radius < DK/2 are convex (1.2.12).
(4) follows easily from (1): if В = B(x, I), where I < DK, then the map Β χ
[О, 1] —> X which associates to (у, t) the point a distance td(x, y) from у along the
geodesic segment [x, y] is a continuous retraction of В to x.
In (1.2.25) we proved that approximate midpoints are close to midpoints in
M%, and having noted this, (5) follows immediately from the CAT(/c) inequality
for A(x, y, m1). D
1.5 Corollary. For к < 0, any CAT(/c) space is contractible, in particular it is simply
connected and all of its higher homotopy groups are trivial.
1.6 Exercise. Let X be a metric space whose metric is convex in the sense of (1.3).
Prove that X satisfies statements (1) to (4) of (1.4) with DK = oo.
Characterizations of САТ(к) Spaces
The CAT(/c) inequality can be reformulated in a number of different ways, all due to
Alexandrov. We note four such reformulations immediately, and in what follows we
shall pass freely between these and definition (1.1), using whichever formulation is
best suited to the situation at hand and referring to that formulation as the CAT(/c)
inequality.
1.7 Proposition. Fix к e K. Let X be a metric space that is DK-geodesic. The
following conditions are equivalent (when к > 0 we assume that the perimeter of
each geodesic triangle considered is smaller than 2DK):
(1) X is a CAT(k) space.
(2) For every geodesic triangle Δ([ρ, q], [q, r], [r, p]) in X and every point χ e
[q, r], the following inequality is satisfied by the comparison point χ e [q, г] С
Δ(ρ, q, г) С Ml:
dip, x) < d(p, x).
(3) For every geodesic triangle Δ([ρ, q], [q, r], [r, p])inX and every pair of points
x e \p, q], у e [ρ, r] with χ φ ρ and у φ ρ, the angles at the vertices
corresponding top in the comparison triangles A(p, q, г) С Ml and A(p, x, у) с Ml
satisfy:
4\x,y)<4\q,r).
(4) The Alexandrov angle (as defined in 1.1.12) between the sides of any geodesic
triangle in X with distinct vertices is no greater than the angle between the
corresponding sides of its comparison triangle in M%.

162 Chapter П. 1 Definitions and Characterizations of С AT(re) Spaces
(5) For every geodesic triangle Δ([ρ, q], [ρ, r], [q, r]) in X with ρ φ q and
ρ Φ r, if γ denotes the Alexandrov angle between [ρ, q] and [p, r] at ρ and if
Αφ, q, г) С Μ2 is a geodesic triangle with d{p, q) = dip, q), d{p, r) = d[p, r)
and Z.p{q, r) = y, then d(q, r) > d(q, r).
Proof. It is clear that (1) implies (2) and that (4) is equivalent to (5). And (4) follows
immediately from (3) and the observation that one can use comparison triangles in
M2 rather than E2 when defining the Alexandrov angle (1.2.15). We shall prove that
(1) and (3) are equivalent, that (2) implies (3), and that (4) implies (2).
Let p, q, г, x, у be as in (3). We write ζ to denote comparison points in Δ =
A(p, q, г) С Μ2, and ζ' to denote comparison points in Δ = A(p, x, у) С М2.
Consider the vertex angles a = Z.^K\q, r) and a' = /!^\x, y) at ρ and p'. According
to the law of cosines (1.2.13), the inequality d(x, y) > d(x, y) = d(x, f) is valid if
and only if a' < a. Thus (1) and (3) are equivalent.
Maintaining the notation of the previous paragraph, we shall now show that
(2) implies (3). Let A(p",x", г") С Μ2 be a comparison triangle for A(p,x, r).
Let a" denote the vertex angle at p". By (2), we have d(x, y) < d(x", y"), where
y" e \p", r"] is the comparison point fory. But d(x, y') = d(x, y), so a' < a". Again
by (2), d(x', J") = d(x, r) < d(x, ?), hence a" < a. Thus a' < a.
Finally we prove that (4) implies (2). Given a geodesic triangle Δ in X, say
Δ = Δ([ρ, q], [q, r], [r,p]), and a point χ e [q, r] distinct from q and r, we fix a
geodesic segment [jc, p] joining χ to ρ in X. Let γ and γ' be the angles at χ which
[x, p] makes with the subsegments of [q, r] joining χ to q, and χ to r, respectively. Let
β be the angle at q between the geodesic segments [q, p] and [q, r]. Let A(p, q, T)
be a comparison triangle for A(p, q, r) in M2 and let β be the angle at the vertex q.
Consider comparison triangles A(p, x, q) and A'(p, x, r) in M2 for the geodesic
triangles A(p, x, q) and A{p, x, r) respectively. We choose these comparison triangles
in such a way that they share the edge [x, p] and q and r lie on opposite sides of the line
which passes through ρ and x. Let γ = Δχ{ρ, q), γ' = Ζχ(ρ, r) and/3 = Zq(p, x). By
1.1.13(2) we havey+y' > я, hence, by(4), γ+γ' > я. Alexandrov's lemma implies
that β < β. Therefore, using the law of cosines, we have d(p, x) > dip, x) = dip, x).
D
1.8 Other Notions of Angle. Let X be a metric space and suppose that we have a
map A which associates a number Aic c') € [0, π] to each pair of geodesies с, с' in
X that have a common initial point. One might regard A as a reasonable notion of
angle if for each triple of geodesies с, с' and c" issuing from a common point we
have:
(1) Aic,c')=Aic',c);
(2) Aic, c") < Aic, c') + Aic', c")\
(3) if с is the restriction of d to an initial segment of its domain then Aic, c') = 0;
(4) if с ·. [—α, α] —> X is a geodesic and c_, c+ '. [0, a] —> X are defined by
C-it) = c(—t) and c+(i) = c(i), then A(c_, c+) = n.

Characterizations of CAT(re) Spaces 163
The Alexandrov angle satisfies these conditions (see 1.1.13 and 14). The Rie-
mannian angle also satisfies these conditions. And a trivial example is obtained by
defining A(c, d) = π unless с and c' have a common initial segment.
If in (1.7) one replaces the Alexandrov angle by a function Л satisfying the above
conditions, then the implications (5) <=£· (4) => (2) remain valid. This observation
provides us with a useful tool for proving that certain spaces are CAT(/c) (see (1 A.2)
and (10.10)).
1.9 Exercises.
(1) Let X be a geodesic metric space. Prove that the following conditions are
equivalent:
(a) X is a CAT(/c) space.
(b) (cf. 1.7(2))ForeverygeodesictriangleA([p, q], [q, r],
[r,p])inX(withperimeter smaller than 2DK if к > 0), the point m e [q, r] with d(q, m) = d(r, m) and
its comparison point m e [q, f] С A(p, q, г) С Μ2 satisfy:
dip, m) < dip, m).
Show further that in the case к = 0 the above conditions are equivalent to:
(c) The CN inequality18 of Bruhat and Tits [BraT72]. For all p,q,r e X and
all m e X with d(q, m) = d(r, m) = d(q, r)/2, one has:
d(p, qf + dip, rf > 2 d(m, pf + i d(q, rf.
(In E2 one gets equality by a simple calculation with the scalar product.)
(2) Condition (1.7(5)) can be reformulated more analytically using the law of
cosines. In the case к = 0, if we let a = dip, q), b = dip, г), с = diq, r) and write
γ for the Alexandrov angle alp between [p, q] and/?, r], then the required condition
is:
c2 > a2 + b2 - 2abcosy.
Find the equivalent reformulations for к < 0 and к > 0.
(3) Let X be a CAT(/c) space. If p, x, у e X are such that dix, p) + dip, y) <
DK, then the geodesic segment [x, y] is the union of [x,p] and [p, y] if and only if
^Pix,y) = π.
The CAT(k) 4-Point Condition
All of the reformulations of the CAT(/c) condition given above concern the
geometry of triangles. There is also a useful reformulation concerning the geometry of
quadrilaterals.
Courbure negative

164 Chapter II. 1 Definitions and Characterizations of С AT(re) Spaces
1.10 Definition. A subembedding in M\ of a 4-tuple of points {x\ ,yi,X2, Уг) from a
pseudometric space X is a 4-tuple of points (xi, yx, x2, y2) in M\ such that dQc;, yj) =
d(xj, yj) for i,j e {1, 2}, andd(xx,x2) < d(xx,x2) andd(y\,y2) < d(yx,y2).
X is said to satisfy the CAT(/c) 4-point condition if every 4-tuple of points
Oi,yi,x2, Уг) with d{x\,y\,) + d(yx, x2) + d(x2, y2) + d(y2, x\) < 2DK has a
subembedding in M^.
Note that every subspace of a CAT(/c) space X satisfies the CAT(/c) 4-point
condition. This condition first appears in the work of Reshetnyak [Resh68]. It was
used extensively by Korevaar and Schoen in their work on harmonic maps [KS93],
and also by Nikolaev [Ni95].
Recall that a pair of points x, у in a metric space X is said to have approximate
midpoints if for every 5 > 0 there exists m! e X such that тах{й?(л:, т'), d(y, m')} <
\d(x, y)+8. If X is complete and every pair of points in X has approximate midpoints,
then X is a length space.
1.11 Proposition. Let X be a complete metric space. The following conditions are
equivalent:
(1) X is a CAT(k) space.
(2) X satisfies the С AT(/c) 4-point condition and every pair of points x,y e X with
d(x, y) < DK has approximate midpoints.
Proof. We first prove that (1) implies (2). The existence of geodesies in X
ensures the existence of approximate midpoints. In order to show that X satisfies the
CAT(/c) 4-point condition we must construct a subembedding in M\ of each 4-tuple
(χι ,yi,x2, Уг) from X with d{xx ,y\,) + d(yx, x2) + d(x2, y2) + d(y2, xx) < 2DK.
Given such a 4-tuple, we consider the quadrilateral Q С М^ formed by
comparison triangles A(5cx,x2,yi) = A(jq,x2,yi) and A(xx, x2, y2) = A(xx, x2, y2) with a
common edge [xx, x2] and with y\ and y2 on opposite sides of the line containing
[xi,x2].
If Q is convex then the diagonals [xx, x2] and \y\, y2] intersect in some point z-
Let ζ € [jci, JC2] be such that d(xx, z) = d(xx,z). From the CAT(/c) inequality (and
triangle inequality) we have:
d(yi, уг) < d(yi, z) + d(z, y2) < d(yi, z) + d(z, y2) = d($i - h)-
By construction, d(xx,x2) = d(xx,x2), so if Q is convex then {x\,y\,xi,y2) is a
subembedding of (x\,y\,x2, Уг) inM\.
If Q is not convex then one of the jcj, say x2, is in the interior of the convex
hull of the other three vertices of Q. By applying Alexandrov's lemma (1.2.16) we
obtain a quadrilateral in M\ with vertices x\,y\,xi,y2 where d(xi,yj) = d(xj,yj)
for i,j = 1,2 and x2 e \yi,y2] is such that d(xx,x2) > d(xx,x2) = d(xx,x2).
And d(yx, y2) = d(yx, x2) + d(x2, y2) = d(yx, x2) + d(x2, y2) > d(y\, y2). Therefore
(xx,yx,x2, y2) is a subembedding of (x\,y\,x2, y2).
We shall now show that (2) implies (1). As a first step, we prove that (2) implies
the existence of midpoints for pairs of points less than DK apart; as X is complete, the

CAT(re) Implies САТ(к') if к < к' 165
existence of geodesies follows immediately. We fix x, у e X with d(x, y) < DK and
consider a sequence of approximate midpoints m; with max{d{x, mi), d(y, mi)} <
\d(x, y) + i. If we can prove that this is a Cauchy sequence then the limit will be a
midpoint for χ and y. We fix an arbitrary ε > 0 and fix λ between d(x, y) and DK.
According to (1.2.25), there exists 5 = δ(κ, λ, ε) such that in M2, if dip, q) < I
and max.{d(p, m'), d(q, m')} < \d{p, q) + δ, then d(m, m') < ε, where m is the
midpoint of [p, q].
If (3c, m,·, у, щ) is a subembedding of (jc, m,, у, иг,), then by definition d{m\, Щ) >
d(m„ щ). Also, d(x, y) < d(x, щ) + й?(т,, у) = d(x, гщ) + d(mi, у) < d(x, у) + 1/г.
Thus, for г and j sufficiently large we have d(x,y) < λ. And if тах{1/г, l/j] <
δ(κ, λ, ε) then d(mj, m) < ε and d(m), m) < ε, where m is the midpoint of [Зс, у].
Since d(mt, mj) > d{nii, щ), we have proved that m,- is a Cauchy sequence, as required.
It remains to prove that triangles in X satisfy the CAT(/c) inequality. Consider a
triangle A(x, y, z) С X of perimeter less than 2DK and let m be a point on [x, y]. Let
(z, x, m, y) be a subembedding of (z, jc, m, y) in M\. As u?(jc, y) < u?(3c, }T) < u?(3c, m) +
u?(m, y) = d(x, y), we see that Д(3с, у, ζ) is a comparison triangle for A(x, y, z) and m
is the comparison point for m e \x,y]. By the definition of subembedding, d(z, m) <
d(z, m). Thus A(x, y, z) satisfies the CAT(/c) inequality (as characterized in 1.7(2)).
D
CAT(k) Implies САТ(к/) if к < к'
The following basic theorem shows in particular that real hyperbolic space H" is a
CAT(O) space.
1.12 Theorem.
(1) IfX is α CAT(/c) space, then it is а САТ(к') space for every к' > к.
(2) If X is a CAT(k') space for every к' > к, then it is a CAT(/c) space.
Proof. We first prove (2). Given jc, у e X with d(x, y) < DK, we have d(x, y) < DK·
for all к' > к sufficiently close to к. Thus if X is DK'-geodesic for all к' > к, then it
is DK -geodesic.
Given a geodesic triangle Δ = A(p, q, r) in X with perimeter < 2DK, we choose
к' > к sufficiently close to к to ensure that the perimeter of Δ is less than 2DK·.
Let a = d{p, q), b = dip, r) and с = diq, r). Let γ be the Alexandrov angle at ρ
between \p, q] and [p, r]. In the light of the law of cosines for Мгк, if к > 0 then
characterization (1.7(5)) of the САТ(к') inequality becomes:
cosic^K') < cosiaVK') cosibV^) + sin(av/ic') sin^Vic') cos(y).
If к > 0, then passing to the limit we get the same inequality with к' replaced by
к. If к = 0, then in the limit we get c2 < a2 + b2 — 2ab cos y. Thus we obtain the
CAT(/f) inequality (in the guise of 1.7(5)). The case к < 0 is similar.

166 Chapter II. 1 Definitions and Characterizations of CAT(re) Spaces
We now prove (1). If к1 > к, then since X is assumed to be DK-geodesic it is a
fortiori DK'-geodesic.
Given a geodesic triangle in X of perimeter less than DK we compare its
comparison triangles Ак С M2K and Ак- С М2К,. It is sufficient to prove that the angles at the
vertices of AK are not smaller than the angles at the corresponding vertices of Ακ·.
Using the law of cosines in M2, we reduce to proving the following lemma. D
1.13 Lemma. Fix к' > к and let A (resp. A') be a geodesic triangle in Ml
(resp. M2,) with vertices А, В, С (resp. А', В', С) and opposite sides of length a, b, с
(resp. a, b, c'), where the side opposite A has length a etc.. Suppose аЛ-ЬЛ-с < 2DK<,
and suppose that the angles at the vertices С and С are equal and lie in (0, π). Then
c' < с
Proof. We introduce polar coordinates (r, Θ) in M2 and M2, (in open balls of radius
DK- = nf-Jlc1 if к' > 0), see (1.6.16). Consider the map h sending the point of M2,
with polar coordinates (r, Θ) to the point with the same coordinates in M2. We take
С and С to be the centres of these polar coordinates (so h(C) = C). Because the
derivative of h at the origin is an isometry, we may assume h(A') = A and h(B') = B.
The image of h is the ball centred at С with radius DK', hence it contains the triangle
Δ.
We wish to show that h increases the length of every path that is not radial in the
polar coordinates. For this we must show that if υ is a vector in the tangent space
TXM2, at a point χ in the domain of h, then the norm of its image Txh(v) by the
differential Txh of h at χ is no smaller than the norm of v, and if ν is not radial then
IIWOII > IIν||.
Recall (1.6.17) that in the polar coordinates on Ml the Riemannian metric is given
by
ds2 = dr2 +f(K, rfdO2,
where the function к ы> /(/с, r) is defined on the interval (—oo, (π/r)2) by
_ ι
Як, r)
-4= sinhtV^-if r) if к < 0
r if/f = 0
1 sin(x/if r) if к > 0.
■Я
It is easy to show that the function к ы> /(/с, г) is continuous and strictly decreasing.
Let ν = vrfj.(r, Θ) + ve-^g(r, Θ) be a tangent vector at a point χ with polar
coordinates (r, Θ) in the domain of h; the square of its norm is v2 +/(/f', r)2vj. The
image of υ under the derivative of h has the same expression in the polar coordinates
on Ml and the square of its norm is v2 +/(/f, r)2i>|, hence \\Txh{v)\\2 > \\v\\2 with
equality only if vg =0. Thus Txh strictly increases the norm of vectors which are not
radial, and so h strictly increases the length of paths which are not radial. It follows
that the segment [А, В] С Δ, which has length c, is the image under h of a path from
A' to B' of length less than c; but d = d(A', B'), so d < с D

Simple Examples of CAT(re) Spaces 167
Simple Examples of САТ(к) Spaces
Pre-Hilbert spaces are obviously CAT(O). Conversely we have:
1.14 Proposition. If a normed real vector space V is САТ(к)/ог some к е K, then
it is a pre-Hilbert space.
Proof. If V is not a pre-Hilbert space then, as in (1.4.5), there exist u, υ e V such that
lim,^o ^o(tu, tv) = lim^o /.^{tu, tv) does not exist, whereas 1.7(3) implies that if
V were CAT(/c) then this limit would exist. D
1.15 Examples
(1) When endowed with the induced metric, a convex subset of Euclidean space
E" is CAT(O). More generally, a subset of a CAT(/c) space, equipped with the induced
metric, is CAT(/c) if and only if it is DK-convex. Here, it is important to distinguish
between the induced metric and the induced length metric: many non-convex subsets
of E" are CAT(O) spaces when endowed with the induced length metric.
(2) Let X be the planar set which is the complement of the quadrant {(x, y) |
χ > 0,y > 0}, endowed with the induced length metric from E2. In (1.3.5) we
described the geodesies in this space. It follows immediately from this description
and Alexandrov's lemma (1.2.16) that X satisfies the angle criterion 1.7(4) for a
CAT(O) space.
A similar argument applies to the complement of any sector in the plane, and a
local version of the same argument shows that the complement of any polygon in
M2 has curvature < к.
(3) If Χι and X2 are CAT(O) spaces, then their product Xt χ Χ2 (as defined in
(1.5.1)) is also a CAT(O) space. This is easily seen using the characterization of
CAT(O) spaces given in (1.9(lc)).
(4) A metric simplicial graph is CAT(/c) if and only if it does not contain any
essential loops of length less than 2DK; in other words, every locally injective path
of length less than 2DK is injective.
(5) An Ж-tree is a metric space Τ such that:
(i) there is a unique geodesic segment (denoted [x, y]) joining each pair of points
x, у e Г;
(ii) if [у, х] П [χ, ζ] = {χ}, then [у, х] U [χ, ζ] = \y,z].
M-trees are CAT(/c) spaces for every к. Indeed in such spaces every geodesic
triangle with distinct vertices x\,x2, хъ is degenerate in the sense that there exists
ν e Τ such that [x-,, xj\ = [x-,, v] U [v, Xj] if i φ j. In particular, the angle at each
vertex is either 0 or n, and a vertex angle of π occurs if and only if one vertex lies
on the geodesic segment joining the other two vertices. Thus (1.7(4)) holds for all
кеК. Conversely, any metric space which is CAT(/c) for all к is an M-tree.
Simply connected metric simplicial graphs, as defined in (1.1.10), are the easiest
examples of M-trees. In general an M-tree cannot be made simplicial. For example,

168 Chapter П. 1 Definitions and Characterizations of CAT(re) Spaces
consider the set [0, oo) χ [0, oo) with the distance d(x, y) between two points χ =
{х\,хг) and у = (y\,yi) defined by d(x,y) = x\ + у ι + \x2 - y2| if χι Ф Уг and
d(x, y) = |*ι - yi | if x2 = уъ
The asymptotic cone of Ш" provides another non-simplicial example. In this
case, the complement of every point in the K-tree has infinitely many connected
components.
The apparently trivial nature of the geometry of R-trees is deceptive and belies
their importance in low-dimensional topology and the delicacy of questions
concerning their isometry groups. We refer the reader to [Sha91] for a survey to 1991 and
[Pau96] for an account of recent developments in this area.
1.16 Exercises.
(1) Let X be the closed subset of E3 which is the complement of the octant
{(x, y,z) | χ > 0, у > 0, ζ > 0}. Show that X, endowed with the induced length
metric, is not a CAT(0) space (or indeed a CAT(/c) space for any κ).
(2) Let к > 0. Prove that Xt χ X2 is a CAT(/c) space if and only if both Xi and
X2 are CAT(/c) spaces.
(3) Give a second proof of (1.14) by showing that if anormed space satisfies the
CAT(/c) inequality then the norm satisfies the parallelogram law.
(4) Let С be a geodesic segment in M\. Consider M\ ч С equipped with the
induced length metric and let X denote its metric completion. Prove using Alexan-
drov's lemma that X has curvature < к but that it is only a CAT(/c) space if к > 0
and С has length DK.
(5) Prove that when endowed with the induced length metric, the complement of
an open horoball in И2 is a CAT(— 1) space. (Hint: Use criterion 1.7(4).)
Historical Remarks
Prior to the work of A.D. Alexandrov, H. Busemann had introduced a weaker notion
of curvature in metric spaces [Bus48,55]. The book of W. Rinow [Rin61] provides
good historical references for the period to 1960, and the report of K. Menger [Men52]
explains the importance of the early work of Wald [Wa36]. We close this chapter
with two remarks concerning the work of Alexandrov and Busemann.
1.17 The Relative Excess of a Triangle. Following Alexandrov, given к е К and
a geodesic triangle Δ = Δ([ρ, q], [q, r], [r, p\) in a metric space X with perimeter
smaller than Δ&, we define the relative excess δκ(Α) to be the difference of the sum
of the vertex angles of Δ and the sum of the corresponding angles for a comparison
triangle in M2.
It follows immediately from (1.7(4)) that 5Κ(Δ) < 0 for any geodesic triangle
Δ in a CAT(/c) space. Alexandrov proved the non-trivial fact that the converse is

Appendix: The Curvature of Riemannian Manifolds 169
also true: in a geodesic space X, if the relative excess 8K of every geodesic triangle
is non-positive, then X is a CAT(/c) space. The proof of this result requires a rather
long argument (see Alexandrov [Ale57a]).
Alexandrov shows also that a locally geodesic metric space X is of curvature < к
if and only if, for each point χ e X and each geodesic triangle Δ in a neighbourhood
of x, the inequality 5ο(Δ)/Α < κ holds, where A is the area of a comparison triangle
for Δ in E2.
1.18 Busemann's Approach to Curvature. Busemann [Bus48] made an extensive
study of spaces X whose metric is locally convex in the sense that every point
has a neighbourhood on which the induced metric is convex in the sense (1.3).
(Equivalently, every point χ e X has an open neighbourhood which is a geodesic
space and for every geodesic triangle Δ = Δ([ρ, q], [q, r], [r, p\) in U the distance
between the midpoints of [p, q] and [p, r] is not bigger than the distance between the
comparison points in a comparison triangle Δ С Е2.) The metric on any space of
non-positive curvature is locally convex, but not conversely. For example, if У is a
normed space whose unit ball is strictly convex in the sense of (1.1.4), then the metric
on V is convex, whereas V is non-positively curved if and only if it is a pre-Hilbert
space (cf. 1.14).
More generally one might define a geodesic space X to be of curvature < k (in a
weak sense) if every point of X has a neighbourhood such that given any three points
p, q, r in this neighbourhood (with d(p, q) = d(q, r) + d(r, p) < DK) and geodesies
[p, q] and [p, r], the midpoints χ e \p, q] and у e\p, r] and their comparison points in
A(p, q, г) С Μ2 satisfy the inequality d(x, y) < d(x, y). We shall see in the Appendix
that this condition is equivalent to Alexandrov's definition of curvature < k if X is a
sufficiently smooth Riemannian manifold.
Appendix: The Curvature of Riemannian Manifolds
The purpose of this appendix is to relate Alexandrov's definition of upper curvature
bounds (1.2) to the classical notion of sectional curvature. In what follows Μ will
be a smooth Riemannian manifold of class C3 and dimension η equipped with its
associated distance function d, as in (1.3.18). We begin with a technical criterion for
deciding when a Riemannian manifold is of curvature < к.
1A.1 Lemma. Fix к e Μ, χ e Μ and о е М". Suppose that for a suitable positive
number ε < DK there exists a diffeomorphism efrom B(o, ε) onto an open setU^M
such that e(o) = χ and
(a) for ally e B(o, ε) and ν e TyMnK, we have \Tye(y)\ > \v\, where Tye : TyM"K ->
Te(y)M denotes the differential of e aty;
(b) |Гз,е(1>)| = \v\ if υ is tangent to the geodesic joining о to y; in particular
T0e : T0M" —> TXM is an isometry.

170 Chapter II. 1 Definitions and Characterizations of CAT(re) Spaces
Then:
(1) U = B(x, ε), indeed, for eachy e B(o, ε), the geodesic segment [о, у] is mapped
isometrically by e onto a geodesic segment joining χ = e(0) to e(y) in M, and
this segment is unique;
(2) for all y,ze B(o, ε/2), we have d(e(y), e{z)) > d(y, z).
Proof. As in (1.3.17), the Riemannian length of a curve с will be denoted Ir(c). The
hypotheses of the lemma imply that for any piecewise C1 curve с : [0,1]—» B(o, ε),
we have 1ц(е о с) > Ir{c) with equality if с is the geodesic path joining о to y. This
together with the fact that any piecewise C1 curve in Μ joining χ to a point outside
of U has length at least ε, implies that d(o, y) = d(x, eiy)), that U = B(x, ε), and
that the image under e of the geodesic segment [о, у] is the unique geodesic segment
joining χ to eiy).
If y, ζ e B(o, ε/2), then d(e(y), e{z)) < ε is the infimum of the lengths lR(c) of
piecewise C1 curves с of length < ε joining eiy) to e(z). Any such curve is the image
under e of a piecewise C1 curve joining у to ζ whose length (by (a)) is not bigger
than Ir(c). Hence d(e(y), e(z)) < d(y, z). This proves (2). D
1A.2 Proposition. Let Μ be a Riemannian manifold and let к е K. Let ρ e Μ
and suppose that ε > 0 is such that for each point χ e B(p, ε/2) one can find a
map ex : B(o, ε) —> B(x, ε) satisfying the hypotheses of(lA.l). Then B(p, ε/2) is a
CAT(k) space.
Proof. Let В = B(p, ε/2). Lemma 1 A. 1 implies that В с М is convex and uniquely
geodesic. It also implies that the Riemannian angles at the vertices of any geodesic
triangle A(x, y, z) С В with distinct vertices are no greater than the corresponding
angles in a comparison triangle Δ = Δ(χ, у, ζ) С М\. As remarked in (1.8), this
implies that В satisfies condition (2) of (1.7). D
1A.3 Jacobi Fields and Sectional Curvature. We recall some classical
constructions and facts from Riemannian geometry; these are proved in any textbook on the
subject (e.g. [Mil63] or [ChEb75]). We consider a Riemannian manifold Μ that is
smooth enough (class C3 suffices) to ensure that the constant speed local geodesies
t ы> c(t) in Μ are the solutions of the differential equation
D
-jc{t) = 0,
dt
where Ц denotes the covariant derivative applied to smooth vector fields along the
path c, and c(t) e TC(t)M is the velocity vector of с at t.
For each vector ν e TXM whose norm \v\ is small enough, there is a unique
constant speed local geodesic c„ : [0, 1] —> Μ such that c„(0) = χ and c„(0) = v;
the point c„(l) is denoted exp(u). We shall need the following classical fact: every
point of Μ has a neighbourhood U for which one can find ε > 0 such that exp is

Appendix: The Curvature of Riemannian Manifolds 171
defined on TU = (JxsUT^M, where Tx = {υ e TXM\ \υ\ < ε}; the restriction ехрл
of exp to each TXM is a diffeomorphism onto an open subset of Μ and for each χ e U
and ν e TXM, the path c„ is the unique constant speed geodesic [0, 1] —» X joining
χ to exp(u).
Given χ e U, a non-zero vector и е Г£М and υ e TXM, there is a one-parameter
family of constant speed geodesies cu+sv, where s varies over a small neighbourhood
of 0 e R. Associated to this family of geodesies one has the vector field J(t) =
jsct(u+sv)(t)\s=o along the geodesic path с = c„. This satisfies the second order
differential equation
—J(t) = (-R(J(t),c(t))) (c(f)),
with initial conditions 7(0) = 0 and jtJ(0) = υ = \imt^.0 J(t)/t, where R( , ) e
End(rM) is the curvature tensor. Any vector field satisfying such an equation is called
a Jacobi vector field along the geodesic path с If 7(0) = 0 and £ is orthogonal to
c(0), then 7(i) is orthogonal to c(t) for every t.
If 7(ί) φ 0 is orthogonal to c(t), the sectional curvature of Μ along the 2-plane
in TC(t)M spanned by the orthogonal vectors 7(i) and c(t) is by definition the number
= (R(J(t), b{t))b{t) 1 7(f))
W |7(i)p |c(i)|2
Given «el, if the sectional curvature of Μ along every 2-plane in TM is < κ, then
one says the sectional curvature of Μ is < к.
The following lemma is due to H. Rauch [Rau51].
1A.4 Lemma. Let J be a Jacobi vector field along a unit speed geodesic с : [0, ε] —>
Μ. IfJ(t) is orthogonal to c(t)for all t, then
\J\"(t)>K(t)\J\(t)
at each point where J(t) φ 0.
If K{t) < к, 7(0) = 0 and | §7|(0) = 1, then for every t < DK in the domain of с
we have
\J\(t)>h(t),
where jK is the solution of the differential equation j'^i) = —KJK{t) with initial
conditions jK(0) = 0,fK(0) = 1.
Proof. For any vector fields X(t) and Y(t) along c, we have |(X(i) | Y(t)) = (jtX(t) |
Y(t)) + (X(t) | jtY(t))- Thus we may write
\J\'(t) = j{ (7(f) | 7(i))1/2 = (jtJ(t) | 7(f)) 17(f)!-1,
hence

172 Chapter II. 1 Definitions and Characterizations of CAT(re) Spaces
\J\"{t) = ( ^J(t) I 7(f)) 1Л0Г1 + (~J(t) I ~J(t)\ №\~l
jtJ(t)\J(t)\ |7(ί)Γ3
■(R(J(t),c(t))c(t)\J(t))\J(t)\~3
D ,.. ., D
+ I I jtJ(t)\'\J{t)\£ - (-7(f) | J{t)f\ \J(t)f3.
Therefore by the Cauchy-Schwarz inequality
|7|"(f)>*(f)|7|(f).
Now assume K(t) < к, 7(0) = Oand |f 7|(0) = 1. The solution of the differential
equation fK(t) = —KJK(t) with initial conditions 7'κ(0) = 0,7^(0) = 1 is jK(t) =
J= sin(v/ici) if к > 0; it is jk(t) = t if к = 0; and jK(t) = -т= sinh(Vc:«:i) iff < 0.
The first part of the lemma implies that
(|7|'(f)/*(f)-|7|(f)^(f))'>0,
and hence \J\'{t)jK{t) - |7|(ί)/κ(ί) > 0 (because this function of t vanishes at t = 0).
In other words, a.s jK(t) > 0 for 0 < t < DK,
(*)
forO < t < DK.
By de l'Hopital's rule,
,. |7|(f)
hm =
<->о Mt)
|7|'(о > т
|7|(f) ~ hit)
|7|'(0) limf_^o|7(i)|/i
/*(0) 1
By integrating (*) we get |7(i)| > jK(t). D
1A.5 Lemma. Let Μ be as in (1A.3). If Μ has sectional curvature < к, then for
every ρ e Μ there exists a neighbourhood V of ρ and a positive number ε > 0 such
that for all χ e V one can find a map e : B(o, ε) —> Μ satisfying the hypotheses of
Lemma 1A.1.
Proof. Given a compact neighbourhood V of ρ one can find a positive number ε such
that, for all χ e V, the exponential map ехрл : Г£М —> 5(jc, ε) is well defined and is
a diffeomorphism. Fix о e M" and χ eV and identify Γ0Μ" with TXM by means of a
linear isometry. Define e = expx о exp~'. This map satisfies condition (b) of Lemma
1A.1.
Let u, υ e T0M"K be such that \u\ = 1, |υ| = 1 and (ν | u) = 0. Let JK(t)
(resp. 7(i)) be the Jacobi field that is associated to the one-parameter family of
geodesies c,(U+sv){t) = exp0(t(u + sv)) (resp. expx(t(u + sv)) as in (1A.3), where s
varies over a small neighbourhood of 0 e K. By construction the differential of e

Appendix: The Curvature of Riemannian Manifolds 173
maps JK(t) to J(t). It follows immediately from the description of the Riemannian
metric on M2 in terms of polar coordinates (cf. 1.6.17), thatjK(i) = \JK\(t), and hence
the second part of Lemma 1A.4 implies that e satisfies condition (a) of (1A.1). D
The following theorem is due to Alexandrov [Ale51]. The fact that non-positive
sectional curvature implies non-positive curvature in the sense of (1.2) was proved
by E. Cartan [Car28].
1A.6 Theorem. A smooth Riemannian manifold Μ is of curvature < к in the sense
of Alexandrov (1.2) if and only if the sectional curvature of Μ is < к.
Proof. (1A.5) and (1A.2) together imply that if the sectional curvature of Μ is
bounded above by к, then Μ is of curvature < к in the sense of Alexandrov.
To prove the converse we use the following classical fact [Car28]. Let χ e Μ
and let u, υ be orthogonal unit vectors in TXM. For t small enough, we have unique
geodesies t м> c„(i) and cv(t) issuing from χ with c„(0) = и and c„(0) = v. Let К
be the sectional curvature of Μ along the 2-plane in TXM spanned by и and v. Let
d(s) = d(cu(s), c„(e)). Then19
d{sf = 2ε2 - f ε4 + 0(ε5).
6
Consider a geodesic triangle in M2 with two orthogonal sides of length ε; let
c(e) be the length of the third side. As Μ is assumed to be of curvature < κ,
we have d(S) > c(s) (see (1.7(5)) and (1A.7)). If к =0, then d(s)2 > φ)2 =
2ε2, hence К < 0. If к < 0, then using the law of cosines we can express
c(s) as cosh(V—к c(s)) = cosh2(*J—K ε). The inequality d(s) > c(s) implies
cosh(V_K d(s)) > cosh(s/—K c(e)), hence
1 - κε2 + -(κ Κ + κ2) ε4 + 0(ε5) > 1 - κ ε2 + -κ2ε4 + 0(ε5),
6 3
which implies К < κ. The case к > 0 is treated similarly. D
The "if implication implies the following result.
1 A.7 Corollary. Let Μ be a smooth Riemannian manifold and let c, c' : [0, ε] —> Μ
be geodesies issuing from a common point χ = c(0) = c'(0). The limit as t,f —> 0
of the comparison angle Zx(c(t), c'(f)) exists and is equal to the Riemannian angle
between the velocity vectors c(0) and c'(0). In other words, the Riemannian angle
between с and d is equal to the Alexandrov angle between them.
Proof. As Μ is locally compact, the point χ is contained in a neighbourhood where the
sectional curvature of Μ is bounded by some number к. Hence χ has a convex
neighbourhood that is a CAT(/c) space. It follows for 1.7(3) that the angle /.<f\c(t), c'{lf))
19 A proof in modern language can be found in unpublished lecture notes of Wolfang Meyer
on "Toponogov's Theorem and Applications".

174 Chapter II. 1 Definitions and Characterizations of CAT(re) Spaces
at χ in a comparison triangle A(x, c(i), c(Y)) С Μ\ is a non-increasing function
of t, f > 0. Hence the desired limit exists — call it a. It follows from (1.2.9) that
Ζχ(φ), c'(f)) also converges to a as t, f —> 0. To compute a we can take t = f;
observe that 2 sin(Z(c(i), c'(i))/2) = d(c(t), c'(t))/t. Let ехрл be the exponential map
defined on a neighbourhood of 0 e TXM. There are unit vectors и, и' е Г*М such
that c(t) = expx(tu) and c'{t) = expx(tu'); we have to prove that a is the angle
between и and u', namely that 2sin(a/2) = \\u — u'\\. As the differential of expA. at
0 is an isometry on the tangent spaces, the argument used in the proof of (1.3.18)
shows that if t is small enough, then lim,^od(c(t), d{i))/\\tu — tu'\\ = 1, hence
2sin(a/2) = limt^0d(c(t), c'{t))/t = \\u - u'\\. Π
ΙΑ. 8 Remark. The proof of (1 A.6) shows that if a smooth Riemannian manifold Μ is
of curvature < к in the weak sense of (1.18), then its sectional curvature is bounded
above by к, and hence Μ is of curvature < к in the sense of Alexandrov (1.2).

Chapter II.2 Convexity and its Consequences
In this chapter we concentrate mainly on С AT(0) spaces, establishing basic properties
that we shall appeal to repeatedly in subsequent chapters. We shall also describe the
extent to which each of the results presented can be extended to the case of CAT(/c)
spaces with к > 0.
The most basic of the properties defining the nature of CAT(O) spaces is the
convexity of the metric (2.2). From this property alone one can deduce a great deal
about the geometry of CAT(O) spaces (cf. [Bus55]). We shall also examine orthogonal
projections onto complete convex subspaces (2.4), and we shall see that the existence
of unique centres for bounded subsets implies that every compact group of isometries
of a complete CAT(O) space must have a fixed point (2.8).
The third section of this chapter begins with the observation (from [Ale51]) that
when considering a triangle Δ in a CAT(O) space X, if one gets any non-trivial equality
in the CAT(O) condition, then Δ spans an isometrically embedded Euclidean triangle
in X. (An analogous result holds for CAT(/c) spaces in general (2.10).) This insight
leads quickly to results concerning the global structure of CAT(O) spaces (e.g. 2.14)
which illustrate the subtlety of Alexandrov's definition (1.1).
Convexity of the Metric
We recall the definition of a convex function.
2.1 Definition. A function/ : / —> M. defined on an interval / (not necessarily closed
or compact) is said to be convex if, for any t,fel and s e [0, 1],
/((i-l)f + Ji/)<(i-iy(f) + tf(//)-
A function / : X —> Μ defined on a geodesic metric space is convex if, for any
geodesic path с : / —> X parameterized proportional to arc length, the function
t м> f(c(t)) defined on the interval / is convex. Equivalently, for each geodesic path
с : [0, 1] —> X parameterized proportional to arc length, we have
f(c(s))<(s-lY(c(0)) + sf(c(l))
for each s e [0, 1].

176 Chapter II.2 Convexity and its Consequences
2.2 Proposition. IfX is a CAT(O) space, then the distance function J:IxX->l
is convex, i.e. given any pair of geodesies с : [0, 1] —> X and c' : [0, 1] —> X,
parameterized proportional to arc length, the following inequality holds for all t e
[0, I]:
d(c(t), c'(t)) < (1 - t)d(c(0), c'(0)) + td(c(l), c'(l)).
Proof. We first assume that c(0) = c'(0) and consider a comparison triangle Δ с Е2
for Д(с(0), c(l), c'(l)). Given ί e [0, 1], elementary Euclidean geometry tells us
that d(cjtj, ~d{f)) = td(c{\), сЧТ)) = td{c{\), d{\)). And by the CAT(O) inequality,
d(c(t), c'{t)) < d(c(t), c'(t)). Hence we obtain d(c(t), c'(t)) < td{c{\), c'{\)).
In the general case, we introduce the linearly reparameterized geodesic c" :
[0, 1] —> X with c"(0) = c(0) and c"(l) = c'(l). By applying the preceding special
case, first to с and c" and then to c' and c" with reversed orientation, we obtain:
d(c(t), c"{t)) < td{c{\), c"{\)) and d(c"(t), d{t)) < (1 - t)d{c"{0), c'(0)). Hence,
rf(c(i), c'(i)) < i?(c(i), c"(i)) + d(c"(t), c(t)) < td(c(l), c'(l)) + (1 - t)d(c(0), c'(0)),
as required. D
2.3 Exercises.
(1) Let X be a CAT(/c) space and fix xq e X. Prove that the restriction of χ м>
u?(jc, xo) t° the open ball of radius DK/2 about xq is convex.
(2) Let X be a CAT(0) space. Prove that for all points p, q, r e X,if с : [0,α] -> Χ
and с' : [0, fe] —> X are the unique geodesies joining q to ρ and r to p, respectively,
then d(c(t), c'(t)) < d(q, r) for all t < minfa, b}.
Convex Subspaces and Projection
In subsequent sections we shall make frequent use of orthogonal projections onto
complete, convex subsets of CAT(0) spaces. Orthogonal projection (or simply
'projection') is the name given to the map π : X —> С constructed in the following
proposition. (A careful inspection of the proof of this proposition shows that,
modulo the usual restrictions on scale, one can also project onto convex subsets of CAT(/c)
spaces when к > 0, see 2.6(1).)
2.4 Proposition. Let X be a CAT(0) space, and let С be a convex subset which is
complete in the induced metric. Then,
(1) for every χ e X, there exists a unique point π(χ) e X such that d(x, π(χ)) =
d(x, C) := mfysCd(x, y);
(2) ifx1 belongs to the geodesic segment [χ, π(χ)], then π{χ!) = π {χ);

Convex Subspaces and Projection 177
(3) given χ φ С and у е С, if у φ π {χ) then20 Ζπ^(χ, у) > π/2;
(4) the map χ ы> π (χ) is a retraction of X onto С which does not increase distances;
the map Η :Xx[0, 1] —» X associating to (x, t) the point a distance t d(x, n(x))
from χ on the geodesic segment [χ, π(χ)] is a continuous homotopy from the
identity map of X to π.
Proof. To show the existence of π(χ), we consider a sequence of points yn e С
such that d(yn, x) tends to d(x, C). We claim that this is a Cauchy sequence. Once
we have proved that this is the case, we can take π{χ) to be the limit point, whose
existence is guaranteed by the completeness of С Moreover, the fact that every such
sequence (yn) is Cauchy also establishes the uniqueness of n(x), because if there
were a second point π(χ)' e С with d(n(x)', x) = d(x, C), then the sequence whose
terms were alternately π(χ) and π(χ)' would satisfy the definition of (yn), but would
not be Cauchy.
Let D = d(x, C) and let ε > 0 be small compared to D. By hypothesis, there
exists N > 0 such that d(y„, x) < D + ε whenever η > N. We fix n, m > N and
consider a comparison triangle A(x, y„,ym) С Е2. We then draw two circles about
x, one of radius D and one of radius D + ε. An elementary calculation in Euclidean
geometry shows that any line segment which is entirely contained in the closed
annular region bounded by these two circles can have length at most 2V2e£> + ε2.
The line segment [y„,ym] must be contained in this annular region, for if it were
not then there would exist ζ e \yn, ym] with d(z, x) < D. But then, by the CAT(O)
inequality, the corresponding point ζ € \yn,ym] С С would satisfy d(x, ζ) < D,
contradicting the definition of D. Hence, if n, m > N, then d(yn, ym) < 2V2e£> + ε2.
Thus the sequence (y„) is Cauchy.
(2) follows from the triangle inequality.
For (3), one observes that if Ζπ(Λ)(χ, у) were less that π/2, then one could find
pointsx' e [π(χ), jt]and/ e [π(χ), у] distinct from π(χ) such that, in the comparison
triangle Δ(χ/, π (χ), у1) the angle at π {χ) would be < π/2. This, together with the
CAT(O) inequality, would imply that for some point ρ € [τι (χ), у'] С С we would
have d{x!, p) < d{x!, π(χ)). But by (2), d{x!, π(χ)) = d(xf, C).
(4) We claim that if x\, xj. e X do not belong to С and if π(χ{) φ n(xi), then
d{x\,xi) > d(H(x\, t), H(x2, i))foralli e [0, 1]. To this end, we write jc,-(r) = #0;, t)
and consider the quadrilateral in E2 which is obtained by adjoining comparison
triangles Δ(*ι, π(χ\), jrfe)) and A(jci, it(xi), xi) along the edge [xt, π(χι)~\ with
xi and π{χ\) on different sides. It follows from (3), together with (1.7(4)), that the
angles at π(χ\) and π(χ2) are not less than 7r/2. Hence d(xi,X2) = d(x\,X2) >
d&jj), ~rtf)) > d(xx {t), x2(t)). О
The first part of the following corollary implies in particular that for any r > 0 the
closed r-neighbourhood of a convex set in a complete CAT(O) space is itself convex.
20 Δπ(Χ)(χ,у) is the Alexandrov angle between the geodesic segments [тг(л),л)] and [n(x),y)]

178 Chapter II.2 Convexity and its Consequences
2.5 Corollary. Let С be a complete convex subset in a CAT(O) space X. Let dc be
the distance function to C, namely dc{x) = d(x, C). Then:
(1) dc is a convex function;
(2) for all x, у e X, we have \dc(x) — dciy)\ < d(x, y);
(3) the restriction of dc to a sphere with centre χ and radius r < dc(x) attains its
infimum at a unique point у and dc(x) = dciy) + r.
Proof Let π be the projection of X onto С
(1) Let с : [0, 1] —> X be a linear parameterization of a geodesic segment
and let d : [0, 1] —> С be a linear parameterization of the geodesic segment
[7г(с(0)), 7г(с(1))]. By the convexity of the distance function we have
dc(c(t)) < d(c(t), c'(t))
< (1 - t)d(c(0), c'(0)) + td(c(l), c'(l))
= (l-f)dc(c(0)) + ftic(c(l)).
For (2), note dc(x) < d(x, тг(у)) < d(x, y) + d(y, тг(у)) = d(x, y) + dc(y).
(3) Let у be the point on [χ, π(χ)] such that d(x, y) = r. Then dc(x) = dc(y) + r
by (2.4(2)). If у is such that d(x, y') = r and ас(У) < dc(y), then
d(x, тг(У)) < d(x, У) + d(y', тг(У)) = r + dc(y) = d(x, π(χ)),
hence я (У) = π (χ) and у' =у. D
2.6 Exercises.
(1) Let X be a CAT(/c) space and let С с X be a complete subset which is DK-
convex. If we replaced by V = {xeX\ dc(x) < DJ2] then parts (1) to (3) of (2.4)
remain valid, as does (2.5). Moreover, the map Η described in (2.4(4)) is a homotopy
from id ν t0 7r.
(2) Let X be a complete CAT(/c) space and let С с X be a closed DK-convex
subspace. Endow Υ = X \ С with the induced path metric, let У be its completion,
and define Bd(Y) = Υ \ Y. Show that any geodesic in Υ with endpoints in Bd(Y)
is entirely contained in Bd(Y). (Hint: Υ is open in У and there is a natural map
φ : Υ —> X; if a local geodesic in У met Bd(F) only at its endpoints, then its image
under φ would be a local geodesic of the same length with endpoints in C.)
The Centre of a Bounded Set
Early in this century Elie Cartan [Car28] proved that in a complete, simply connected
manifold Μ of non-positive curvature, given any finite subset [x\, ..., xn], the
function χ м> 53; d(x, Xj)2 has a unique minimum which can usefully be regarded as the
"centre" of the subset. Using this idea, he proved the existence of a fixed point for
the action of any compact group of isometries of M. Using essentially the same idea

The Centre of a Bounded Set 179
(but with a different notion of centre) Bruhat and Tits proved a fixed-point theorem
for group actions on Euclidean buildings [BruT72]. In this section we explain how
this idea applies to complete CAT(O) spaces, and we also explain how to construct a
centre for suitably small bounded sets in complete CAT(/c) spaces with к > 0.
Given a bounded subset У of a metric space X, the radius of Υ is by definition
the infimum of the positive numbers r such that Υ с B(x, r) for some χ e Χ.
2.7 Proposition. Let X be a complete САТ(к) space. If Υ с X is a bounded set of
radius ry < DK/2, then there exists a unique point су е X, called the centre21 ofY,
such that Υ с B(cy, ry).
Proof. Let (x„) be a sequence of points in X with the property that Υ с В(х„, r„) and
rn —> ry as η —> oo. We shall prove that (jc„) is a Cauchy sequence; the idea of the
proof is very similar to that of (2.4). Since X is complete, this sequence will have a
limit point, and such a limit has the property required of cy. The fact that every such
sequence (x„) is Cauchy also establishes the uniqueness of cy, as in (2.4).
We fix a basepoint О e M2. Given ε > 0 we choose numbers R e (ry, DK/2) and
/?' < ry such that any geodesic segment which is entirely contained in the annular
region A = B(0, R) \ B(0, R') has length less than ε.
For sufficiently large n, ri we have r„,r„- < R. Let m be the midpoint of the unique
geodesic segment j oining x„ to xn>. For each у е Y, we consider a comparison triangle
Ay = А(0,хЦ,х^) С M^ forA(y, χ„, χηι). If it were the case that for every у е У the
midpoint of [хЦ, х^] С Δ (у) belonged to 5(0, /?'), then by the CAT(/c) inequality we
would have У С 5(m, /?'), contradicting the fact that R' < ry. Therefore, there exists
у & Υ such that the midpoint of [хЦ, 3ζ/] С Δ^ lies in the annulus A. This implies
that at least half of \x^, x#] lies in A, and hence \x^, x^] has length less than 2ε. D
For an alternative proof in the case к = 0, see [Bro88, p. 157]. Finer results of a
similar nature can be found in the papers of U. Lang and V. Schroeder [LS97a].
2.8 Corollary.
(1) IfX is a complete CAT(O) space and Г is a finite group of isometries ofX or,
more generally, a group of isometries with a bounded orbit, then the fixed-point
set of Τ is a non-empty convex subspace ofX.
(2) If a group Γ acts properly and cocompactly by isometries on a CAT(O) space,
then Γ contains only finitely many conjugacy classes of finite subgroups.
Proof. In order to see that the fixed point set of Γ is non-empty, one simply applies
the preceding proposition with the role of Υ played by a bounded orbit of Γ; since
Υ is Г-invariant, so is its centre. If an isometry fixes p, q e X, then it must fix the
unique geodesic segment [p, q] pointwise. Hence the fixed-point set of Γ is convex.
Part (2) follows immediately from (1) and 1.8.5(5). D
more precisely, circumcentre.

180 Chapter Π.2 Convexity and its Consequences
Flat Subspaces
Most of the results that we have presented so far follow fairly directly from the
definition of a CAT(/c) space. In this paragraph we begin to strike a richer vein of
ideas, based on the observation (from [Ale51]) that when considering a triangle Δ
of perimeter at most 2DK in a CAT(/c) space X, if one gets any non-trivial equality in
the CAT(/c) condition then the natural map Δ —> Δ from any comparison triangle
Δ С Μ2 extends to give an isometric embedding of the convex hull of Δ into X (see
(2.9) and (2.10)).
Flat Triangles
By definition, the (closed) convex hull of a subset Л of a geodesic space X is the
intersection of all (closed) convex subspaces of X containing A. The convex hull of
a geodesic triangle A(p, q, г) С X coincides with the convex hull of its vertex set
{p, q, r}; in a general CAT(O), for example in the complex hyperbolic plane CH2 the
convex hull of three points in general position is 4—dimensional (see (10.12)).
2.9 Proposition (Hat Triangle Lemma). Let A be a geodesic triangle in a CAT(O)
space X. If one of the vertex angles of A is equal to the corresponding vertex angle
in a comparison triangle Δ с Ε2 for Δ, then "Δ is flat", more precisely, the convex
hull ο/Δ in X is isometric to the convex hull of A in E2.
Proof. Let Δ = A(p, q, q") and suppose that Zp(q, q') = Zp(q, q')- Let r be any
point in the interior of the segment [q, q']. The first step of the proof is to show that
our hypothesis on Zp(q, r) implies that for all such r we have equality in the CAT(O)
condition, i.e. d(p, 7) = d(p, r)^Let Δ' = A(p, q, r), and let^V' = A(p, q', r). We
consider comparison triangles Δ' = A(p, q, r) for Δ', and Δ" = A(p, q', r) for Δ"
in E2, and assume that these are arranged with a common side [p, r] so that q and q'
are not on the same side of the line through [p, r]. Let r be the comparison point in
Δ for r. As the sum of the angles at r of Δ' and Δ" is not less than π, we can apply
Alexandrov's lemma (1.2.16). We have
Zp(q, q') < Zp(q, r) + Zp{c(, r) < Z-p(q, r) + Zp(q', r) < Zjfq, q'),
where the second inequality follows from the CAT(O) condition (1.7(4)). By
assumption Z.p{q, q1) = Zp(q, q'), hence we have equality everywhere, in particular
^р(Я' r)+^p(<l'> r) — ^p(5, q')- Therefore, by Alexandrov's lemma again, d(p, r) =
dip, r) = d(p, r), as desired.
Let j be the map from the convex hull C(A) of Δ С Ε2 to X which, for every
r e [q, g'], sends the geodesic segment [p, r] isometrically onto the geodesic segment
[p, r]. We claim that j is an isometry onto its image; it then follows that the unique
geodesic joining any two points of the image of j will be contained in the image,
so j maps C(A) onto C(A). Consider two points χ e [p, r] and x7 e [p, F] in
C(A), where r and r' lie on [q, q'], and r is between q and У. Let χ = j(x), xf =
jix1), r = j(r), r1 = j(r'), and let 5i, 52, 53 be the angles Zp(q, r), Zp(r, r1), Z^r1, q'),

Flat Subspaces 181
respectively; Si, S2, <$з will denote the corresponding angles in Δ. The first step of
the proof shows that Αφ, q, 7) is a comparison triangle for A(p, q, r), hence 51 < Si.
Similarly, S2 < δ2 and S3 < S3. But Zp(q, q") < St + S2 + S3 < Si + S2 + S3 =
Z^(g, q') = Z.p(q, q1), so in fact S2 = S2. Hence d(x, x1) = d(x, 3?), as required. D
2.10 Exercises.
(1) Let A(p, q, r) be a geodesic triangle in а С AT(0) space, and let Αφ, q, r) С Е2
be its comparison triangle. Use (2.9) to prove that if there exists a point χ in the interior
of [p, f] and a point ;y e [p, g] with ;y φ ρ such that ύ?(χ, у) — d(x, у), then the triangle
A(p, q, r) is flat.
(2) Generalize Proposition 2.9 and (1) to arbitrary CAT(/c) spaces.
Flat Polygons
2.11 The Flat Quadrilateral Theorem. Consider four points ρ, q, r, s in a CAT(O)
space X. Let a = ^P(q, s), β = Z.q{p, r), γ = Z.r{q, s) and δ = Zs(r, p). If a + β +
γ + δ > 2π, then this sum is equal to 2π and the convex hull of the four points
p, q, r, s is isometric to the convex hull of a convex quadrilateral in E2.
Proof. The method of proof is similar to that of the preceding proposition. Let Δ ι =
A(p, q, s) and Δ2 = Δ(γ, q, s). To begin, we construct a quadrilateral in E2 by
joining comparison triangles Δι = A(p, q, s) and Δ2 = Δ(γ, q, s) along the edge
[q, J] so that ρ and r lie on opposite sides of the line that passes through q and s.
We denote the angles of Δ] at the vertices;?, q, s by α, β{, δι, respectively, and
those of Δ2 at the vertices f, q, s by ψ, β2, δ2. From (1.7(4)) and (1.1.14) we have
a < α, β <β~{+β~2, δ < 5ι + δ2, γ <у-
We are assuming that a + β + γ + δ > 2π, and the sum of the angles of a Euclidean
quadrilateral is 2π, so we conclude that in the above expression we have equality
everywhere. In particular βι + β2 = β < π and δι +δ2 = δ < π; therefore
the Euclidean quadrilateral Q with vertices {p, q, r, s} is convex. The preceding
proposition then implies that Δι and Δ2 bound flat triangles. Letj be the map from
the convex hull С of Q to X whose restrictions, j\ andy2, to the convex hulls C{A\)
and C(A2) of the triangles Δι and Δ2 are isometries onto the convex hulls of Δι
and Δ2. To check that j is an isometry, we have to prove that, for all x\ e C(Ai)
and x2 e C(A2), ifj(xi) = χι andj(x2) = x2, then d(x\,x2) = d(x\,x2). This will
follow once we have proved that the angle μ = /Lq{x\,x2) is equal to the angle
μ = /.q{x\,x2).
To check this equality, we let β[ = ^qip, x\), β" = ^q(*i, s), β2 = ^q(s, x2)
and β2 = ^q(x2, r). As above, one sees that each of these angles is equal to the
corresponding angle in the comparison figure. Hence
β<β[+μ + β2<β[+β'{ + βϊ + β2=-βι+-β2 = β.
And since β'χ—βχ and β2 = β2, we deduce μ = ~β. Π

182 Chapter Π.2 Convexity and its Consequences
2.12 Exercises.
(1) Let X be a CAT(O) space and consider η distinct points {/>(}"= ι in X- Let
a, = ZP;(pi-i,Pi+i), where indices are taken mod n. Prove that if Σ ai — (n — 2)π
then the convex hull in X of {/>,'}<"= ι is isometric to a convex и-gon in E2.
(2) (The Sandwich Lemma) Let X be a CAT(O) space. We write dc(x) =
inf {ύ?(χ, с) | с е С} to denote the distance of a point from a closed subspace С СХ.
Let Ci and C2 be two complete, convex subspaces of X. Prove that if the restriction
of dCi to Cj. is constant, equal to a say, and the restriction of dCl to Cj is constant,
then the convex hull of C\ U Cj. is isometric to C\ x [0, a].
(3) With the hypothesis of (2), let ρ : X -> Ct be orthogonal projection. Let С
be an arbitrary subspace of X. Prove that if the restriction of ρ to C" is an isometry
onto C\ then the restriction of dCl (x) to С is constant, equal to b say, and there is a
unique isometry 7 from С χ [0, b] to the convex hull of Ci U С such thatyXx, 0) = χ
and 7'(x, b) = p(x) for all χ e С".
Flat Strips
Recall that a geodesic line in a metric space X is a map с : К —> X such that
d(c(t), c(O) = I* — f\ for all i, f7 e M. Two geodesic lines с, с' in X are said to be
asymptotic if there exists a constant К such that d(c(t), c'(t)) < К for all (el.
We remind the reader that if a function Ж —» Ж is convex and bounded, then it is
constant.
2.13 The Flat Strip Theorem. Let X be a CAT(O) space, and let с : К -> X and
с' : Μ —> Χ be geodesic lines in X.Ifc and c' are asymptotic, then the convex hull of
c(K) U c'(K) /j isometric to aflat strip Κ χ [0, D] с Е2.
Proo/ Let π be the projection of X onto the closed convex subspace c(K) (cf. (2.4)).
By reparameterizing if necessary, we may assume that c(0) = 7r(c'(0)), i.e., that c(0)
is the point on c(K) closest to c'(0).
The function ί м> d(c(t), c'(t)) is convex, non-negative and bounded, hence
constant, equal to D say. Similarly, for all a e K, the function t м> (c(t + a), c'(t)) is
constant. In particular d(c(a + t), c'{t)) = d(c(a), c'(0)) > d(c(0), c'(0)), and hence
7r(c'(0) = c(i) for all i. This same inequality shows that the projection π' onto c'(K)
maps c(i) to c'(i)·
Given ί < f, we consider the quadrilateral in X which is the union of the geodesic
segments [c(t), c(f)], [c(f), c'(f)l [c'(0. c'(01 and [c'(i). c(01· According to (2.4),
all of the angles of this quadrilateral are at least π/2. By (2.11), the convex hull of
(c(i), c'(t), c(f), c'if)} is isometric to a Euclidean rectangle, and therefore the map
j:lx[0,D]^X which sends (t, s) to the point on the geodesic segment [c(t), c'(t)]
a distance s from c(t) is an isometry onto the convex hull of c(M.) U c'(K). Π
In view of the Rat Strip Theorem, the terms parallel and asymptotic are
synonymous when used to describe geodesic lines in a CAT(O) space. Both are in common
use.

Flat Subspaces 183
2.14 A Product Decomposition Theorem. Let X be a CAT(O) space and let с :
К —» X be a geodesic line.
(1) The union of the images of all geodesic lines d : К —> X parallel to с is a
convex subspace Xc ofX.
(2) Let ρ be the restriction to Xc of the projection from X to the complete convex
subspace c(W). Let X^ = p~](c(0)). Then, X°C is convex (in particular it is a
CAT(O) space) andXc is canonically isometric to the product X°c χ Κ.
Proof. Given two points x\,x2 e Xc we fix geodesic lines c\ and c2 parallel to с
such that x\ lies in the image of c\ and x2 lies in the image of c2. Because c\ is
parallel to c2, we can apply the Hat Strip Theorem and deduce that the convex hull
of C] (K) U с2(Ж) is isometric to a flat strip; in particular, it is the union of images of
geodesic lines parallel to c. This proves that Xc is convex, hence it is a CAT(O) space.
In order to prove (2), we restrict our attention to those geodesic lines c' which
are parallel to с and for which p(c'(0)) = c(0). Every x e XC lies in the image of a
unique such geodesic line, which we denote cx.
Let 7 :XjxR-> Xc be the bijection defined byj(x, t) = cx{t). Using the Flat Strip
Theorem, it is clear that this map is an isometry provided d(x, x1) = d(cx(R), с^(Щ)
for ζΆχ,χ1 eXj. Thus the following lemma completes the proof of the theorem. D
2.15 Lemma. Consider three geodesic lines c\ : К —» X, i = 1,2, 3 in a metric
space X. Suppose that the union of each pair of these lines is isometric to the union
of two parallel lines in E2. Letpij be the map that assigns to each point ofq(№.) the
unique closest point on c,(R). Thenp\ 3 opi2 °Pi,\ — P\,\, the identity ofc\(K).
Proof. This proof was simplified by use of an idea suggested to us by Phil Bowers and
Kim Ruane (cf. [BoRu96b]). Ifpi,3 °Рз,2 °Pi,\ were not the identity on c\ (K), then it
would be a translation by a non-zero real number, b say, so/?i3 оръ 2 °Pi,\{c\{t)) =
ci(t + b)foTzRteR.
If we reparameterize C2 and C3 so that P2,i(c\(0)) = сг(0) and /?з,2(сг(0)) =
c3(0), then pi,3(c3(0)) = cx(b). Let α, = d(c'i(K), c2(K)), α2 = d(c2(K), c3(K)),
α3 = d(ci(M), сз(М)), and let a = ax + a2 + аъ. By hypothesis, d{c\{t), c2{t + s)) =
^a\ + s2, d(c2(t), c3(i + s)) = Ja\ + s2 and d{a{t), c\(t + s)) = J a2 + (s- b)2.
Therefore, for all s,
d(ci(0), ci(as + b))
< d(a(0), c2(a]s)) + d(c2(a]s), c3((ai + a2)s))
+ d(c3((ai + a2)s), c\((ax +a2 + a3)s + b))
= αι-v/l +s2 + а2л/\ + s1 + аЪу/\ + s2 = ay/1 +s2.
But since c\ is a geodesic, d(ci(0), ci(as + b)) = \as + b\, so we have (as + b)2 <
a2(\ + s2) for all seK, which is impossible if b φ 0. D

Chapter II.3 Angles, Limits, Cones and Joins
In this chapter we examine how upper curvature bounds influence the behaviour of
angles, limits of sequences of spaces, and the cone and join constructions described
in (1.5). We then use the results concerning limits and cones to describe the space of
directions at a point in a CAT(/c) space.
Angles in САТ(к) Spaces
In CAT(/c) spaces, angles exist in the following strong sense.
3.1 Proposition. Let X be a CAT(/c) space and let с : [0, a] —> X and c' :
[0, a'] —> X be two geodesic paths issuing from the same point c(0) = c'(0).
Then the к-comparison angle Z*,L(c(f), c'(f)) is a non-decreasing function of both
t,f>0, and the Alexandrov angle Z.{c, c') is equal to lim,,,<_».o Z^0){c{t), c'(f)) =
lim^o 4*0)(c(0. c'(0)· Hence, in the light of (1.2.9),
Z(c, c) = lim 2 arcsin — d(c(t), c'(t)).
Proof. Immediate from 1.7(3) and the fact that one can take comparison triangles in
M2 instead of E2 in the definition (1.1.12) of the Alexandrov angle (see 1.2.9). D
3.2 Notation for Angles. For the convenience of the reader we recall the following
notation. Let p, x, у be points of a metric space X such that ρ φ χ, ρ фу.
Zj?\x, у) denotes the comparison angle in M2. (see 1.2.15);
^P(x< У) = ^-f\xi У) denotes the comparison angle in E2 (see 1.1.12);
if there are unique geodesic segments \p, x] and [p, y], then we write Zp(x, y)
to denote the (Alexandrov) angle between these segments (see 1.1.12).
Recall that a real-valued function/ on a topological space Υ is said to be upper
semicontinuous if/(y) > limsupn^00/(y„) whenever y„ —> у in Y.
3.3 Proposition. Let X be a CAT(/c) space. For all points p,x,y e X with
max{d(p, x), d(p, y)} < DK,

Angles in CAT(re) Spaces 185
(1) the function (p, x, y) i-> Zp(x, y) is upper semicontinuous, and
(2) for fixed ρ e X, the function (x, y) \-> Z.p(x, y) is continuous.
Proof. (1) Let (xn), (yn) and (pn) be sequences of points converging to x, у and ρ
respectively. Let c, c', c„ and c'„ be linear parameterizations [0, 1] —> X of the
geodesic segments \p,x], \p,y], [ρη,χη] and [p„,y„] respectively. Fori e (0, 1], let
α(ί) = 4*>(c(f), C(t)) and let α„(ί) = 4*>(c„(f), c'n(t)). According to (3.1), a(t)
and a„(t) are non-decreasing functions of t and α := Zp(x, y) = lim(_>.o oc(t) and
cen := ZPn(xn, y„) = lim,_>.o ot„(t). And for fixed t we have a„(t) -> α(ί) as и -> oo.
Given ε > 0, let Γ > 0 be such that a(t) — ε/2 < a for all t e (0, Γ]. Then for η big
enough, α„(Γ) < a(T) + ε/2, therefore an < a„(T) < a(T) + ε/2 < a + ε. Thus
lim sup an < a as required.
(2) We keep the above notations, but we assume that pn = ρ for all n. Let β„ =
Zp(x,xn) and y„ = Zp(y,yn). By (1.7(4)), βη -+ 0 and y„ -► 0 as и -^ oo. By the
triangle inequality for Alexandrov angles, (α-α„| < β„ + γ„. Hence lim^^ce,, = a.
D
3.4 Remark. With regard to part (1) of the preceding proposition, we note that in
general it will not be true that ρ м> Zp(x, у) is a continuous function. For example,
consider the CAT(O) space X obtained by endowing the subset {(jci , 0) ( χι eK}U
{(χι, x2) ( χι > 0, x2 € Щ of the plane with the induced length metric from E2. Let
ρ = (0, 0) and let p„ = (— l/n, 0). Given any χ = (x\, x2) and у = (y\, y2) e X such
that x\ > 0 and yi > 0, the angle which χ and у subtend at ρ is equal to the usual
Euclidean angle, but Z.Pn{x, y) = 0 for all n.
The following addendum to (3.3) will be useful later.
3.5 Proposition. Let Xbea САТ(к) space, let с : [0, ε] —> X be a geodesic segment
issuing from ρ = c(0) and let у be a point ofX distinct from ρ (with ε and dip, y)
less than DK if к > 0). Then,
lim Zp(c(s), y) = Ζρ(φ), у).
s-i-0
Proof. Asjh- ZpK\c(s), y) is non-decreasing, γ := lims^o ZpK\c(s), y) exists. By
(1.2.9), we have γ = lims_*o Z.p(c(s), y). This last expression is, by definition, the
strong upper angle between [p, y] and c, which we showed in (1.1.16) to be equal to
the Alexandrov angle. D
3.6 Corollary (First Variation Formula). With the notation of the preceding
proposition,
.. d(c(0),y)-d(c(s),y)
hm
s^-0 S
exists and is equal to cos Zp(c(s), y).
Proof. This follows from the preceding proposition and the Euclidean law of cosines.
D

186 Chapter II.3 Angles, Limits, Cones and Joins
4-Point Limits of САТ(к) Spaces
We now turn our attention to limits of sequences of CAT(/c) spaces. We want a
notion of convergence for metric spaces that is strict enough to ensure that if a
complete geodesic space is a limit of CAT(/c) spaces then it is itself a CAT(/c) space.
On the other hand, we would like a notion of convergence that is weak enough to
allow a wide range of applications. We would also like a definition that allows us
to construct a "tangent space" at each point in a CAT(/c) space X as a limit of the
sequence (X, d„), where dn(x, y) = η d(x, y) (see 3.19). Since the local structure of a
CAT(/c) space can differ wildly from one point to another, this seems a lot to ask, but
in our discussion of ultralimits (1.5) we saw that a single sequence of spaces can have
a wide variety of "limits" if the notion of limit is defined in terms of approximations
to finite configurations of points. Since the CAT(/c) inequality can be characterized
by a condition on 4-point configurations (1.10), we are led to the following definition
(cf. [Ni95]).
3.7 Definition of 4-Point Limits. A metric space (X, d) is a 4-point limit of a
sequence of metric spaces (Xn, dn) if, for every 4-tuple of points {х\,хг,хъ,м)
from X and every ε > 0, there exist infinitely many integers η such that there is
a 4-tuple (xi(n), xj.(n), χ-$(η), x^{n)) from Xn with \d{xt, Xj) — d„(xi(ri), Xj(n))\ < ε for
1 < i,j < 4.
3.8 Remark. The unrestrictive nature of 4-point limits means that if (X, d) is a limit
of a sequence of metric spaces (Xn, dn) in most reasonable senses (for instance, a
Gromov-Hausdorff limit or an ultralimit) then it is a 4-point limit of the sequence.
A single sequence (Xn, dn) may have a wide variety of 4-point limits. For example,
given a metric space Y, every subspace X С Υ is a 4-point limit of the constant
sequence X„ = Y.
The final sentence of the preceding remark shows that if one wishes to deduce
that a 4-point limit of a sequence of CAT(/c) spaces is CAT(/c) then one must impose
an hypothesis on the limit to ensure the existence of geodesies. The most satisfactory
way of doing this is to require that the space be complete and have approximate
midpoints (cf. 1.11).
3.9 Theorem (Limits of CAT(/c) Spaces). Let (X, d) be a complete metric space.
Let (X„, d„)be a sequence о/САТ(к„) spaces. Suppose that (X, d) is a 4-point limit
of the sequence (Xn, dn) and that к = lim„^oo κ η- Suppose also that every pair of
points x, у e X with d(x, y) < DK has approximate midpoints. Then X is a CAT(/c)
space.
Proof. According to (1.12), it suffices to show that X is а САТ(к') space for every
к' > к, and according to (1.11) for this it suffices to show that X satisfies the САТ(к')
4-point condition. Fix к' > к. For η sufficiently large we have κη < κ', so by (1.12)
we may assume that all of the Xn are CAT(/c') spaces.

4-Point Limits of С AT(«) Spaces 187
Let (x\,у\,хг, уг) be a 4-tuple of points from X with d{x\,y\,) + d(y\, xj) +
d(xi,yi) + а{уг,х\) < 2DK>. According to the definition of a 4-point limit, there
is a sequence of integers и, and 4-tuples {х\{щ),ух(щ),Х2(пд,Уг{щ)) from X„;
such that, for j, k e {0, 1}, as щ —> oo we have d(xj(rij), хк{щ)) —> d(jj, jc*) and
d(yj(n,), yk(n,)) -> а{уь yk), and d(xj(n), yk(n)) -* d(xj, yk).
Each Хщ is a CAT(/c') space, so the 4-tuple {х\{п{),у\{щ),Х2{щ),у2{гц)) has a
subembedding (xi(«,·). ίι^ί).^^·),^^)) ш М%,. We may assume that all of the
points x\{rii) are equal, so these 4-tuples are all contained in a compact ball in M^,.
Passing to a subsequence if necessary, we may then assume that the sequences I2(ij)
and yk(n,) converge, to хг and yk say. Clearly (xi,y~i,X2, У2) ls a subembedding of
(χι, y\, x2, уг)- Thus we have shown that X satisfies the CAT(/c') 4-point condition,
and we conclude that X is CAT(/c') for every κ' > κ. D
We articulate some special cases of (3.9) that are of particular interest.
3.10 Corollary. Fix κ el. Let (X, d) be a complete metric space and let (X„, d„) be
a sequence o/CAT(/c) spaces.
(1) If(X, d) is a (pointed or unpointed) Gromov-Hausdorff limit of(X,„ d„), then
(X, d) is a CAT(k) space.
(2) If(X, d) is an ultralimit of(X„, d„), then (X, d) is a CAT(/c) space.
Let (Y, 8) be a CAT(k) space and let ω be a non-principal ultrafilter on N. Let
Y„ = Y, let δ„ = η.δ and let δ'η = ±S.
(3) Ιϊπιω(Υη, δ„) is a CAT(0) space.
(4) If к = 0, then СопешУ = Иш^У,,, 5,',) is a CAT(0) space.
(5) If к < 0, then СопешУ is an R-tree.
Proof. In each case the existence of approximate midpoints follows easily from the
hypothesis. For(3), (4) and (5) we recall that an ultralimit of geodesic spaces is always
complete and geodesic, and that if (Y, d) is CAT(/c) then (Y, X.d) is САТ(Х~2к). D
Metric Completion
For the most part, in this book we work with spaces that are assumed to be complete
and we only resort to the additional hypothesis of local compactness when it appears
unavoidable. One advantage of this approach is that any CAT(/c) space can be realized
as a dense, convex subset of a complete САТ(к) space, namely its metric completion.
3.11 Corollary. The metric completion (X1, d') of a CAT(/c) space (X, d) is а САТ(к)
space.
Proof. It is clear that (X1, d') is a 4-point limit of the constant sequence (Xn, dn) =
(X, d) and that it has approximate midpoints. It is complete by hypothesis, so we can
apply (3.9). D

188 Chapter Π.3 Angles, Limits, Cones and Joins
3.12 Remark. We saw in (1.3.6) that the completion of a geodesic space need not be
geodesic, so our appeal to (3.9) in the above proof masked the fact that the curvature
hypothesis was being used to prove the existence of geodesies.
3.13 Exercises
(1) Give an example of a space X such that X and its completion X' are locally
compact geodesic metric spaces and X is non-positively curved but X' is not.
(2) Let U С Ml be a bounded subset that is open and convex. Let с : [0, /] —> Ml
be an arc-length parameterization of the (rectifiable) boundary curve of U. Let P„ с
M\ be the polygonal disc bounded by the concatenation of the geodesic segments
[c(il/n), c((i + 1)//«)], i = 0,..., η - 1. Let Xn = M2K ч Pn. Prove that Μ ч U is the
(4-point) limit of the sequence Xn (where all of the spaces considered are endowed
with the induced path metric from Ml).
(3) Fix к < 0, consider an open convex subset U С М2К, let U be its closure and
let Bd(i/) = V ч U. Prove the following facts and use (2) and 1.15(2) to deduce that
Ml ч U has curvature < /c. (Hint: One can reduce to the case where U is bounded
by restricting attention to a suitable ball in M2K.)
(i) Fix ρ e U. Show that for all χ e Ml ч (7 the geodesic segment [jc, /?] intersects
Bd(t/) in exactly one point, which we denote x1.
(ii) Show that χ м> χ1 is a continuous map and that Bd(t/) is a 1-manifold,
(iii) Show that if U is bounded and с is a simple closed curve enclosing U, then
the length of Bd(t/) (which is homeomorphic to a circle) is less than that of c.
(Hint: Consider the orthogonal projection onto U.)
(iv) Show that if U is unbounded then each component of its boundary is the image of
a map с : К —» Мгк whose restriction to each compact subinterval is rectifiable.
In the case к = 0, prove that if Bd(t/) is not connected then it consists of two
parallel lines.
Cones and Spherical Joins
The purpose of this paragraph is to prove the following important theorem of
Berestovskii [Ber83] (see also [AleBN86]). This result provides a basic
connection between CAT(1) spaces and CAT(/c) spaces in general. The first evidence of
the importance of this link will be seen in the next paragraph, where we shall
combine (3.14) with results on limits in order to prove that the 'tangent space' at any
point of a CAT(/c) space has a natural CAT(O) metric. We shall make further use
of Berestovskii's theorem in Chapter 5, when we discuss curvature in polyhedral
complexes.
The definition of a /c-cone was given in Chapter 1.5.
3.14 Theorem (Berestovskii). Let Υ be a metric space. The к-cone X = CKY over
Υ is a CAT(/c) space if and only if Υ is a CAT(1) space.

Cones and Spherical Joins 189
Proof. First we assume that У is a CAT(l) space and prove that X is a CAT(/c)
space. By 1.5.10, we know that any two points xi,x2 e X with d{x\,x2) < DK are
joined by a unique geodesic segment [xt, хг]. We have to check that the CAT(/c)
inequality is satisfied for any geodesic triangle with vertices jc, = ?;у,, i = 1, 2, 3
(assuming that its perimeter is less than 2DK). If one of the i, is zero, then the triangle
[jei,*2] U [x2, хз] U [хз, xi], with its induced metric, is isometric to its comparison
triangle in M2, so we may assume each /, > 0.
Following Berestovskii [Be83], we consider three cases:
(a) d(yu y2) + d(y2, уз) + d(y3, yi) < 2π.
(b) d(yi, y2) + d(y2, уз) + d(y3, yx) > 2π but d{yt, yj) < π for all i,j = 1, 2, 3.
(c) One of the d(yt, yj) is > π.
Case (a). Let Δ = A(\yi,y2], \y2, уз], \уз,у\]) С Υ. We fix a comparison
triangle Δ in M\ = S2 with vertices yx, y2, уъ. The comparison map Δ —» Δ extends
naturally to a bijection from CKA с M\ to CKA. Let χ = ty be a point on the
segment [x2, хз\ and let у be the comparison point for у in [^ггУз]· The triangle
Δ(χι,^2.^3) С Мък, with χι = tiyir can be viewed as a /c-comparison triangle for
Α(χχ,χ2, Χ3), with χ = ty as the comparison point for*. As the CAT(l) inequality
holds for Δ, we have d(y\, y) < d(yx,y). From this we see (using the formula 1.5.6
defining d{x\, x)) that d{x\, x) < d(x\, x).
Case (b). Let Δ(0,x.\,x2) and Δ(0,х\,хз) be comparison triangles in M2. for
Δ(0, x\, x2) and Δ(0, x\, Χ3), respectively, chosen so that x2 and X3 are on different
sides of [χι, 0] (where, as usual, 0 denotes the cone point of X). From the definition
of the metric on CKY we have Zq(x\,5c2) = d(y\,y2), /5(^1,^3) = а{у\,уъ), and
^ϊι(0,x2) = ZXl(0,x2) and /^,(6,^3) = Zt,(0, jc3). And by hypothesis we have
<ХуиУ2) + d(y\,y3) > π, hence
^ofe, h) = 2π - Z-0(x2, χι) - Z-0(x3, x\) < d(y2, y3) = Z0(x2, хз)
and d{x2, хз) < d(x2, X3). Therefore, in a comparison triangle Δ = Δ(χι, χ2, χ3) in
M2. for Α(χχ, jt2, хз), we have:
^ϊ, fa, хз) > ^ϊ, (h, h) = ^i, (h, 0) + А (б, хз)
= ZXt (x2, 0) + ZXl (0, хз) > ZXi {x2, хз)
Hence condition (1.7(4)) is satisfied.
Case (c). Suppose d{y\, уз) > π. Then the geodesic segment [*ι, *з] is the
concatenation of [xi, 0] and [0, JC3]. Let Δι = Δ(0, x\,x2) and Δ3 = Δ(0,хз,х2)
be comparison triangles in M2 for the triangles Δι = Δ(0, χι, хг) and Δ3 =
Δ(0, хз, x2), chosen so that the vertices x\ and X3 lie on different sides of the
common segment [0, x2]. (In the case к > 0 such comparison triangles exist because the
perimeter of Δ(*ι, χ2, Χ3) is assumed to be smaller than 2DK). The sum of the angles
at 0 in Δι and Δ3 is d„(yi,y2) + ал{у2,уз) > π. When we straighten the union
of these two triangles to get a comparison triangle for Δ(*ι, x2, X3), Alexandrov's
lemma (1.2.16) ensures that the angles do not decrease, and therefore А(х1гх2,хз)
satisfies the CAT(/c) inequality.

190 Chapter Π.3 Angles, Limits, Cones and Joins
It remains to be proved that if X is a CAT(/c) space then У is a CAT(l) space.
From 1.5.10(1) we know that pairs of points a distance less than π apart in Υ are
joined by a unique geodesic segment. Let A(\y\, y2], \уг, Уз], \Уз, У ι]) be a triangle
in Y of perimeter less than 2π and let Δ = A(y{, y2, у·}) be a comparison triangle in
М\. Given у e \y%, y^], let у е \y2, У3] be its comparison point. On the subcone CK Δ,
we consider three points jc, = eyi, i= 1,2,3, where ε is positive (and small enough
to ensure that the perimeter of the triangle with vertices х\,хг, хъ is less than 2DK).
The cone CKA is a subcone of CKM\ с Мък and the points 5c, = syh i= 1, 2, 3, are
the vertices of a comparison triangle Δ for the geodesic triangle Δ' with vertices
*ι,*2, JC3. If χ = ty e [jc2, JC3], then its comparison point is χ = ty. By hypothesis
d(x\, jc) < d(x\, I). Hence d{y\, y) < d(yi, y), by the definition of the metric on
CKY. D
3.15 Corollary. The join Y\ * Yi of two metric spaces Y\ and Y2 is a CAT(l) space
if and only if Y[ and Yj. are CAT(l) spaces.
Proof. This follows immediately from the theorem, because we saw in (1.5.15) that
C0(Yi * Y2) is isometric to C0Yi χ C0Y2, and C0Y\ χ C0Y2 is a CAT(O) space if and
only if C0Yi and C0Y2 are CAT(O) spaces (1.16(2)). D
3.16 Corollary. Let Υ be a metric space. The following conditions are equivalent.
(1) CKY is а СΑΎ(κ) space.
(2) С к Υ has curvature < к.
(3) A neighbourhood of the cone point 0 e CKY is a CAT(/c) space.
Proof. Trivially, (1) =φ· (2) =>· (3). And (3) =>■ (1) is a consequence of the
preceding theorem. D
3.17 Example. If S is isometric to a circle of length a, then its cone CKS is а САТ(/с)
space if and only if a > 2π.
The Space of Directions
3.18 Definition (The Space of Directions and the Tangent Cone). Let X be a metric
space. Two non-trivial geodesies с and d issuing from a point ρ e X are said to define
the same direction at ρ if the Alexandrov angle between them is zero. The triangle
inequality for angles (1.1.14) implies that [с ~ d iff Z(c, d) = 0] is an equivalence
relation on the set of non-trivial geodesies issuing from p, and Ζ induces a metric
on the set of equivalence classes. The resulting metric space is called the space of
directions at ρ and is denoted Sp(X). (Note that two geodesies segments issuing from
ρ may have the same direction but intersect only at p.)
CqSp(X), the Euclidean cone over Sp(X), is called the tangent cone at p.

The Space of Directions 191
If X is a Riemannian manifold, then Sp(X) is isometric to the unit sphere in the
tangent space of X at ρ and CqSp{X) is the tangent space itself. If X is a polyhedral
complex, then Sp(X) is isometric to Lk(p, X), the geometric link of p. In a Riemannian
manifold with non-empty boundary, if ρ is a point on the boundary then Sp(X) may
not be complete; for instance if X is a closed round ball in E" and ρ belongs to the
boundary of this ball, then Sp(X) is isometric to an open hemisphere in §""'.
The following theorem is due to I. G. Nikolaev [Ni95].
3.19 Theorem. LetK e M..Ifa metric space X has curvature < к, then the completion
of the space of directions at each point ρ eX is a CAT(l) space, and the completion
of the tangent cone at ρ is a CAT(O) space.
Proof. In the light of Berestovskii's Theorem (3.14), it suffices to prove that the
completion of Co(Sp(X)) is a CAT(O) space. Following the outline of a proof by
B. Kleiner and B. Leeb [K1L97], we shall prove this by showing that C0(Sp(X))
satisfies the CAT(O) 4-point condition and has approximate midpoints (cf. (1.11) and
(3.9)). In the light of (3.16), it suffices to establish these conditions in a neighbourhood
of the cone point.
Since Sp(X) depends only on a neighbourhood of p, we may assume that X is a
CAT(/c) space of diameter less than DK/2, in which case there is a unique geodesic
segment [p, x] for every χ e X.
Let j : X —> CqSp(X) be the map that sends ρ to the cone point 0 and sends each
x φ ρ to the point a distance d{x, p) from 0 that projects to the class of [p, jc] in Sp(X).
And for each t e [0, 1 ], let tx denote the point of X a distance td(p, x) from ρ on the
geodesic segment [p, x].
For each ε e (0, 1], define de to be the following pseudometric on X:
1
ds(x, y) = - d(sx, sy).
ε
Note that ds satisfies the CAT(e2/c) 4-point condition (and hence the CAT(O)
4-point condition if к < 0).
Fix x, у е X distinct from p, and let γε = Ζρ(εχ, sy). By the Euclidean law of
cosines,
dE{x,yf = \x\2 + \y\2 - 2 Μ. Μ cosy,,
where \x\ = dip, x) and \y\ = dip, y). And from (3.1) we have Zpix, y) = lim^o Υε-
By letting ε —» 0 in the above formula, we see that doix, y) := lim^o de(x, y)
exists for all x, у e X, and that j : (X, do) —» CoSpiX) is a distance-preserving map
of pseudometric spaces. Moreover, (X, do) satisfies the CAT(O) 4-point condition,
because it satisfies the CAT(e2/c) 4-point condition for every ε. The image of j
contains a neighbourhood of the cone point in CoSpiX), so this neighbourhood also
satisfies the CAT(O) 4-point condition. In order to complete the proof of the theorem,
it only remains to establish the existence of approximate midpoints forpairs of points
j\x),jiy) e C0SpiX).
We first consider the case к < 0. Let mE be the midpoint of the segment [εχ, εγ].
We claim that when ε is small, -j(me) is an approximate midpoint for jix),jiy). To

192 Chapter II.3 Angles, Limits, Cones and Joins
see this, note that by the convexity of the distance function on X, we have do(x, y) <
d(x, y) for any x, у еХ, and hence
1 1 1
d(j{x), -j(me)) = -d{sj{x),j{me)) = -d0(sx, me)
ε ε ε
< -ά(εχ, mt) = —ά{εχ, sy) = ~de(x, у).
ε 2ε 2
Similarly d(j(y), \j{ms)) < ~de{x, у). And as d(j(x),j(y)) = lim£^0 de(x, y), we are
done.
In the case к > 0, the argument is entirely similar except that one compensates for
the invalidity of the inequality do(x, y) < d(x, y) by noting that for any x, у е B(p, r)
and ε > 0, we have de(x, y) < C(r)d(x, y), where C(r) —> 1 as r —> 0. The existence
of this constant is a consequence of the following lemma and the CAT(/c) inequality
for Δ(ρ, χ, у). О
3.20 Lemma. For each к > 0 there is a function С : [0, DK) —> Ж such that
lim^o C{r) = 1 and for all ρ e M\ and all χ, у е В(р, r)
ά{εχ, εγ) < гС(г) d(x, у),
where εχ denotes the point a distance ε d(p, x)from ρ on the geodesic [p, x].
Proof Regard B(p, г) С М\ as the ball of (Euclidean) radius r in K2 with the Rieman-
nian metric gK expressed in polar coordinates (ρ, Θ) by ds2 = dp2+~ sin2(^/Kp)de2.
The associated metric on B(p, r) is denoted dK. Because εχ is also the point a distance
εά{ρ, χ) from ρ along the Euclidean geodesic from ρ to x, we have
(1) dE2 (εχ, ey) = ε dp. (x,y).
And by the compactness of B(p, r), there is a constant C(r) > 1, which goes to 1 as
r —> 0, such that the norm of any tangent vector υ in the Euclidean metric is related
to the norm || \\K for gK by:
VC(r)
This gives a corresponding Lipschitz relation between the lengths of curves in the
two metrics, and hence
1
■ dK(x,y) < d&(x,y) < ^C(r)dK(x,y)
for all x, у e B(p, r). And this, together with (1), proves the lemma. Π
3.21 Remark. The above proof applies more generally to any Riemannian metric for
which one has normal coordinates.
3.22 Exercise. Let (X, d) be a space of curvature < к. For each positive integer и,
define dn(x,y) := nd(x,y). Prove that the tangent cone C0S(X) of X at ρ (and its
completion) is a 4-point limit of the sequence of metric spaces (X, dn).

Chapter II.4 The Cartan-Hadamard Theorem
Requiring a Riemannian manifold to have non-positive sectional curvature is a
restriction on the infinitesimal geometry of the space. Much of the power and elegance
of the theory of such manifolds stems from the fact that one can use this infinitesimal
condition to make deductions about the global geometry and topology of the
manifold. The result which underpins this passage from the local to the global context
is a fundamental theorem that is due to Hadamard [HI898] in the case of surfaces
and to Cartan [Car28] in the case of arbitrary Riemannian manifolds of non-positive
curvature. The main purpose of this chapter is to show that the Cartan-Hadamard
Theorem can be generalized to the context of complete geodesic metric spaces. We
shall see in subsequent chapters that this generalization is of fundamental importance
in the study of complete (1-connected) metric spaces of curvature < к, where к < 0.
Related results concerning non-simply connected spaces and CAT(/c) spaces with
к > 0 will also be presented in this chapter. For a more complete treatment of the
case к > 0, see Bowditch [Bow95c].
Local-to-Global
The proof of the Cartan-Hadamard theorem that we shall give is very close to that
given by S. Alexander and R.L. Bishop [AB90]. The main argument is best viewed
in the general setting of spaces in which the metric is locally convex in the sense
of (1.18). We remind the reader that the metric on a space X is said to be convex
if X is a geodesic space and all geodesies c\ : [0, a{\ —> X and cj. : [0, aj\ —> X
with ci(0) = сг(0) satisfy the inequality d{c\{ta\), cxitai)) < td{c\(a\), сгОг)) for
all t e [0, 1]. The metric on a space is said to be locally convex if every point has a
neighbourhood in which the induced metric is convex. If the metric on X is locally
convex then in particular X is locally contractible, and therefore has a universal
covering ρ : X —» X. In Chapter 1.3 we showed that there is a unique length metric
on X making ρ a local isometry — this is called the induced length metric.
4.1 The Cartan-Hadamard Theorem. Let Xbea complete connected metric space.
(1) If the metric on X is locally convex, then the induced length metric on the
universal covering X is (globally) convex. (In particular there is a unique

194 Chapter Π.4 The Cartan-Hadamard Theorem
geodesic segment joining each pair of points in X and geodesic segments in X
vary continuously with their endpoints.)
(2) IfX is of curvature < к, where к < 0, then X (with the induced length metric)
is a CAT(/c) space.
This theorem is a variation on a result of M. Gromov ([Gro87], p.l 19), a detailed
proof of which was given by W. Ballmann in the locally compact case [Ba90].
S. Alexander and R.L. Bishop [AB90] proved (4.1) under the additional hypothesis
that X is a geodesic metric space.
4.2 Remarks
(1) The Cartan-Hadamard Theorem is of considerable interest from a purely
topological viewpoint, because it provides a tool for showing that the universal
coverings of many compact metric spaces are contractible.
(2) We emphasize that in (4.1) we do not assume that X is a geodesic space. Thus,
although there exist (non-simply connected) complete length spaces of non-positive
curvature which are not geodesic spaces,22 (4.1) shows that the universal covering
of such a space (with the induced length metric) is always a geodesic space.
The second part of (4.1) is deduced from the first by a patchwork process that
will be described in the next section. In this section we shall prove 4.1(1). The main
work is contained in the following three lemmas.
In what follows, it is convenient to use the term "(local) geodesic" in place of
"linearly reparameterized (local) geodesic " and we shall do so freely throughout
this chapter.
4.3 Lemma. Let Xbea metric space. Suppose that the metric on X is locally complete
and locally convex. Let с : [0,1] —> X be a local geodesic joining χ to y. Let ε > 0
be small enough so that for every t e [0, 1] the induced metric on the closed ball
B(c(t), 2ε) is complete and convex. Then:
(1) For all x, у e X with d(x, χ) < ε and d(y, y) < ε, there is exactly one 23 local
geodesic с : [0, 1] —> X joining χ to у such that t м> d(c(t), c(t)) is a convex
function.
(2) Moreover,
1(c) < 1(c) + d(x, x) + d(y, y).
Proof. We first prove that if с exists then it is unique. Note that if с does exist, then the
convexity of t м> d(c(t), c(t)) implies that d(c(t), c(t)) < ε for every t e [0, 1]. Let
c', c" : [0, 1] -> X be local geodesies such that d(c(t), c'(t)) < ε and d(c(t), c"(t)) <
For instance the graph with two vertices joined by countably many edges the rc-th of which
has length 1 + ^.
There may be many local geodesies joining 3c to y, but only one has this property.

Local-to-Global 195
ε for all t e [0, 1]· Because the metric on each of the balls B(c(t), 2ε) is convex,
the function t м> d(c'(t), c"(t)) is locally convex, hence convex. In particular, if
c'(0) = c"(0) and c'(l) = c"(l), then d = c".
Next we prove that (1) implies (2). To this end, we continue to consider c' and
c" with d(c(t), c'{t)) < ε and d(c(t), c"{t)) < ε for all t e [0, 1], and we now assume
that c'(0) = c"(0). By convexity, d(c'(t), c"(t)) < t d(c('l), c"(D). Combining this
with the fact that l(c'\{0,t]) = t l(c') and l(c"\[0j]) = d(c"(0), c"{t)) for small t > 0,
we have:
tl(c") = d(c"(0),c"(t)) = d(c'(0), c"(t))
<</(c'(0),c'(f)) +^(0,^(0)
<fZ(c') + fd(c'(l),c"(l)),
and hence l(c") < 1(d) + d(c'(l), c"(l)).
Let c' be the unique local geodesic from χ to у that satisfies the conditions of the
lemma. By applying the argument of the preceding paragraph with d = с' and с" — с
we get 1(c) < l(c') + d(y, y). And by applying the argument with c'(t) := c(l-t) and
c"(t) := c'(l - t) we get l(c') < 1(c) + d(x, x). Thus 1(c) < 1(c) + d(x, x) + d(y, y).
It remains to prove the existence of c. Let A > 0 and consider the following
statement:
P(A) For all a,be [0, 1] with 0 < b - a < A and allp, q e X with d(c(a),p) < ε
and d(c(b), q) < ε, there is a local geodesic с : [a, b] —> X such that c(a) = p,
c(b) = q and d(c(t), c(t)) < ε for all t e [a, b].
If Л < ε/1(c) then P(A) is clearly true, so it suffices to prove:
Claim. IfP(A) is true then P(3A/2) is also true.
Let a, b e [0, 1] be such that 0 < b - a < 3A/2. Divide [a, b] into three equal
parts with endpoints a < a\ <b\ < b. Letp, q e X be such that d(c(a),p) < ε and
d(c(b), q) < ε. Proceeding recursively, we shall construct Cauchy sequences (p„)
and (q„) in the ε-neighbourhoods of c(a\) and c(b\) respectively (figure 4.1). We
shall then construct a local geodesic from a to b, as required in P(3A/2), by taking
the union of a local geodesic joining ρ to the limit of the sequence (q„) and a local
geodesic joining the limit of the sequence (pn) to q.
To this end, we define po := c(a{) and go := c(b\). Then, assuming that p„-\
and g„_i have been defined, we use P(A) to construct local geodesies c„ : [a,b\\ —>
X and c'n : [αϊ, b] —> X joining ρ to o„_i and pn-\ to q with d(c(t), cn(t)) < ε
for t e [a,b\] and d(c(t), c'n(t)) < ε for t e [a\,b\. Let pn := cn(a\) and let
qn := dn(b\). By convexity we have d(p0,pi) < ε/2 and d(q0,q\) < ε/2. More
generally, the convexity of the metric in the balls B(c(t), ε) tells us that on [a, b\]
the function t м> d(c„(t), c„+\(t)) is locally convex, hence convex, and therefore
d(p„,p„+i) < d(q„-.i,q„)/2. Similarly, d(qn, q„+l) < d(p„-i,p„)/2. It follows that

196 Chapter II.4 The Cartan-Hadamard Theorem
a a, b, b
. 1 1 .
c(a) c P0=c(fl1) 90=Φχ) Φ)
Fig. 4.1 Constructing с
d{p,u Ρη+ι) < ε/2η+ι and d{qn, qn+\) < ε/2"+1 for all η e N, and therefore (pn) and
(q„) are Cauchy sequences in B(po, ε) and 5(tfo, ε) respectively.
Now, the function ί м> d(cn(t), cn+\(t)) is convex and bounded by d{qn^\,qn) <
ε/2". Thus (c„(i)) is a Cauchy sequence in the complete ball Β(φ),ε) for each
t e [a, b\\. Similarly, (c'n(f)) is aCauchy sequence in B(c'(f), ε) for each f e [a\,b\.
It follows that the local geodesies c„ and c'n converge uniformly to local geodesies
whose restrictions to the common interval [αϊ, &ι] coincide. The union of these two
local geodesies gives the local geodesic с : [a,b] —*■ X required to complete the
proof of the Claim. D
An Exponential Map
In the case of 1-connected Riemannian manifolds of non-positive curvature, the
Cartan-Hadamard Theorem follows easily from the fact that the exponential map
from the tangent space at each point is a covering map. In the more general context
of (4.1) one uses the following analogue of the exponential map.
4.4 Definition. Let X be a metric space and let jeo e X. We define XXo to be the
set of all (linearly reparameterized) local geodesies с : [0, 1] —> X issuing from jeo,
together with the constant map xq at jeo. We equip XXo with the metric
d(c, c) := sup{d(c(t), c'(t)) | t e [0, 1]}.
And we define exp : Xxo —> X to be the map с м> с(1).
4.5 Lemma. Let X be a metric space and suppose that the metric on X is locally
complete and locally convex.
(1) XXo is contractible (in particular it is simply connected).
(2) exp : XXt) —» X is a local isometry.
(3) There is a unique local geodesic joining x.q to each point ofXXo.

An Exponential Map 197
Proof. (1) There is a natural homotopy Хт χ [0, 1] —> XXo from the identity map of
XXo to the constant map at xq given by (c, s) м> rs(c), where rs(c) : [0, 1] —» X is the
path t н-* c(st).
(2) Part (1) of Lemma 4.3 implies that for every с e Хло there exists ε > 0 such
that the restriction of exp to the ball B(c, ε) is an isometry onto B(c(l), ε).
(3) Because exp is a local isometry, a continuous path in XXo is a local geodesic
if and only if its image under exp is a local geodesic in X. In particular, с м> exp ос
is a bijection from the set of local geodesies с : [0, 1] —> Xj0 issuing from jco to the
set of local geodesies с : [0,1] -> X issuing fromx0· Thus for each с eXi0, the path
с : [0, 1] —> Xj0 defined by c(i) = rs(c) is the unique local geodesic joining x0 to с
inXX0. D
4.6 Lemma. Lei X fee α metric space and letxo e X. If the metric on X is complete
and locally convex, then the metric on XXo is complete.
Proof. Let (c„) be a Cauchy sequence in Xxo. Because X is complete, for every
t e [0, 1] the Cauchy sequence (c„(i)) converges in X, to c(t) say. Part (2) of Lemma
4.3 shows that the lengths l(cn) are uniformly bounded, and hence the pointwise limit
t M* c(t) of the curves c„ is a local geodesic. D
4.7 Corollary. Lei X be a connected metric space and letxo e X. If the metric on X
is complete and locally convex, then:
(1) exp : XXo —> X is a universal covering map (in particular it is surjective);
(2) there is a unique local geodesic joining each pair of points in XXo.
Proof. Assertion (1) follows immediately from (1.3.28), the preceding lemma, and
the fact that XXo is simply connected (4.5(1)).
We claim that every path in X is homotopic (rel endpoints) to a unique local
geodesic. Since xq e X is arbitrary, it suffices to consider continuous paths с :
[0, 1] —> X issuing from jco. And since XXo is a universal covering, the set of paths
in X that are homotopic to с (rel endpoints) is in 1-1 correspondence with the set of
paths in X^ that issue from x0 and have the same endpoint as the lifting of с that
issues from ico. By (4.5(3)), the latter set of paths contains a unique local geodesic.
This proves the claim.
Let p,q e XXo. Because XXo is simply connected, there is only one homotopy
class of continuous paths joining ρ to q. The projection exp : XXo —> X sends this
class of paths bijectively onto a single homotopy class (rel endpoints) of paths in X,
and the argument of the preceding paragraph shows that the latter class contains a
unique local geodesic. D

198 Chapter II.4 The Cartan-Hadamard Theorem
The Proof of Theorem 4.1(1)
Let X be as in 4.1(1) and fix x0 e X. We have shown that exp : X^ -> X is a
covering map. In (1.3.25) we showed that (under suitable hypotheses) the induced
length metric on a covering space is the only length metric that makes the covering
map a local isometry. Thus (4.5(1)) implies that the length metric which X induces
on XXo is the same as the length metric associated to the metric defined in (4.4).
According to (4.7(2)), there is a unique local geodesic joining each pair of points in
XXa. And (4.3(1)) shows that these local geodesies must vary continuously with their
endpoints. To complete the proof of Theorem 4.1(1), we apply the following lemma
with 7 = ^.
4.8 Lemma. Let Υ be a simply connected length space whose metric is complete
and locally convex. Suppose that for every pair of points p, q e Υ there is a unique
local geodesic cpq : [0,1] —> Υ joining ρ to q. If these geodesies vary continuously
with their endpoints then:
(1) each срл is a geodesic;
(2) the metric on Υ is convex.
Proof. (1) It suffices to show that for every rectifiable curve γ : [0, 1] —> Υ and
every t e [0, 1] we have l(cY(o)yY(t)) 5· Ky\[o,i])· Because the metric on Υ is locally
convex, for sufficiently small t the local geodesic cy(0),y(() is actually a geodesic.
Thus the subset of [0, 1] consisting of those f for which the above inequality holds
for all t < f is non-empty. It is clear that this subset is closed. We claim that it is also
open, and hence is the whole of [0, 1 ]. Indeed if i0 is such that /(cy(o), y((>) < l(y | [o,/])
for all t < to, then (4.3(2)) tells us that when ε is sufficiently small,
Kcy(0),y(to+e)) < '(Cy(0),y(i0)) + '(yl[(o,(o+e])
< '(yl[0,(o]) + Κγ\ΐΐο,ιΒ+ε]) = 1(γ\ΐ0,ι<>+ε]),
as required.
(2) We have shown that У is a complete uniquely geodesic space in which
geodesies vary continuously with their endpoints. In order to prove that the metric on
У is convex, it suffices to showthatd,(cp9o(l/2), cp,9l(l/2)) < (l/2)d(q0, gOforeach
pair of geodesies cpi9o, cp<qi : [0, 1] —» Y. Suppose that go and q\ are joined by the
geodesic s i-> qs. Because the metric on Υ is locally convex, we know from (4.3(1))
that when s and У are sufficiently close d{cpqs{\/2), cpAs,(l/2)) < (l/2)d(qs, qs·).
Partition [0, 1] finely enough 0 = sq < ■ ■ ■ < sn = I so that the above inequality
holds with {s, s'} = {s\, ji+1} for i = 0, ..., η — 1. By adding the resulting
inequalities we get d(cM0(l/2), cMl(l/2)) < (l/2)d(q0, qi). □

Alexandrov's Patchwork 199
Alexandrov's Patchwork
4.9 Proposition (Alexandrov's Patchwork). Let X be a metric space of curvature
< к and suppose that there is a unique geodesic joining each pair of points that
are a distance less than DK apart. If these geodesies vary continuously with their
endpoints, then X is a CAT(/c) space.
This proposition is an immediate consequence of the following two lemmas and
the characterization of CAT(/c) space in terms of angles (1.7(4)). The idea of the
proof is portrayed in figure 4.2.
4.10 Gluing Lemma for Triangles. Fix к e K. Let X be a metric space in which
every pair of points a distance less than DK apart can be joined by a geodesic. Let
Δ = Δ([ρ, q{\, [ρ, q2], [q\, #2]) be a geodesic triangle in X that has perimeter less
than 2DK. Suppose that the vertices of A are distinct. Consider r e [q\, q2] with
r ψ q\ and r φ q2. Let [p, r] be a geodesic segment joining ρ to r.
Let Δ, be a comparison triangle in M^for Δ, = Δ,-([ρ, qi\, [ρ, r], [g,-, r]). If for
i = 1,2, the vertex angles ο/Δ; are no smaller than the corresponding angles ο/Δ„
then the same is true for the angles of any comparison triangle for Δ in M2K.
Proof. Let Δι(ρ, ql, T) and Α2φ, q2, >") be comparison triangles in M^ for Δι and
Δ2, glued along \p, 7] so that g, and q2 do not lie on the same side of the line through
ρ and r. The sum of the angles of Δι and Δ2 at r is at least π, so the sum of the
comparison angles is as well. Therefore we can apply Alexandrov's Lemma (1.2.16).
D
4.11 Lemma. Пхк е Ж. LetXbe ametric space of curvature < K.Letq : [0, 1] -» X
be a linearly reparameterized geodesic joining two distinct points qo = q(0) and
ql = q(l) and let ρ be a point ofX which is not in the image of q. Assume that for
each s e [0, 1] there is a linearly reparameterized geodesic cs : [0, 1] —> X joining
ρ to q(s), and assume that the function s \-> cs is continuous. Then the angles atp, qo
and q\ in the geodesic triangle Δ with sides co([0,1]), ci([0,1]) and g([0, 1])) are
no greater than the corresponding angles in any comparison triangle Δ с Μ2Κ. {If
к > 0, assume that the perimeter of A is smaller than 2DK.)
Proof. By hypothesis, every point of X has a neighbourhood which (with the induced
metric) is a CAT(/c) space, and the map с : [0, 1] χ [0, 1] -> X given by с : (s, t) м>
cs(t) is continuous. Hence there exist partitions 0 = so < s\ < ■ ■ ■ < sk = I and
0 = ίο < t\ < ■ ■ ■ < tk = 1 such that for all i and j there is an open ball Uij с X of
radius < DK/2 which is a CAT(/c) space and which contains c([j/_i, s{\ χ [/μι, /,]).
Consider the sequence of adjoining triangles Δι,..., Δ* where Δ, is the union
of the geodesic segments cSi_, ([0, 1]), cs,([0, 1]) and g([s,_ 1, J,-]). Repeated use of the
Gluing Lemma (4.10) reduces the present proposition to the assertion that the angles
at the vertices of each Δ; are no greater than the corresponding angles in comparison
triangles Δ,- С М\. In order to prove this assertion, we subdivide each Δ, and make

200 Chapter Π.4 The Cartan-Hadamard Theorem
further use of the Gluing Lemma (see figure 4.2). Specifically^for each i we^consider
the sequence of 2k — 1 adjoining geodesic triangles Α], Δ?, Δ?,..., Δ*·', Δ*, where
Δ! is the geodesic triangle in U\\ with verticesp, cs,_t(t\), cSi(t\) , AJ; the geodesic
triangle in U,j with vertices cSi.^{tj-.\), cSj(tj-i), cs,.^{tj) and Δ^· the geodesic triangle
in Ujj with vertices cSi.t (0, cs,(/,_i), cSj(tj). Since U;j is a CAT(/c) space, each vertex
angle in these small triangles is no greater than the corresponding angle in a
comparison triangle in M\. By making repeated use of the gluing lemma for triangles,
we see that the vertex angles of Δ,- also satisfy the desired inequality. Π
Fig. 4.2 Alexandrov's Patchwork
4.12 Corollary. Fix /eel, Let X be a metric space of curvature < к. Suppose that
all closed balls in X of radius less than DK/2 are compact. Then X is а С AT(/c) space
if and only if every pair of points a distance < DK apart can be joined by a unique
geodesic segment.
Proof. Immediate from (4.9) and (1.3.13). D
Local Isometries and πι-Injectivity
We point out some topological consequences of the Cartan-Hadamard Theorem.
4.13 Theorem. Let X be a complete, non-positively curved metric space, and fix a
basepoint xq e X.

Local Isometries and πι-Injectivity 201
(1) Every homotopy class γ e π\(Χ,xq) contains a unique local geodesic cy :
[0, 1] —»■ X. (The uniqueness implies that a non-constant locally geodesic loop
cannot be homotopic to a constant loop.)
(2) Every non-trivial element ofn\ (Χ, xq) has infinite order.
Proof. Consider the universal covering ρ : X —»■ X endowed with the induced length
metric. Fix xq € X with p(xq) = xq. The fundamental group π\ (X, xq) acts on X by
deck transformations; the action is free and by isometries. Each loop с : [0, 1] —*■ X
in the homotopy class γ e π\{Χ, xq) lifts uniquely to a path с : [0, 1] —*■ X such that
c(0) = xq and c(l) = γ.χο. Because ρ is a local isometry, с will be a local geodesic
if and only if с is a local geodesic.
The Cartan-Hadamard Theorem tells us that X is a CAT(O) space. Hence there is
a unique local geodesic cy : [0, 1] —»■ X joining xq to y.XQ. This proves (1).
Because X is a CAT(O) space, any isometry of finite order will have a fixed point
(2.8). Thus (2) follows from the fact that the action of it\ (X, xq) is free. D
Recall that a map f : Υ —> X between metric spaces is said to be locally an
isometric embedding if for every у e Υ there is an ε > 0 such that the restriction of
/ to B(y, ε) is an isometry onto its image.
4.14 Proposition. Let X and Υ be complete connected metric spaces. Suppose that
X is non-positively curved and that Υ is locally a length space. If there is a map
f : Υ —»■ X that is locally an isometric embedding, then Υ is non-positively curved
and:
(1) For every yo e Y, the homomorphismf* : ni(Y,yo) —*■ πi(X,f(yo)) induced by
f is injective.
(2) Consider the universal coverings X and Υ with their induced length metrics.
Every continuous lifting f : Υ —*■ X off is an isometric embedding.
Proof. First we prove that Υ is non-positively curved. Each point у e Υ has a
neighbourhood B(y, ε) that is isometric to its image in X. Shrinking ε if necessary, we
may assume that the image of B(y, ε) under/ is contained in a neighbourhood of/(у)
that is CAT(O) in the induced metric. Fix p,q e B(y, ε/З). Because Υ is locally a
length space, there is a sequence of paths joining ρ to q in B(y, 3ε/4) whose lengths
converge to d(p, q). Parameterize these paths by arc length. It follows from (1.4(5))
that the points halfway along the images of these paths form a Cauchy sequence in
the complete space f(B(y, 3ε/4)). The limit/(m) of this sequence is a midpoint for
f(p) and f(q) in f(B(y, 3ε/4)). Because we are in a CAT(O) neighbourhood of/Су),
the point/(w) must lie in the open ball of radius ε/З about f(y). Thus we obtain
a midpoint m between ρ and q in B(y, ε/З). Since Υ is complete, we deduce that
B(y, ε/З) is a geodesic space. And since geodesic triangles in B(y, ε/З) are isometric
to geodesic triangles in a CAT(O) neighbourhood of/tv), it follows that B(y, ε/З) is
a CAT(O) space. Thus Υ is non-positively curved.
Υ is complete and non-positively curved, so by (4.13) every non-trivial element
γ e π\(Υ, yo) is represented by a local geodesic cy : [0, 1] —»■ Y. The image of cy

202 Chapter II.4 The Cartan-Hadamard Theorem
under/ is again a local geodesic (of the same length), so it is not homotopic to a
constant map (4.13) and therefore/*(y) φ 1.
Let и : X —»■ X and υ : Υ —>- Υ be the universal covering maps. Let/ : Υ —>- X be
a lifting of/. By the Cartan-Hadamard Theorem, both X and У are CAT(O) spaces,
so the distance between each pair of points is the length of the unique local geodesic
joining them. But if с is a local geodesic in Y, then/ о с is a local geodesic of the
same length in X, because/ о с is a lift of/ ο υ о с, and и, υ and/ all carry local
geodesies to local geodesies of the same length. D
Injectivity Radius and Systole
We now turn our attention to the study of spaces which contain closed geodesies
(isometrically embedded circles).
4.15 Definition. Recall (1.7.53) that the injectivity radius, injrad(X), of a geodesic
space X is the supremum of the non-negative numbers r such that if d(x, у) < r then
there is a unique geodesic segment joining χ to у in X. If X contains any isometrically
embedded circles then the injectivity radius of X is obviously bounded above by one
half of the infimum of the lengths of such circles. (An isometrically embedded circle
of length I is, by definition, the image of an isometric embedding Mxp —»■ X, where
*/p = 2π/ί.) If X contains such circles then Sys(X), the systole ofX, is defined to
be the infimum of their lengths.
We urge the reader to consider the many phenomena which may cause X not to
contain any isometrically embedded circles.
The following result, which appears in [Gro87], complements Proposition 4.9.
We follow a proof of R. Charney and M. Davis [CD93]. Recall that a metric space
X is said to be cocompact if there exists a compact subset К с X such that X —
\J{Y.K | γ e Isom(X)}.
4.16 Proposition. A cocompact, proper, geodesic space X of curvature < к fails
to be a CAT(/c) space if and only if it contains an isometrically embedded circle of
length < 2DK. Moreover, if it contains such a circle then it contains a circle of length
Sys(X) = 2injrad(X).
Proof. The sufficiency of the condition in the first assertion is clear; we shall prove its
necessity. Let r = injrad(X). As X is assumed to be cocompact and locally CAT(/c)
we have r > 0. Suppose that X is not a CAT(/c) space. Then, by (4.12), we have
r <DK.
We shall use the term (non-degenerate) digon to mean the union of two distinct
geodesic segments [x, y] and [x, y)', called its sides, joining two points χ and у in X.
We shall refer to χ and у as the vertices of the digon. The first step of the proof is
to show that there exists a digon of perimeter 2r and the second step is to show that
such a digon is an isometrically embedded circle.

Injectivity Radius and Systole 203
By (1.3.13), geodesies of length < r emanating from a fixed point in X vary
continuously with their terminal endpoint. Thus we may apply (4.11) to any geodesic
triangle Δ in X whose perimeter is less than 2r and conclude that the angles at the
vertices of Δ are no bigger than the corresponding angles in a comparison triangle
inM*.
By the definition of r, there is a sequence of digons in X whose perimeter tends
to 2r. Because X is proper and cocompact, we can translate these digons into a
compact subspace and extract a subsequence D(ri) converging to the union of two
geodesic segments [x, y] and [x, y]' of length r; we have to show that these two
segments are distinct. If this were not the case, then for η big enough the digon
D{ri) = [xn, y„] U [xn, y„]' would be very narrow, consequently if m is the midpoint
of [*„,)'„] and m' is the midpoint of [x„,y„Y, the geodesic triangles Δι and Δ2
with vertices x„, m, m! and yn, m, m!, respectively, would each have perimeter less
than 2r. The, by (4.11), the vertex angles in Δι and Δ2 would be less than the
corresponding angles in their comparison triangles in M%. But the comparison triangle
for Δ = [x„, m] U [m,yn] U [x„,yn]' is degenerate, so the angles at the vertices
corresponding to x„ and y„ are zero. Alexandrov's lemma shows that these angles are
not smaller than the angles at the vertices corresponding to x„ and y„ in comparison
triangles for Δι and Δ2 respectively. Therefore these latter comparison triangles
are also degenerate, so [m, xn] = [m',xn] and [m,yn] = [m',yn], contradicting the
hypothesis that [xn, yn] is distinct from [xn,yn]'. Thus D = [x, y] U [x, y]' is a (non-
degenerate) digon.
We say that a point г e [x,y] is opposite the point z' e [x, y]'iid(z, x)+d(z',x) =
r. We claim that d(z, z') = r. If not, then the geodesic triangles with vertices ζ,ζ',χ
and z,z!,y, respectively, would be of perimeter smaller than 2r and we could apply
the same argument as above to conclude that the geodesic segments [jc, y] and [x, y]'
were the same. It follows that D is an isometric image of a circle. D
We saw in Chapter 1.7 that in many respects MK-polyhedral complexes with only
finitely many isometry types of cells behave like cocompact spaces. Here is another
important instance of this phenomenon.
4.17 Proposition. Let К be an MK-polyhedral complex with Shapes(if) finite and
suppose that К has curvature < к. If К is not a CAT(/c) space, then К contains an
isometrically embedded circle of length Sys(K) = 2 injrad(K). If к > 0 and К is not
a CAT(/c) space then Sys(K) < 2π/^/κ.
Proof. Since К is locally CAT(/c) it is a fortiori locally geodesic. So by (1.7.55), we
have r = injrad(K) > 0. In order to apply the arguments of (4.16) it is enough to
establish the existence of a non-degenerate digon whose sides have length r. In the
present setting, as on previous occasions, we shall use the hypothesis that Shapes(if)
is finite to reduce to consideration of the compact case.

204 Chapter Π.4 The Cartan-Hadamard Theorem
First we choose a sequence of (non-degenerate) digons D(n) with endpoints xn
and yn such that d(xn, yn) approaches r. Since the lengths of the sides of the digons
D(n) are uniformly bounded, (1.7.30) yields an integer N such that each D(ri) is
contained in a subcomplex L„ of К that can be expressed as the union of at most N
closed cells. Because Shapes(if) is finite, there are only finitely many possibilities
for Ln up to isometric isomorphism, so by passing to a further subsequence we may
assume that all of the Ln are isometrically isomorphic. In other words there exists
a finite MK -polyhedral complex L with Shapes(L) с Shapes(iT) such that for every
positive integer η there is a bijective map φ„ : L —»■ L„ that sends cells to cells and
restricts to an isometry on the individual cells equipped with their local metrics.
Each of the maps <p„ is a length-preserving map of geodesic spaces, so d(xn, y„) =
ά(φ~ι(χ„), ф~1(у„)) and $~'D(«) С L is a geodesic digon with sides of length
d(xn,yn). Since L is compact, we may pass to a subsequence and assume that these
digons converge in L to a geodesic digon D with endpoints χ and y. The argument
given in (4.16) shows that the width of the digons D{n) is bounded away from zero,
so since φη : φ~ιϋ(η) —> D(ri) does not increase distances, D is non-degenerate.
However, we are not quite done at this stage, because in general the sides of φη0
will not be geodesies in K: we have ά(φη(χ), ф„(у)) < d(x, у) = г for all и, but we
do not get equality in general. However we do know that
d(<f>„(x), фп(у)) - r < ά{φη{χ), χη) + d(xn, y„) + d{yn, ф„(у)) - г
< d(x, ф~\х„У) + а(ф~1{уп), у) + d(x„, yn) - г,
which tends to zero as η —> сю.
Let χ e S and уел be points in the model cells S,S e Shapes(L) с Shapes(iT)
such that fs(x) = χ andfB>(y) = у for some closed cells S э χ and S' э у in L. By
construction, for all η we have φ„(χ) e Χχ and ф„(у) e Υ κ, where Χχ and Υ χ are as
in (1.7.59). We proved in (1.7.59) that the set of numbers {d(x, y) | χ e Хк, у е Υχ]
is discrete. Hence for all sufficiently large η we have ά{φη(χ), ф„(у)) = r, and φ,,Φ)
is a non-degenerate geodesic digon with sides of length r. Π

Chapter II.5 M^-Polyhedral Complexes
of Bounded Curvature
In this chapter we return to the study of metric polyhedral complexes, which we
introduced in Chapter 1.7. Our focus now is on complexes whose curvature is bounded
from above. The first important point to be made is that in this context one can
reformulate the CAT(/c) condition in a number of useful ways. In particular, Gromov's
link condition (5.1) enables one to reduce the question of whether or not a complex
supports a metric of curvature < к to a question about the geometry of the links in the
complex. This opens the way to arguments that proceed by induction on dimension,
as we saw in (1.7). If the cells of the complex are sufficiently regular then the link
condition can be interpreted as a purely combinatorial condition; in particular this is
the case for cubical complexes and many 2-dimensional complexes.
This chapter is organized as follows. We begin by establishing the equivalence of
various characterizations of what it means for a polyhedral space to have curvature
bounded above. We then address the question of which complexes have the property
that their local geodesies can be prolonged indefinitely, and we describe some general
results of a topological nature. There then follows a discussion of flag complexes and
all-right spherical complexes. Two important results arising from this discussion are:
there is no topological obstruction to the existence of CAT(l) metrics on simplicial
complexes (5.19), and there is a purely combinatorial criterion for deciding whether
or not a cubical complex is non-positively curved (5.20). In the fourth section we
describe some constructions of cubical complexes and outline Davis's construction of
aspherical manifolds that are not covered by Euclidean space. The remaining sections
deal with two-dimensional complexes. Two-dimensional complexes provide a rich
source of examples of non-positively curved spaces, and their usefulness is enhanced
by the close link between geometry and group theory in dimension two (cf. 5.45).
Two-dimensional complexes also enjoy some important properties which are not
shared by their higher dimensional cousins (see 5.27).
Further explicit examples of non-positively curved polyhedral complexes will be
described in Chapters II. 12 and Ш.Г, and the techniques described in Chapters II. 11,
11.12 and III.С provide means of constructing many other examples.

206 Chapter П.5 MK-Polyhedral Complexes of Bounded Curvature
Characterizations of Curvature < к
We begin by describing criteria for recognizing when an MK -polyhedral complex is
a CAT(/c) space. In the course of our discussion we shall need to call on a number of
facts from Chapter 1.7 concerning MK-polyhedral complexes with only finitely many
isometry types of cells. Most of these facts can be proved much more easily in the
special case where К is locally compact, as the interested reader can verify.
The following criterion for an MK -complex to have curvature < к is due to
M. Gromov [Gro87, p. 120]. A proof in the locally compact case was given by
Ballmann [Ba90] and in the general case by Bridson [Bri91].
5.1 Definition. An MK -polyhedral complex satisfies the link condition if for every
vertex ν e К the link complex Lk(u, K) is a CAT(l) space.
5.2 Theorem. An MK-polyhedral complex K, with Shapes(iT) finite, has curvature
< к if and only if it satisfies the link condition.
Proof. Let υ be a vertex of K. According to (1.7.39), there exists ε(υ) > 0 such
that B(v, ε(ν)) is convex and isometric to the ε(ν) neighbourhood of the cone point
in CK(Lk(v, K)). Thus, in the light of Berestovskii's theorem (3.14), we see that К
satisfies the link condition if and only if every vertex has a neighbourhood that is a
CAT(/c) space. To complete the proof of the theorem one simply notes that, given
any χ e K, if υ is a vertex of supp(jc) and η > 0 is sufficiently small, then B(x, η) is
isometric to B{x*, η) for some x1 e B(v, ε(ν)) (see 1.7.56). D
5.3 Remark In the above theorem one can weaken the hypotheses and require only
that the number ε(χ) defined in (1.7.39) is positive for every χ e K.
5.4 Theorem. Let К be an MK-polyhedral complex with Shapes(K) finite. If к < 0
then the following conditions are equivalent:
(1) К is a CAT(/c) space;
(2) К is uniquely geodesic;
(3) К satisfies the link condition and contains no isometrically embedded circles;
(4) К is simply connected and satisfies the link condition.
If к > 0 then the following conditions are equivalent:
(5) К is a CAT(/c) space;
(6) К is (πI~/K)-uniquely geodesic;
(7) К satisfies the link condition and contains no isometrically embedded circles
of length less than 2π/«/κ.
Proof. The implications (1) => (2) and (5) => (6) are clear. (4.17) together
with (5.2) shows that (3) is equivalent to (1) and that (5) is equivalent to (7). As a
consequence (1.7.1), the Cartan-Hadamard Theorem and (5.2), we have (4) => (3).

Extending Geodesies 207
We argue by induction on the dimension of К in order to see that (2) => (4)
and (6) => (7). For complexes of dimension 1 these implications are clear. If К
satisfies (2) or (6) then in particular its injectivity radius is positive, so by (1.7.55)
Lk(x, K) is uniquely 7r-geodesic for every χ e K. Hence, by induction, Lk(jc, K) is
a CAT(l) space. (5.2) completes the proof of (6) => (7). To complete the proof of
(2) => (4), we need the additional fact that if (2) holds then by (1.7.58) geodesies
vary continuously with their endpoints in K, so К is contractible (hence simply
connected). Indeed, for fixed xq e K, we obtain a contraction Κ χ [0, 1] —»■ К by
sending (χ, t) to the point a distance td(xo, x) from χ along the unique geodesic from
jetojeo. D
The same arguments are enough to prove a local version of the above theorem:
5.5 Theorem. Let К be an Μκ-polyhedral complex with Shapes(AT) finite. The
following conditions are equivalent:
(1) К has curvature < к;
(2) К satisfies the link condition;
(3) К is locally uniquely geodesic;
(4) К has positive injectivity radius.
The 2-DimensionaI Case.
Let К be a 2-dimensional MK -complex. The link of a vertex υ e К is a metric
graph whose vertices correspond to 1 -cells incident at υ and whose edges correspond
to corners of the 2-cells S incident at υ; the length of an edge is the vertex angle at
the corresponding corner of the model simplex S e Shapes(AT)· A metric graph is a
С AT (к) space if and only if every locally injective loop in the graph has length at
least 2π/^/κ, therefore:
5.6 Lemma. A 2-dimensional MK-complex К satisfies the link condition if and only
if for each vertex υ e К every injective loop in Lk(u, K) has length at least 2π.
Extending Geodesies
Many important results concerning the global geometry of simply connected Rie-
mannian manifolds X of non-positive curvature rely in an essential way upon the
fact that if the manifold is complete then one can extend any geodesic path in X to a
complete geodesic line. When seeking to generalize such results to CAT(O) spaces,
it is important to understand which spaces share this extension property. We shall
address this question in the case of polyhedral complexes and topological manifolds.
It is useful to phrase the extension property locally.

208 Chapter Π.5 MK-Polyhedral Complexes of Bounded Curvature
5.7 Definition. A geodesic metric space X is said to have the geodesic extension
property if for every local geodesic с : [a, b] —»■ X, with α φ b, there exists ε > 0
and a local geodesic c' : [a, b + ε] —»■ X such that с'|[ай] = с
5.8 Lemma. Let X be a complete geodesic metric space.
(1) X has the geodesic extension property if and only if every local geodesic с :
[a, b] —> X with α ψ b can be extended indefinitely, i.e. there exists a local
isometry с : К —»■ X such that с\[а.щ = с.
(2) If X is a CAT(O) space, then X has the geodesic extension property if and only
if every non-constant geodesic can be extended to a geodesic line №. —*■ X.
Proof. For (1), one simply notes that if X has the geodesic extension property then, for
every local geodesic с : [a, b] —»■ X, the set of numbers t > 0 such that с : [a, b] —»■ X
can be extended to a local isometry [a — t, b + t] —> X is open; the completeness of
X assures that it is also closed, and hence the whole of [0, oo).
Assertion (2) is an immediate consequence of the fact that in a CAT(O) space
every local geodesic is in fact a geodesic (1.4(2)). D
In the case of polyhedral complexes, the geodesic extension property can be
rephrased in terms of the combinatorial structure of the complex.
5.9 Definition. Let К be an M*-polyhedral complex. A closed и-cell В in К is said
to be Ά free face if it is contained in the boundary of exactly one cell B' of higher
dimension and the intersection of the interior of B' with some small neighbourhood
of an interior point of В is connected. (The second clause in the preceding sentence
is necessary in order to avoid suggesting, for example, that if both ends of a 1-cell
are attached to a single vertex in a graph then that vertex is a free face.)
5.10 Proposition. Let К be an MK-polyhedral complex of curvature < к with
Shapes(iT) finite. К has the geodesic extension property if and only if it has no
free faces-
Proof By (1.7.39), a small neighbourhood of each point χ e К is isometric to the
к -cone on Lk(jc, K). It follows from (1.5.7) that in any cone CK Y, a local geodesic
from ty incident at the cone point 0 can be extended past 0 if and only if there exists
У e Υ with d(y, y') > it. Therefore К has the geodesic extension property if and
only if for every χ e К and every и e Lk(jc, K) there exists u' e Lk(jc, K) with
d(u, u') > it. We call u' an antipode for u.
Suppose that К has a free face of dimension n. If χ is an interior point of this face,
then Lk(jc, K) is isometric to a closed hemisphere in S". In particular, some points of
Lk(jc, K) do not have antipodes, and К does not have the geodesic extension property.
Suppose now that К does not have any free faces. We shall prove that every point
in every link Lk(jc, K) has an antipode. We argue by induction on the dimension
of K; the 1-dimensional case is trivial. Given χ e К and и e Lk(jc, K), we choose

Extending Geodesies 209
и' e Lk(jc, K) with и1 фи (note и must exist, for otherwise χ would be a free face of
dimension 0). If и' is in a different connected component to и then they are an infinite
distance (greater than π!) apart and we are done. If not then we connect и to u' by a
geodesic in Lk(jc, K). Since К has no free faces, neither does Lk(x, K), so using our
inductive hypothesis we may extend the geodesic [и, и']. Indeed, because Lk(x, K)
is complete (7.19), we may extend [и, и'] to a local geodesic с : Ж —»■ Lk(jc, К) with
c(0) = и.
By hypothesis, Lk(jc, K) is a CAT(l) space. In a CAT(l) space any local geodesic
of length π is actually a geodesic (1.10), thus c(n) is the desired antipodal point
for u. D
5.11 Exercise. Give an example to show that the curvature hypothesis in the preceding
proposition is necessary.
Here is another important class of spaces with the geodesic extension property.
5.12 Proposition. If a complete metric spaceXhas curvature < к and is homeomor-
phic to a finite dimensional manifold, then it has the geodesic extension property.24
Proof. Let η be the dimension of X. Suppose that a certain local geodesic с cannot
be extended past χ e X. We fix a small CAT(/c) neighbourhood В of χ and a point
у e Bin the image of c. Because the unique geodesic from у to χ cannot be extended
locally, no geodesic to у from any point of В passes through x. Thus the geodesic
retraction of В to у restricts to give a retraction of В \ [x] within itself. It follows
that the local homology group H„(B, В \ {x}) is trivial. On the other hand, we may
choose a homeomorphism from the и-dimensional disc Dn onto a neighbourhood V
of χ in B, and by excision, H„(B, Β χ {χ}) = Hn(V, V χ {χ}) = H„(D", D" χ {0}) = Ζ
(see [Spa66]). D
At this point, when we have temporarily found the need to use some non-trivial
topology, we note a result that relates complexes to more general spaces of bounded
curvature.25
5.13 Proposition. Let X be a compact metric space. If there exists /eel such that
X has curvature < к, then X has the homotopy type of a finite simplicial complex.
(In fact, X is a compact ANR.)
Proof. Every point of X has a neighbourhood which is CAT(/c) in the induced metric.
Since X is compact, it follows that there exists ε e (0, π/6«/κ) such that B(x, 3ε)
is CAT(/c) for every χ e X. According to (1.4), geodesies in B(x, 3ε) are unique
and vary continuously with their endpoints, and B(x, ε) is convex. Lemma I.7A. 19
24 Experts will notice that all that is actually required in the proof of (5.12) is that X is a
homology manifold.
25 An equivariant version of this result is proved in a recent preprint of Pedro Ontaneda entitled
" Cocompact CAT(0) spaces are almost extendible".

210 Chapter Π.5 MK-Polyhedral Complexes of Bounded Curvature
implies that X is homotopy equivalent to the nerve of any covering of X by balls of
radius ε.
The parenthetical assertion in the statement of the proposition is a special case
of [Hu(S), III.4.1]. (A deep theorem of Jim West [WestW] states that every compact
ANR has the homotopy type of a finite simplicial complex, so this parenthetical
assertion is stronger than the main assertion of the proposition.) D
5.14 Remark. The preceding theorem does not say that every compact non-positively
curved space is homotopy equivalent to a compact non-positively curved simplicial
complex. In fact this is not true even for compact Riemannian manifolds of non-
positive curvature, as was recently shown by M. Davis, B. Okun and Ε Zheng, and
(independently) by B. Leeb.
Flag Complexes
In the introduction to this chapter we alluded to the usefulness of having a purely
combinatorial criterion for checking whether complexes satisfy the link condition
and we indicated that such a criterion exists for cubed complexes. The purpose of
this section is to describe this condition.
5.15 Definition. Let L be an abstract simplicial complex and let V be its set of vertices.
L is called г. flag complex (alternatively, "L satisfies the no triangles condition")
if every finite subset of V that is pairwise joined by edges spans a simplex. In other
words, if {υ,·, ц} is a simplex of L for all i,j e {1,... , n}, then {vi,... , v„} is a
simplex of L.
5.16 Remarks
(1) If L is a flag complex, then so is the link of each vertex in L.
(2) The girth of a simplicial graph is the least number of edges in any reduced
circuit in the graph. A simplicial graph L is a flag complex if and only if its girth is
at least 4.
(3) The barycentric subdivision of every simplicial complex is a flag complex.
(4) (No triangles condition). Let L be a simplicial complex. Then L is a flag
complex if and only if the statement Ф„ (L) is false for all η > 0, where the statement
Φ„ is defined inductively as follows: &o(L) is the statement that there exist vertices
i>o, "1,^2 ei such that {υ,·, υ,·} is a simplex for all i,j e {0, 1, 2} but {do, ub v2} is
not a simplex of L; and Φη+ι is the statement that there exists a vertex vq e L such
that Ф„(Ьк(и0, L)).
(5) The simplicial join of two flag complexes is again a flag complex.
5.17 Definition. An all-right spherical complex is an Mi-simplicial complex each
of whose edges has length π/2.

Flag Complexes 211
The link of every vertex in an all-right spherical complex is itself an all-right
spherical complex.
The following theorem is due to M. Gromov [Gro87].
5.18 Theorem. Let L be a finite dimensional all-right spherical complex. LiiCAT(l)
if and only if it is a flag complex.
Proof. First we observe that if there exist vertices vq,V\,V2 € L that are pairwise
joined by edges but do not span a simplex, then [vq , ν ι ] U [ν ι, 1>г] U [υ2, vq ] is a locally
geodesic loop in L. To see this, note that in Lk(i>i, L), which is an all-right complex,
the vertices corresponding to the edges [vq, υ ι ] and [υ ι, 1>г] are not joined by an edge,
so the distance between them is at least π and therefore [do, V[] U [иь иг] U [i>2, vq]
is geodesic in a neighbourhood of i>i (by 5.4).
If L is CAT(l), then so is the link of every vertex ν e L, the link of every vertex
v' e Lk(u, L), the link of every vertex v" e Lk(u', Lk(u, L)), etc.. In particular, none
of these successive links can contain a geodesic circle of length less than 2π. Each
of these links is an all-right spherical complex. Thus, applying the argument of the
preceding paragraph to these successive links, we deduce that the statement Φ„(Χ)
defined in (5.16(4)) is false for all η > 0, and therefore L is a flag complex.
It remains to prove that if L is a flag complex then it is CAT(l). Proceeding by
induction on the dimension of L, we may assume that Lk(u, L) is CAT(l) for every
vertex υ e L, and therefore it suffices (by 5.4(7)) to prove that any curve l С L
which is isometric to a circle must have length at least 2π.
We consider the geometry of geodesic circles in arbitrary all-right complexes K.
We claim that if υ is a vertex of К and £ is a geodesic circle, then each connected
component I' of Ι Π Β(υ, η/2) must have length π. To see this, consider the
development of I' in S2 (Fig. 1.7.1): this is a local geodesic (with the same length as I')
in an open hemisphere of S2 (the ball of radius π/2 about the image of υ). Since the
endpoints of this local geodesic lie on the boundary of the hemisphere, it must have
length π.
Now suppose that the length of £ is less than 2π. In this case I cannot contain two
disjoint arcs of length > π and therefore cannot meet two disjoint balls of the form
Β(υ, π/2). Thus the set of vertices ν e К such that I meets Β(υ, π/2) is pairwise
joined by edges in K. If К were a flag complex, this set of vertices would have to
span a simplex S and I would be contained in S, which is clearly impossible. Thus we
conclude that all-right flag complexes do not contain isometrically embedded circles
of length less than 2π. D
The following corollary was proved previously by Berestovskii [Ber86]. It shows
that there is no topological obstruction to the existence of a CAT(l) metric on a finite
dimensional complex.
5.19 Corollary. The barycentric subdivision of every finite dimensional simplicial
complex К supports a piecewise spherical CAT(l) metric.

212 Chapter II.5 MK-Polyhedral Complexes of Bounded Curvature
Proof. Consider the barycentric subdivision K' of the given complex — this is a flag
complex. The canonical all-right spherical metric on K' makes it CAT(l). D
We refer the reader to (1.7.40) for an explanation of the terminology "cubed"
versus "cubical". The following theorem is due to M. Gromov [Gro87].
5.20 Theorem. A finite dimensional cubed complex has non-positive curvature if
and only if the link of each of its vertices is a flag complex.
Proof. According to (5.2), a cubed complex has non-positive curvature if and only
if the link of each vertex is CAT(l). Such a link is an all-right spherical complex, so
we can apply Theorem 5.18. D
5.21 Moussong's Lemma. There is a useful generalization of (5.18) due to Gabor
Moussong [Mou88]. A spherical simplicial complex L is called a metric flag complex
(see [Da99]) if it satisfies the following condition: if a set of vertices {vq, ..., Vk] is
pairwise joined by edges ец in L and there exists a spherical ^-simplex in S* whose
edges lengths are l(e;j), then {do, ..., vk} is the vertex set of a ^-simplex in L.
Moussong's Lemma asserts that if all the edges of a spherical simplicial complex
L have length at least π/2, then L is CAT(1) if and only if it is a metric flag complex.
This lemma has been used by Moussong [Mou88] (resp. M. Davis [Da98]) to
prove that the appropriate geometric realization of any Coxeter system (resp.
building) is CAT(O). See also [CD95a].
Constructions with Cubical Complexes
The following construction is essentially due to Mike Davis.
5.22 Proposition. Given any finite dimensional simplicial complex L, one can
construct a cubical complex К such that the link of every vertex in К is isomorphic to
L. The group of isometries of К acts transitively on the set of vertices ofK, and ifL
has a finite number m of vertices, then К has 2m vertices.
Proof. Let V be the vertex set of L and let £ be a Euclidean vector space with
orthonormal basis {es \ s e V}. We shall construct К as a subset of the following
union of cubes С — {Jt{J2s<=tXses I x* e ДО> 1]}, where Τ varies over the finite
subsets of V. A face of С belongs to К if and only if it is parallel to a face spanned
by basis vectors eso,..., e St, where sq, ..., s^ are the vertices of a ^-simplex in L.
Note that the vertices of К are just the vertices of С The link in К of the vertex at
the origin is isomorphic to L.
Let rs be the reflection in the hyperplane that passes through es/2 and is orthogonal
to es, and let G be the subgroup of Isom(£') generated by these reflections. G is the

Constructions with Cubical Complexes 213
direct sum of the cyclic groups of order two generated by the rs (so if V is finite then
\G\ = 2|Vi). The action of G leaves К invariant and acts simply transitively on the
set of vertices of K. Since we already know that the link of one vertex is isomorphic
to L, this proves the proposition. D
According to (5.20), the cubical complex К constructed above will be non-
positively curved if and only if L is a flag complex. Thus, for example, if L is the
barycentric subdivision of any simplicial complex, then the universal cover К of
the complex constructed above will be a CAT(O) space, and in particular it will be
contractible.
5.23 A Compact Aspherical Manifold Whose Universal Cover is Not a Ball. In
1983 Mike Davis solved an important open question in the study of topological
manifolds. Until that time it was unknown whether or not there existed closed aspherical
manifolds whose universal cover was not homeomorphic to Euclidean space. Davis
constructed the first such examples [Da83]. We shall sketch his construction,
proceeding in several steps from the construction in (5.22).
Step 1: Let L, К and G be as in (5.22) but assume now that L is a finite complex. Let
F be the intersection of К with the cube {Ylsxses e Ε \ xs e [0, 1/2]}. Note that F is
a cubical complex (with edges of length 1 /2). Note also that F is a strict fundamental
domain for the action of G on K, i.e. each G-orbit meets F in exactly one point. Given
a (k — 1 )-simplex σ of L with vertices s ι,..., Sk, we write Ga to denote the subgroup
of G generated by the reflections rS],..., rSk and F" to denote the intersection of F
with the affine subspace defined by the equations xS] — 1/2,... ,xSt = 1 /2. We also
write ea = 5Z*=1 eSj. Note that G„ is the stabilizer of F" in G.
We endow G with the discrete topology. Thus GxFis the disjoint union of
copies of F that are indexed by the elements of G. We identify К with the quotient
of G χ F by the equivalence relation [(g, x) ~ (gh, x) if χ e F" and h e GCT], where
σ ranges over the simplices of L. We write this identification (g, x) м> g.x.
An important point to note is that F can be identified with the simplicial cone
CL' over the barycentric subdivision L' of L. Indeed one obtains a bijection CLI —> F
that is affine on each simplex by sending the cone vertex of CL' to the origin of Ε
and the barycentre of each simplex σ of L to jea e F. This isomorphism identifies
F" with the subcomplex L" of V С CL that is the intersection of the stars in 11 of
the vertices s, e σ.
Step 2: Let Μ be a compact piecewise-linear26 и-manifold with non-empty
boundary dM and let L be a triangulation of dM (compatible with the given PL-
structure). We mimic the construction of К in Step 1, replacing G χ F by G χ Μ.
Explicitly, we define N to be the quotient of G χ Μ by the equivalence relation
[(g, x) ~ {gh, x) if χ e L" с дМ and h e Ga], where σ ranges over the simplices
ofL.
henceforth abbreviated to PL

214 Chapter II.5 MK-Polyhedral Complexes of Bounded Curvature
We claim that N is a compact и-manifold without boundary. It is sufficient to check
that the equivalence class of a point of the form (1, x) e G χ дМ has a neighbourhood
homeomorphic to an и-ball. Because dM is a PL-manifold of dimension η — 1, if
σ С L is a (k — l)-simplex then IF is PL-homeomorphic to an (n — fc)-ball. The star
in U of the barycentre of σ is the simplicial join (see 1.7 A.2) of L" and the boundary
of the simplex σ. It follows that for each point χ in the interior of L", there is a
PL-homeomorphismy from a neighbourhood U of χ in Μ to a neighbourhood of 0
in (R+f χ Ж"~к, vnthj(U П L) С Э(К+)* χ Μ""*: the map; sends χ to 0 and sends
U Π Ζ/', for each vertex st of σ, into the subspace consisting of vectors whose г'-th
coordinate is 0.
Consider the homomorphism a : Ga —»■ Isom(E") sending each rSj to the
reflection in the г'-th coordinate hyperplane of W. The map sending (g, y) e Ga χ U to
a(g).j{y) induces a homeomorphism from a neighbourhood in N of the equivalence
class of (1, x) onto a neighbourhood of 0 e Μ* χ W~k.
Step 3: Let К be as in Step 1 and let Μ be as in Step 2, and assume now
that the manifold Μ is contractible. In this case there is a homotopy equivalence
/o : CL —»■ Μ which restricts to the identity on L = 3M and hence on 11. The map
(g, x) м> (g,/o(*)) is compatible with the equivalence relations on G χ CL' = GxF
and G χ Μ and thus we obtain a G-equivariant homotopy equivalence/ : К —»■ N.
If the triangulation L of 3M is a flag complex (this can always be achieved by
passing to a barycentric subdivision), then by (5.20) the cubical complex К is non-
positively curved. In particular, its universal cover is contractible, so all of its higher
homotopy groups are trivial, i.e. К is aspherical. Since N is homotopy equivalent to
K, it too is aspherical (so its universal cover is contractible).
In dimensions 4 and above it is possible to construct compact contractible n~
manifolds whose boundary is not simply connected (the first examples were given
by B. Mazur [Maz61] and V. Poenaru [РоебО]). In this case Davis shows that the
universal covering of N, though contractible, is not simply connected at infinity 2?
and hence is not homeomorphic to W.
In the above construction N is not endowed with any geometry — we merely
related its topology to a space К where we could use geometry (specifically, the
Cartan-Hadamard theorem) to prove that the space was aspherical. Remarkably,
if the dimension of Μ (and hence N) is at least 5, then Ancel, Davis and Guibault
[AnDG97] prove that N can always be metrized as a cubical complex of non-positive
curvature.
This means that N cannot be exhausted by a sequence of compact sets С such that any loop
in the complement of С is contractible in N ч С.

Two-Dimensional Complexes 215
Two-Dimensional Complexes
We now focus our attention on 2-dimensional M^-complexes of curvature < к. There
are three main reasons for doing so. First of all, in the 2-dimensional case it is easy
to check if an explicitly described M^-complex has curvature < к. Secondly, there
are many natural and interesting examples. And thirdly, 2-dimensional complexes
enjoy some important properties that fail in higher dimensions (for example Theorem
5.27). We refer the reader to [Hag93], [BaBr94,95], [BaBu96], [BuM97], [BriW99],
[Sw98] and [Wi96a] for examples of recent results in this active area.
5.24 Checking the Link Condition. As we noted earlier, it is easy to check whether
or not a given 2-dimensional MK -complex Ко has curvature < κ, because the link
condition for such a complex admits a simple reformulation:
(5.6). A 2-dimensional MK-complex Ко satisfies the link condition if and only if for
each vertex υ e Kq every injective loop in Lk(u, Ко) has length at least 2π.
Thus, if we label each of the corners of the 2-cells of Ко with the angle a, at
that corner, then the link condition for Ко is equivalent to a system of simultaneous
inequalities of the form a,·, + · · · + a,„, > 2π — there is one inequality for each
injective loop in the link of each vertex. Moreover, if к < 0 then for each и-sided
face of Kq we know that the sum of the vertex angles of that face is bounded by the
sum of the angles in a Euclidean и-gon: a,< + · · · + ацт < (и — 2)7r. If к < 0 then
each of this last set of inequalities will be sharp.
Changing perspective, we may view the above system of inequalities as giving a
necessary (but not sufficient) condition for a given combinatorial 2-complex to admit
the structure of an M^-complex of non-positive curvature. For suppose that we are
given a connected, combinatorial 2-complex К such that the attaching map of each
2-dimensional cell is a closed locally injective loop in the graph Kw; we say that
the cell is η-sided if this loop crosses exactly η edges. A corner is defined to be a
pair of successive 1-cells in the boundary of a 2-cell (this corresponds to an edge in
the link of the vertex at which the 1-cells are incident). To each corner we assign a
real variable a, and consider the simultaneous system of inequalities requiring that
the sum of the a, around any injective loop in the link graph of each vertex of К is
at least 27Г, and the sum of the a,· around each и-sided face of К is at most (и — 2)π.
If this system of inequalities does not admit a solution with each a, e (0, π], then К
does not support an MK -polyhedral structure of non-positive curvature (cf. [Ger87]).
There is a partial converse: if there is a solution such that all of the corners of each
face have the same angle, then one can obtain a metric of non-positive curvature by
metrizing all of the 2-cells as regular Euclidean и-gons of side length 1. However, the
existence of a less symmetric solution to the above system of inequalities does not
guarantee the existence an M^-complex structure of non-positive curvature (although
it does imply that К is aspherical [Ger87], [Pri88]). The key point to observe is that
while a solution appears to allow one to assign appropriate angles to the corners of K,
in general one cannot assign edge lengths in a consistent way that is compatible with

216 Chapter Π.5 MK-Polyhedral Complexes of Bounded Curvature
metrics on the 2-cells that have the desired angles. Readers wishing to understand
this difficulty should consider the 2-complex with one vertex, two directed 1-cells
labelled a and b respectively, and one 2-cell, whose boundary traces out the path a
(in the positive direction), then b twice, then a in the reverse direction, then b in the
reverse direction. The fundamental group of this complex is solvable but not virtually
abelian, and therefore it is not the fundamental group of any compact non-positively
curved space (cf. 7.5).
We note some situations in which the link condition is immediate. (These are
analogous to the small cancellation conditions of combinatorial group theory, see
[LyS77].)
5.25 Proposition. Let К be a combinatorial 2-complex and suppose that there is an
integer N such that К contains no η-sided faces with η > N. Let ρ and q be positive
integers such that every face of К has at least ρ sides and every simple closed loop
in the link graph of every vertex of К has combinatorial length at least q.
(1) If(p,q) e {(3, 6), (4,4), (6, 3)} then К can be metrized as a piecewise
Euclidean complex of non-positive curvature.
(2) If(p, q) e {(3, 7), (4, 5), (5,4), (6, 3)} then К can be metrized as a piecewise
hyperbolic complex of curvature < — 1.
Proof. For (1) it suffices to metrize each и-sided 2-cell of К as a regular Euclidean
и-gon with sides of length 1.
For every ε > 0 and N e N there exists δ(ε, Ν) > 0 such that if η < N then
the vertex angles in a regular и-gon in H2 with sides of length < 5 are at least
[π(η — 2)/и] — ε. For (2) it suffices to metrize each и-sided 2-cell of К as a regular
hyperbolic и-gon with sides of length 5(тг/21, Ν). Ο
5.26 Exercise. Let К be a piecewise Euclidean 2-complex each of whose faces is
metrized as a regular и-gon, where и may vary from face to face but is uniformly
bounded. Prove that if every injective loop in the link of each vertex of К has length
> 2я, then К can be metrized as an MK -complex of curvature < к for all к < 0.
Subcomplexes and Subgroups in Dimension 2
This section is dedicated to the proof of the following theorem.
5.27 Theorem. If К is a 2-dimensional MK-complex of curvature < к, then every
finitely presented subgroup of π\(Κ) is the fundamental group of a compact, 2-
dimensional, MK-complex of curvature < к with the geodesic extension property.
This theorem exemplifies a principle which can be applied to any class of
combinatorial 2-complexes that is closed under passage to subcomplexes and covering
spaces. This principle is founded upon the concept of a tower, which was developed

Subcomplexes and Subgroups in Dimension 2 217
in the context of 3-manifolds by CD. Papakyriakopoulos [Papa57] and adapted to
the setting of combinatorial complexes by Jim Howie [How81]. Roughly speaking,
Papakyriakopoulos's insight was that a large class of maps can be expressed as the
composition of a series of inclusions and covering maps, and this enables one to
exploit the fact that by lifting a map to a proper covering space one can often simplify
it (in some measurable sense). The relevance of this idea in the present context was
brought to our attention by Peter Shalen.
5.28 Definition. Given connected CW-complexes F and Z, with Υ compact, a map
g : Υ —*■ Ζ will be called an admissible tower if it can be written
g = k °P\ oi'i о ·· · °Ph °k
where each pr : Zr —*■ Fr_i is a connected covering of a compact complex, Zr has
the induced cell structure, and each ir : Yr —»■ Zr is the inclusion map of a compact
connected subcomplex. F/, = Υ and Zq—Z.
A map of CW complexes is said to be combinatorial if it sends open cells
homeomorphically onto open cells. (Note that a tower is a combinatorial map.) Let
X be a compact CW-complex. By a tower lift of a combinatorial map / : X —»■ Ζ
we will mean a decomposition/ = g of, where/' : X —»■ Υ is a combinatorial
map and g : Υ —»■ Ζ is an admissible tower. In this setting it is convenient to write
f:X^Y-^Z.
The following result is a special case of Lemma 3.1 in [How81].
5.29 Lemma. IfX and Ζ are connected CW-complexes andX is compact, then every
f g
combinatorial mapf : X —*■ Ζ has a maximal tower lift/ : X —»■ Υ —»■ Ζ such that
f : π\Χ —*■ π\Υissurjective.
Proof. Let Yq denote the image of/. We measure the complexity of/ by means of
the non-negative integer c(f) := d(X) — d(Yo), where d(K) denotes the number of
0-cells of a compact complex K.
Let i'o : Fq —>■ Ζ be the inclusion map and consider/ : X —*■ Yq —*■ Z. If the
map π\Χ —»■ π\Υ induced by /o is not a surjection then by elementary covering
space theory (see [Mass91] for example) there exists a proper connected covering
px : Z\ -»■ Fq such that/ = i0 opxo ц of, where f : X -»■ Z\ is a lifting of/o with
image Ki and ц :Υ\-*-Ζ\ is the inclusion map.
We claim that since p\ is a non-trivial connected covering, c(f\) < cifo). Note
first that {p\ ο ι'ι), which is surjective, is not injective (for otherwise its inverse would
give a section of p\). Hence there is a non-trivial deck transformation τ of Z\ such
that τ.Υ\ Π Υ\ φ 0. This intersection is the union of closed cells, hence there exist
0-cells v e Yi such that τ.ν = v. Sinceρ\(τ.υ) = ρι(υ), we have d(Y\) > d(Yo) and
hence c(/i) < c(/o).
If (/i)* : Tt\X —»■ 7TiF is not surjective then there exists a proper connected
covering/^ : Z^ —*■ Y\ such that/i = i\ ορ2 ο i2 of, where/2 : X —»■ Z2 is a lifting

218 Chapter II.5 MK-Polyhedral Complexes of Bounded Curvature
of/i with image Yj. and ii : F2 —*■ Z^ is the inclusion map. And с(/г) < c(f\) < c(fo).
Proceeding in this manner, since cif) is a non-negative integer, after a finite number
of steps we obtain a tower lift/ : X -A- F„ —»■ Ζ of/ such that/, induces a surjection
ОП7Г1. D
5.30 Proposition. Let Έ, be a class of (not necessarily compact) combinatorial
complexes that is closed under the operations of passing to finite subcomplexes and
to connected covers. Let Υ be a finitely presented group. If there exists an injective
homomorphism of groups φ : Γ —»■ π\Kfor some К € К, then there exists a compact
complex K' e K, of dimension at most 2, such that Γ = π\Κ'.
Proof. Our goal is to construct a combinatorial map of complexes / : X —»■ К
where X is compact, π\Χ = Γ, the image of/ is contained in 2-skeleton of К and
/* : π\Χ —> π\Κ is an injection (that represents φ). The preceding lemma implies
that such a map/ has a tower lift/ : XX 4 ГХ Л £ with/; : τηΧ -* π{Κ'
surjective. Since/,, = g„ o/| is injective, it then follows that/; is an isomorphism.
The compact complex if' is obtained from К by repeatedly taking subcomplexes and
coverings, hence К' е К; and because the image of/ is contained in the 2-skeleton
of K, the complex K' is at most 2-dimensional. Thus it suffices to construct X and
the combinatorial map/ : X —»■ К representing φ.
To say that Γ is finitely presented means that there is a surjection F(A) —*■ Γ
from the free group on a finite set Λ = {a\, ... , am] such that the kernel N of
this surjection is the normal closure of a finite subset R с F(A). Without loss of
generality we may assume that the number of generators m is chosen to be minimal,
in which case each a, determines a non-trivial element of Г, which we again denote
a;.
Let Xo be a graph with one 0-cell xq, and one l-cell for each of the generators
a,, oriented and labelled a,. (This labelling identifies 7Г1Х0 with the free group F.)
We fix a basepoint ко е К and for each generator α, ς Γ we choose a locally
injective loop in the 1-skeleton of К that represents $(α,·) &π\(Κ, ко). We then define
/ : Xq —»■ Kby sending the oriented edge labelled at to a monotone parameterization
of this loop. We introduce a new combinatorial structure on Xq by decreeing that
each point inf~l(K(0)) is a 0-cell;/ then becomes a combinatorial map. Note that
ft:F = 7Г1Х0 —*■ п\К is the composition of F —»■ Г and 0 : Г —»■ π\Κ, and so has
image φ(Γ) and kernel ЛЛ
Let S be an oriented circle with basepoint. Each of the defining relators r € R
determines an edge path cer : S —»■ X0 which begins at the vertex xq e Xq and then
proceeds to cross (in order) the oriented 1-cells whose labels are the letters of the
word r e F.
Because the word r in the generators a, represents the identity in Г, the loop
/ о cer is null-homotopic in K. So by van Kampen's lemma [K33b, Lemma 1] there
exists a simply connected, planar, 2-complex ΔΓ, a map βΓ : S —>■ Ar that is a
monotone parameterization of the boundary cycle of Ar, and a combinatorial map
fr : Ar -» ίΤ{2) such that/, ο βΓ =f о ar (cf. I.8A).

Subcomplexes and Subgroups in Dimension 2 219
Let X be the combinatorial 2-complex obtained by taking the quotient of Xq U
JJr ΔΓ by the equivalence relation generated by ar(x) ~ βΓ(χ) for all χ e S and all
r e R. Let ρ : Xq —»■ X be the quotient map. Let / : X —»■ # be the combinatorial
map induced by/ : Xq -»■ if and/r : ΔΓ -»■ if. By the Seifert-van Kampen theorem,
p* : Tt\Xo -> π\Χ is onto. And by construction r e ker/?*forallr e R,soN с kerp*.
But/* = /„ op* has kernel N, so/, : 7Г1Х —»■ π\Κ is an isomorphism onto the image
of/,, which is ф(Г). D
In order to apply the preceding proposition we need the following easy
observation.
5.31 Lemma. Fix iteR and let K(/c) denote the class of connected MK-complexes
К of dimension at most 2 that satisfy the link condition and for which Shapes(if)
is finite. If К € K(/c) then, when equipped with the induced length metric, every
connected subcomplex of К is in K(/c) and every connected cover of К is in К(к).
Proof. The asserted closure property of Щк) with respect to covers is not special to
dimension 2. Indeed, if К is an и-dimensional MK-complex with Shapes(if) finite
and К is a connected cover of К (with the induced path metric) then К is an MK-
complex with Shapes(iT) = Shapes(iT). And since the covering map К -»■ К is a
local isometry, К has curvature < к (i.e., satisfies the link condition) if and only if
К does.
The asserted closure property of К(к) with respect to subcomplexes clearly fails
above dimension 2. But if К is 2-dimensional and υ is a vertex of the subcomplex
L с К then Lk(u, L) is (locally isometric to) a subgraph of Lk(u, K). So if Lk(u, K)
contains no closed injective loops of length less than 2π, then neither does Lk(u, L).
The absence of such loops is equivalent to the link condition. D
5.32 Remark. One of the central problems in low-dimensional homotopy theory is
the Whitehead conjecture, which asserts that if the second homotopy group of a
2-dimensional CW complex is trivial, then so is the second homotopy group of each
of its connected subcomplexes. The preceding result, in conjunction with the Cartan-
Hadamard theorem, shows that MK -complexes of non-positive curvature satisfy the
Whitehead conjecture.
We now turn our attention to showing that if a compact 2-dimensional MK-
complex satisfies the link condition then it collapses onto a subcomplex with the
geodesic extension property. When combined with the preceding two results, this
completes the proof of Theorem 5.27.
We recall Whitehead's notion of an elementary collapse (which is the starting
point for simple-homotopy theory [Coh75]). Recall that a cell e of a combinatorial
complex К is, by definition, a free face if it lies in the boundary of exactly one cell e'
of higher dimension and the intersection of the interior of e' with some small
neighbourhood of an interior point of e is connected. We define an elementary collapse of К

220 Chapter П.5 MK-Polyhedral Complexes of Bounded Curvature
to be any subcomplex obtained by removing a free face and the interior of the unique
higher dimensional cell in whose boundary the free face lies. We say that К collapses
onto a subcomplex L if there is a sequence of complexes К = Ко,..., Κ„ = L such
that each Ki+\ is an elementary collapse of K·,.
It is clear that an elementary collapse does not alter the fundamental group of a
complex, so the following lemma completes the proof of Theorem 5.27.
5.33 Lemma. Let К be a compact two dimensional MK-complex. If К satisfies the
link condition, then it collapses onto a subcomplex which (when equipped with its
intrinsic metric) has the geodesic extension property.
Proof. According to (5.10) an MK-complex which satisfies the link condition has the
geodesic extension property if and only if it has no free faces. К has only finitely many
cells, so after finitely many elementary collapses we obtain the desired subcomplex.
D
5.34 Exercises
(1) How unique is the subcomplex yielded by the preceding lemma?
(2) If К is non-positively curved, compact, simply connected and two dimensional,
then it collapses to a point.
Knot and Link Groups
We began our discussion of 2-complexes by asserting that there are many
interesting examples. In Chapter 12 we shall describe how to construct examples using
complexes of groups and in Chapter Ш.Г we shall give explicit examples that are
of interest in group theory. In the present section we shall describe a construction
adapted from classical considerations in 3-manifold topology, in particular Dehn's
presentation for the fundamental group of the complement of a knot in S3. The
geometric construction here is similar to earlier work of Weinbaum [We71] and others
who studied these groups from the point of view of small cancellation theory (see
[LyS77, p.270]). The adaptation to the non-positively curved case was first noticed
by I. Aitchison and (independently) by D. Wise [Wi96a].
5.35 Theorem. 7/X с К3 is an alternating link then πι (Κ3 — /С) is the fundamental
group of a compact 2-dimensional piecewise-Euclidean 2-complex of non-positive
curvature.
5.36 Remark. This result is not the strongest of its ilk, but the degree of generality
chosen allows a straightforward and instructive proof. In fact, all link groups are
fundamental groups of non-positively curved spaces: this is a consequence of Thurston's
geometrization theorem for Haken manifolds [Mor84], [Thu82] and the fact that link

Knot and Link Groups 221
complements are irreducible [Papa57]; cf. [Bri98b] and [Le95]. In addition to (5.35),
we shall cover hyperbolic knots (Theorem 11.27) and torus knots (11.15).
For clarity of exposition we shall consider only knots; the adaptation to the case
of links is entirely straightforward.
5.37 Definitions. We consider smooth embeddings / : S1 ^ K3. Two such
embeddings/o and/i are said to be equivalent if there is a smooth-embedding
F : Sl χ [0, 1] -> Κ3 χ [0, 1] such that F(x, t) = (f(x, t), t) for all χ e Sx and
t e [0, 1], where/(x, 0) =fo(x) andf(x, 1) = fi(x). A (tame) knot is, by definition,
an equivalence class of smooth embeddings Sl ^> R3. It is convenient to work with
a definite choice of representative for the class under consideration, or even just the
image of this representative, which we write К, С К3. We may refer to K, itself as
'the knot'.
Let π : Κ3 —»■ Μ.2 = К2 х {0} be the obvious projection. Every knot has a
representative/ : S1 -4^cl3 such that π of has a nowhere zero derivative, all
of its self-intersections are transverse, and for every χ e im(7r of) the cardinality of
Κ, Π я-1 (χ) is at most two. Π = π of is called a (regular) projection of /C. Let Λ
denote the image of Π. The points χ e Λ for which Κ.(Ίπ~ι(χ) has cardinality two
are called its double points. If χ is a double point, then the elements of Κ, Π η~ι(χ)
are called the overpass and underpass at x, according to which has the greater third
coordinate. Π is called an alternating projection if when traversing K, in a monotone
manner one encounters underpasses and overpasses alternately. By definition, an
alternating knot is one that admits an alternating projection.
Π is called a prime projection if it does not admit a reducing circle, i.e, an
embedded circle S С M.2 such that S intersects Л in exactly two points, neither of
which is a double point, and both components of Μ.2 \ S contain a double point of
Л. Let Σ be a smoothly embedded 2-sphere in R3 that intersects K, transversally at
exactly two points ρ ι and p2. Let 1 and О be the connected components of Κ3 χ Σ.
Let [ρι, p2\ be an arc joiningp\ top% on Σ. Let K\ and KLi be the knots obtained by
smoothing \р\,рг\ U (Κ, Π Ο) and \р\,рг\ U (/С П J) respectively. In this situation,
K, is said to be the connected sum of KL\ and K,2, written K, = /Ci#/C2-
5.38 Remark. The above definition of connected sum can be viewed as an operation
# : {knots} χ {knots} —»■ {knots}. This is well-defined (in particular it is independent
of the choices of /C, KL\ and /C2 within their equivalence classes) and makes the set
of knots into an abelian monoid (see [BurZ85], [Rol90]).
5.39 Lemma. Every alternating knot K, is the connected sum of finitely many knots
each of which has an alternating prime projection.
Proof. Let Π = π of be an arbitrary alternating projection of /C, with image Λ.
If Π is not prime, then there exists a reducing circle id2. Let p~[ and p~i be the
points at which S intersects Λ, and let Λι and Λ2 be the components into which S
separates Л. Let/?i = π~~ι(ρ~\) Π Κ, and pi = Tt~xQp~j) Π /С. By closing the cylinder

222 Chapter II.5 MK-Polyhedral Complexes of Bounded Curvature
S χ К with horizontal discs well above and below K, we obtain an embedded 2-sphere
Σ с К3 that is transverse to /C and meets it only at p\ and pi- Σ separates К, into
two components, parameterized by the restriction of/ : S1 —*■ K, to the connected
components of Sl \ Τί~ι{ρΊ,Ρ2}', we call these two components of the circle C\
and Ci- Thus K, = K,i#K,2, where (interchanging the indices as necessary) /C,- has
projection Π,·, which begins atp,, then describes Λ, as parameterized by Π|<~, then
follows the arc of the reducing circle S joining p, to pt±ι ■
In order to complete the proof we must check that the projection Π,- is alternating.
The double points of Π, are precisely those of П|с;, so since Π = π о/ is alternating
it only remains to prove that if the first double point along Π is encountered as an
overpass, then the last is encountered as an underpass, and vice versa. But this is
easily seen by combining the fact that underpasses and overpasses alternate along
f(C;) with the observation that the number of underpasses on /(C,·) is equal to the
number of overpasses, because there is one of each for each double point in П(С,)-
D
5.40 Corollary. If К, с К3 is an alternating knot then the fundamental group of
(R3 ν KL) is an amalgamated free product of the form
G0 *Cl Gi *c2 ■ ■ · *c„ G„,
where each Gi is the fundamental group of the complement of a knot with a prime
alternating projection and each C, is infinite cyclic.
Proof. This follows from (5.39) upon application of the Seifert-van Kampen theorem
(see [BurZ85]). D
5.41 The Dehn Complex of a Projection.
We maintain the notation and conventions established in (5.37). In particular,
Λ is the image of the regular projection Π of the knot 1С. We need an additional
definition: fix a neighbourhood Vx about each double point χ of Λ, small enough so
that Vx intersects Λ only in two small arcs meeting transversally at x; a connected
component of (Λ Π Vx) χ {χ} is called a germ. Thus each edge of the graph Λ has
two germs, one at each end. We label germs о or и according to whether they are the
image of an arc in K, going through an overpass or an underpass. When embellished
with this extra information, Л is called a knot diagram.
The Dehn complex Т>(П) of the projection Π is a 2-complex with two vertices
v+, υ- (which are thought of as lying, respectively, above and below the plane of
projection). The 1-cells of T>(Tl) ^e in 1-1 correspondence with the connected
components Ло, A\, ... , A„ of Κ2 χ Λ, the г'-th l -cell being oriented from v+ to υ_ ^nd
labelled A,; the2-cellsof Τ>(ΐϊ) are in 1-1 correspondence with the double points of Λ.
They are attached as follows: given a double point χ e Λ, let Α,,ο), Α^,Α^,Α^μ)
be the (not necessarily distinct) components of Κ2 χ Λ that one encounters as one
proceeds anticlockwise around x, beginning at a germ labelled o; for each x, a 2-cell
is attached to the 1-skeleton of T>(JJ) by a locally injective map from its boundary to

Knot and Link Groups 223
the edge-path labelled Aix(i)AJ~J2)Aix0)AJ~J4), where A"1 is A,- traversed in the opposite
direction.
Choosing v+ as basepoint and writing a,- for the homotopy class of the loop
A,Aq , we see that the fundamental group of Τ>(Π) has presentation
"Pn = (<2o, ■ ■ ■, an | a0 = 1, aix(i)a7,(2)ah(3)a7x(4) = l f°r everv double point x).
This is the well-known Dehn presentation of the fundamental group of Κ3 χ /С. The
following exercise is standard.
5.42 Exercise. Show that Vn is indeed a presentation of 7Τι(Κ3 χ /С).
(Hint: Isotop К, to lie in the union of Κ2 χ {0} and the spheres of some small
radius r about the double points of the projection. Place a ball of radius 2r about
each double point and apply the Seifert-van Kampen theorem.)
Henceforth we view T>(SJ) as a piecewise Euclidean complex with the intrinsic
metric obtained by metrizing each 2-cell as a unit square.
5.43 Proposition. THJl) is non-positively curved if and only if Π is a prime
alternating projection.
Proof. We first describe the links Lk(i>±, T>(JJ)). Each edge in one of these link graphs
has length π/2, so our concern will be to determine when they contain circuits of
length less than 4. The vertex set of Lk(u±, Χ>(Π)) is in 1-1 correspondence with
the 1-cells of Τ>(Π); from which it inherits a labelling A0, ■ ■ ·, A„. The edges of
Lk(v+,V(n.)) (resp. Lk(i>_, Χ>(Π))) are in 1-1 correspondence with pairs (x, g),
where χ is a double point of Λ and g is a germ incident at χ labelled и (resp. o). The
edge corresponding to (x, g) joins the vertices labelled by the regions which meet
along g.
If the projection Π is not prime, then there exists a reducing circle S. The two
components of S χ Λ lie in distinct connected components of Κ2 χ Λ, A; and Aj
say. These components meet along two distinct edges, which means that they meet
along four distinct germs. At least two of these germs must carry the same label, о
or u, and therefore give rise to two edges joining the vertices labelled A,- and Aj in
the corresponding link graph — whence a circuit of length two.
To say that the projection Π is not alternating means precisely that the germs at
either end of some edge e in Λ carry the same label, и say. These two germs give rise
to two distinct edges in Lk(i>+, T>(TI)) connecting the vertices labelled by the two
components of Κ2 χ Λ that abut along e.
If Π is alternating then one can divide the components of Κ2 χ Λ into two types, let
us call them black and white: a bounded component is said to be white (respectively,
black) if the anticlockwise orientation of its boundary orients each of the edges in
its boundary from the germ labelled о to the germ labelled и (respectively, и to o);
the unbounded region Ao is coloured according to the opposite convention. Note
that if two regions abut along an edge of Л then they have opposite colours. If we

224 Chapter II.5 MK-Polyhedral Complexes of Bounded Curvature
give the vertices of Lk(i>±, T>(TI)) the induced colouring, then the preceding sentence
implies that the endpoints of each edge in Lk(u±, V(T1)) are coloured differently. It
follows that all closed circuits in Lk(i>±, V(F1)) are of even length, so to check the
link condition we need only rule out circuits of length two.
There are two edges connecting the vertices of Lk(u±, Τ>(Π)) labelled A, and Aj
if and only if the regions A; and Aj meet across two germs of edges in Λ that carry
the same label. If Π is alternating, the germs at either end of an edge carry different
labels, whence A,· and Aj meet across two distinct edges. But then, by connecting the
midpoints of these edges with arcs in the interiors of A, and Aj we obtain a reducing
circle showing that Π is not prime. D
Proof of Theorem 5.35. According to Corollary 5.40, the fundamental group Γ of
the complement of an alternating knot К is of the form Go *c, G\ *c, ■ ■ ■ *c„ Gn,
where each G,- is the fundamental group of a knot with a prime alternating projection,
and each C, is infinite cyclic. The preceding proposition shows that each G, is the
fundamental group of a compact, piecewise Euclidean 2-complex of non-positive
curvature. Itfollows from (11.17 and 1.7.29) that Г is also the fundamental group of
such a complex (but not necessarily a complex built from squares). D
5.44 Exercises
(1) Consider the projection of the trivial knot given in polar coordinates by the
equation r = (1/V2) + cos#. Show that the Dehn complex of this projection is a
M'obius strip. Which projections of the trivial knot have non-positively curved Dehn
complexes?
(2) The Hopf link consists of two unknotted circles in K3 linked in such a way
that they cannot be separated but have a planar projection with only two crossings
(figure 5.1)? What is the Dehn complex of this minimal projection of the Hopf link?
(3) The Borromean rings is a link with three components such that if one removes
any one of the components then the other two can be unlinked (fig. 5.1). What is the
Dehn complex of the projection shown in (fig. 5.1).
From Group Presentations to Negatively Curved 2-Complexes
In [Rips82] Rips introduced a construction that associates to any finite group
presentation a short exact sequence 1—►Ν—>-Γ—>-G—>- 1, where N is a finitely
generated group, Г is a finitely presented group satisfying an arbitrarily strict small
cancellation condition (cf. 5.25) and G is the group denned by the original
presentation. We shall modify Rips's construction so as to arrange that G is the fundamental
group of a negatively curved 2-complex. An earlier modification of this sort was
given by Daniel Wise [Wi98a].
5.45 Theorem. There is an algorithm that associates to every finite presentation a
short exact sequence

From Group Presentations to Negatively Curved 2-Complexes 225
Fig. 5.1 Hopf link and Borromean rings
1 -+ Ν -* ττχΚ^ G^ 1,
where G is the group given by the presentation, N is a finitely generated group and
К is a compact, negatively curved, piecewise hyperbolic 2-complex.
Proof. Given a finite presentation (χι, ... ,xn | r\, ..., rm) of G, let ρ = max |r,| + 2
and let Μ be the least integer such that M1 > (ЮпМ + 5m) + M.
The idea of the proof is to expand the given presentation by adding a finite number
of new generators ay, one uses these new generators to 'unwrap' the relations in G,
replacing r, = 1 with a relation of the form r, = v„ where υ, is a word in the
generators a,; then one forces the subgroup generated by the a, to be normal by
adding extra relations of the formxf αμ~ε = иц^, where the words и,^ involve only
the letters a^. The art of the construction comes in choosing the words υ, and M;j.e.
We shall choose them in a way that allows us to metrize the 2-cells of the standard
2-complex К of the presentation (1.8 A) as right-angled hyperbolic pentagons with
the words r, completely contained in the interior of one side. We then arrange for the
2-complex to have negative curvature.
К is defined as follows: ^
• К has one 0-cell.
• К has (M + n) 1-cells, labelled x\,... ,xn,a\,..., ам- (The loops labelled a,
generate the normal subgroup N in the statement of the theorem.)
• К has (m + 2nM) 2-cells, which are of two types:
• the 2-cells of the first type are in 1-1 correspondence with the original relators
T,
• the 2-cells of the second type are in 1-1 correspondence with triples τ =
(xi, α.), ε), where ε = ±1 , i = 1,..., и and j = I,... ,M.
• The attaching map of the 2-cell corresponding to r, traces out the path labelled
ρ—\Γλ—ι ρ ρ ρ ρ
where each a,_; e {ai,..., ам] — the exact choice of a,,; is deferred.

226 Chapter П.5 MK-Polyhedral Complexes of Bounded Curvature
• The 2-cell corresponding to τ = (χ,, щ, ε) is attached by a path labelled
βοΛα,χΤεβξ?βϊ.τβΙΛτβΖ,τ·
where each βιτ e {a\,... ,ам) — again, the exact choice of /3,,r is deferred.
• Let L be the length of each side in a regular right-angled hyperbolic pentagon.
Each 1-cell in К is metrized so that it has length L/p.
• Each 2-cell is metrized as a regular right-angled hyperbolic pentagon, so that
the arcs of the boundary that are labelled afj (in the case of 2-cells of the first
type) or β? (in the case of 2-cells of the second type) become sides of the
pentagon.
We claim that we may choose the a,;j and the βίτ so as to ensure that the piecewise
hyperbolic complex К described above satisfies the link condition (5.1). How could
it fail to do so? Well, the vertex angles at the corners of the 2-cells are all either я/2 or
я (we call the former 'sharp corners') and at corners with a vertex angle π we either
have one edge labelled jc, and one labelled Oj, or else we have both the incoming and
outgoing edge labelled by the same a,. Each 2-cell has only five sharp corners and
at each of these corners we have arranged that the labels on both the incoming and
outgoing edge are drawn from {αϊ,..., ам}·
In order to ensure that there are no circuits of length less than π in the link of the
vertex of K, it suffices to arrange that no ordered pair of labels (α,, a,-) occurs at more
than one sharp corner, and that at no sharp corner do we have a, = a,<. This amounts
to choosing a family of (m + 2nM) words α,,οα;. ια,^α,^α,^ and βο.τβ\.τβι.τβ\τβΑ.τ
with the following properties: no 2-letter subword сцщ appears more than once; there
is no subword of the form сца^ and for all j,/ there is at most one word of the form
a, * * * af. We shall call a family good if it has these properties.
We seek a good family words. To this end, following Wise [98a], we list all of
the words ща? with i < i' in lexicographical order:
(aia2),..., (aiaM), (a2, a3),..., (а2ам), (азсц), , {aM-\aM)-
We then concatenate the words in this list to form one long word WM. By construction,
no 2-letter subword щщ appears more than once in WM and there is no subword of
the form ciidi. Moreover, for each fixed j,/, the word Wm has at most one subword
of length 5 that is of the form cij***cif.
WM has length M2 - M, and we chose Μ so that M2 - Μ > 5(m + 2nM). Thus
we can divide WM into (m + 2nM) disjoint subwords of length 5:
{αχαιαχα^αχ), {α^αια^αχαβ),
This is the good family of words that we were seeking. Π
5.46 Remarks
(1) We emphasize that К was obtained from the given presentation in an
algorithmic way: the integer Μ depended in a simple way on the number of generators and

From Group Presentations to Negatively Curved 2-Complexes 227
relators; the metric depended only on p, which is an invariant of the presentation;
and the "good" family of words used to attach the 2-cells of К were derived from Μ
by a simple algorithm.
(2) Notice that all of the 1-cells of К are geodesic circles and that they are the
shortest homotopically non-trivial loops in K.
5.47 Exercise (Incoherence). Robert Bieri [Bi76a] showed that if Η and Γ are
finitely presented groups of cohomological dimension at most two, and if Я С Г is
normal, then Η is either free or else it has finite index in Γ. Prove that the group N
constructed in (5.45) is not free, and hence deduce that if G is infinite then N is not
finitely presented.
Groups that contain subgroups which are finitely generated but not finitely
presented are called incoherent. (See [Wi98a] for further results in this direction.)

Chapter II.6 Isometries of CAT(O) Spaces
In Chapters 2 and 6 of Part I we described the isometry groups of the most classical
examples of CAT(O) spaces, Euclidean space and real hyperbolic space. Already in
these basic examples there is much to be said about the structure of the isometry group
of the space, both with regard to individual isometries and with regard to questions
concerning the subgroup structure of the full group of isometries. More generally, the
study of isometries of non-positively curved manifolds is well-developed and rather
elegant. In this chapter we shall study isometries of arbitrary CAT(O) spaces X.
We begin with a study of individual isometries γ e Isom(X). We divide isometries
into three types: elliptic, hyperbolic and parabolic, in analogy with the classification
of isometries of real hyperbolic space discussed in Chapter 1.6. We develop some
basic properties concerning the structure of the set of points moved the minimal
distance by a given isometry γ.
In the second part of this chapter we consider groups of isometries. After noting
some general facts, we focus on the subgroup of Isom(X) consisting of those
isometries which move every element of X the same distance — Clifford translations. These
translations play an important role in the description of the full isometry groups of
compact non-positively curved spaces (6.17). In the last section of this chapter we
shall prove a theorem (6.21) that relates direct product decompositions of groups
which act properly and cocompactly by isometries on CAT(O) spaces to splittings of
the spaces themselves.
We refer to Chapter 1.8 for basic facts and definitions concerning group actions. In
particular we remind readers that in the case of spaces which are not locally compact,
the definition of proper group action that we shall be using is non-standard. \
Individual Isometries
In order to analyze the behaviour of isometries of a CAT(O) space, we first divide
the isometries into three disjoint classes. Membership of these classes is defined in
terms of the behaviour of the displacement function of an isometry. This function is
defined as follows.

Individual Isometries 229
The Displacement Function
6.1 Definitions. Let X be a metric space and let у be an isometry of X. The
displacement junction of у is the function dy : X —> K+ defined by dy(x) = d(y.x, x).
The translation length of у is the number \y\ := inf{dY(x) \ χ e X}. The set of
points where dY attains this infimum will be denoted Min(y). More generally, if Γ
is a group acting by isometries on X, then Μϊη(Γ) := Пу(=г Min(y).
An isometry у is called semi-simple if Min(y) is non-empty. An action of a
group by isometries of X is called semi-simple if all of its elements are semi-simple.
6.2 Proposition. Let X be a metric space and let γ be an isometry ofX. Let Υ be a
group acting by isometries on X.
(1) Min(y) is γ-invariant and Μϊη(Γ) is Y-invariant.
(2) If a is an isometry ofX, then \γ\ = |ауа_1|, andMin(aya~l) = a.Min(y).
In particular, if a commutes with γ, then it leaves Min(y) invariant. If N is a
normal subgroup ofY, then Min(7V) is Y-invariant.
(3) IfX is a CAT(O) space, then the displacement function dy is convex, and hence
Min(y) is a closed convex set.
(4) If С С X is non-empty, complete, convex, and γ-invariant, then \γ\ = \y\c\ and
γ is semi-simple if and only ify\cis semi-simple. Thus Min(y) is non-empty if
and only if CD Min(y) is non-empty.
Proof. The proofs of (1) and (2) are easy, and (3) follows from the convexity of the
distance function in а С AT(0) space (2.2). To prove (4), consider the projection p of
X onto C. By (2.4) we have ρ(γ.χ) = γ.ρ{χ) and d(y.x, x) > d{y.p(x), p(x)), for all
χ e X, hence p(Min(y)) = Min(y) DC = Min(y|c). D
6.3 Definition. Let X be a metric space. An isometry у of X is called
(1) elliptic if у has a fixed point,
(2) hyperbolic (or axial) if dY attains a strictly positive minimum,
(3) parabolic if dY does not attain its minimum, in other words if Min(y) is empty.
Every isometry is in one of the above disjoint classes and it is semi-simple if and
only if it is elliptic or hyperbolic. If two isometries of X are conjugate in Isom(X),
then they have the same translation length and lie in the same class.
6.4 Examples
(1) Consider the half-space model {x = {x\,... ,x„) e W \ x„ > 0} for
hyperbolic space W. If λ > 0 and λ φ 1, the map χ м> λχ is a hyperbolic
isometry у with |y| = | logλ|. In this case Min(y) is a single geodesic line, namely
{x e W | x„ > 0, x; = 0 if i < ή]. Every hyperbolic isometry of Ш" with the
translation length | logX| is conjugate in Isom(H") to the composition of у and an
orthogonal transformation that fixes the axis of у pointwise.

230 Chapter Π.6 Isometries of CAT(O) Spaces
An orientation preserving isometry of Ш2 is parabolic if and only if it is conjugate
in Isom(H2) to an isometry of the form γ : (x\ , x2) м> (χι + 1, x2). Note that | γ | = 0
and that the level sets of dY are the subsets x2 = const.
The half-space model for H2 can be described as the space of complex numbers ζ
whose imaginary part lm(z) is positive. In (1.6.14) we showed that the full isometry
group for this model can be identified with PGL(2, K), the quotient of the group of
real (2, 2)-matrices by the subgroup formed by scalar multiples of the identity. It is
easy to show that the isometry determined by an element of GL(2, K) is semi-simple
if and only if this element is semi-simple in the classical sense, i.e. it is conjugate
over С to a diagonal matrix. More generally, in Chapter 10 we shall describe the
action of GL(«, K) by isometries on the symmetric space P(n, K) of positive definite
(n, «)-matrices and prove that an element of GL(«, K) acts as a semi-simple isometry
if and only if it is a semi-simple matrix in the classical sense.
(2) Let γ be an isometry of E". We have seen (1.2.23 and 1.4.13) that γ is of
the form χ н> Ах + b, where A e 0{n) and b e K". If γ does not fix a point
then — b is not in the image of A — /, so A — / is not invertible. Thus Α ν = υ for
some и 6 1" \ (0|. Let V\ be the vector subspace generated by υ and let V2 be its
orthogonal complement. Then E" is isometric to the product V\ xV2, and because γ
maps each line that is parallel to V\ to another such line, it follows from 1.5.3(4) that
γ splits as the product of a translation γχ of V\ and an isometry γι oiV2. Because
V2 is isometric to E"_1, arguing by induction on η it is easy to prove the following:
6.5 Proposition. Every isometry o/E" is semi-simple. More precisely, an isometry γ
o/E" is either elliptic, or else it is hyperbolic and there is an integer к with 0 < k < η
such that Min(y) is an affine subspace E\ of dimension к. And in the second case, if
E2 is an affine subspace which is an orthogonal complement of Ε ι (i.e. Ε" = Ει χ Ε%),
then γ is the product of a non-trivial translation γ\ ofE\ and an elliptic isometry γ2
ofE2.
In contrast to the finite dimensional case, infinite dimensional Hilbert spaces do
admit parabolic isometries (see Example 8.28).
6.6 Exercises
(1) Let X be a metric space and let γ be an isometry of X. Show that
lim -d(x, γ".χ)
π-»οο η
exists for all χ e X. Show that lim^oo ^d(x, γ".χ) is independent of x, and if γ is
semi-simple then \γ\ = limMOO \d(x, γ".χ).
(Hint: η м> d(x, γ".χ) is a subadditive function. A function / : N —»■ N
is subadditive if f(m + ri) < f(m) + f(n) for all m and n. It is a classical fact
that Нтп^оо/(и)/и exists. Indeed for fixed d > 0, any integer η can be written
uniquely as η = qd + r with 0 < r < d. The subadditivity condition implies that

Individual Isometries 231
f(ri)/n < f(d)/d +f(r)/n, hence limsupn^cxf(n)/n < f(d)/d for any d. Therefore
limsup,MCO/(«)/« < limmid^xif(d)/d.)
(2) Let К be a connected Μκ-simplicial complex with Shapes(if) finite (in the
sense of Chapter 1.7). Show that every simplicial isometry of К is semi-simple.
Show further that if a group Г acts by cellular isometries on K, then the set of
translation lengths {|y| : γ e Γ} is a discrete subset of the real line. (Hint: Given
an isometry y, there exists a model simplex S e Shapes(if), a sequence of points
x„ e S and simplices <pn(S) С К such that άγ{φη{χη)) —»■ \γ\ as и —»■ oo. Combine
this observation with (1.7.59).)
(3) Recall that an M-tree is a geodesic space that is CAT(/c) for all ке1. Prove
that every isometry у of an M-tree is semi-simple. (Hint: Let χ e X and let m be the
midpoint of [χ, γ.χ]. Prove that d(m, y.m) = \y\.)
Semi-Simple Isometries
In this paragraph we note some basic facts about semi-simple isometries. First we
give an easy characterization of elliptic isometries.
6.7 Proposition. Let X be a complete CAT(O) space, and let γ be an isometry ofX.
Then, γ is elliptic if and only if γ has a bounded orbit. And if γ" is elliptic for some
integer η φ 0, then γ is elliptic.
Proof. The first assertion is a special-case of (2.7). If y" is elliptic, fixing χ e X say,
then the orbit of χ under γ is finite, hence bounded. D
The following result concerning the structure of Min(y) when γ is hyperbolic
will be used repeatedly both in the remainder of this chapter and in subsequent
chapters.
6.8 Theorem. Let X be a CAT(O) space.
(1) An isometry γ of X is hyperbolic if and only if there exists a geodesic line
с : К —»■ X which is translated non-trivially by γ, namely y.c(t) = c(t + a),for
some a > 0. The set c(K) is called an axis of γ. For any such axis, the number
a is actually equal to\y\.
(2) If X is complete and γ"1 is hyperbolic for some integer m φ 0, then γ is
hyperbolic.
Let γ be a hyperbolic isometry ofX.
(3) The axes of γ are parallel to each other and their union is Min(y).
(4) Min(y) is isometric to a product Fxl, and the restriction of γ to Min(y) is
of the form (y, f) t-> (y, t + |y|), where у eY,t eR.
(5) Every isometry a that commutes with γ leaves Min(y) = Υ χ Μ. invariant, and
its restriction roFxl is of the form (a', a"), where a' is an isometry of Υ and
a" a translation o/ffi.

232 Chapter II.6 Isometries of CAT(O) Spaces
Proof. The "if" direction in (1) is a special case of 6.2(4). Conversely, we claim
that if у is hyperbolic then every point of Min(y) lies on an axis of y, namely the
union of the geodesic segments [y".x, y"+1 .x]n^z· Since local geodesies in CAT(O)
spaces are geodesies, it suffices to show that [x, y2.x] is the concatenation of [χ, γ.χ]
and [γ.χ, γ2.χ], and this is equivalent to showing that d(m, γ.ηί) = d(m, γ.χ) +
ά{γ.χ, γ.ητ) = 2 d(x, m), where m is the midpoint of the geodesic segment [χ, γ.χ].
As Min(y) is convex, it contains m, hence d(m, γ.ηί) = d(x, γ.χ). But d(x, γ.χ) =
2 d(x, m), so d(m, γ.ηί) = 2 d(x, m), as required.
To complete the proof of (3) we must show that if c, d : К —> X are axes for у
then they are parallel. Since γ.φ) = c(t + \γ\) and γχ'(ί) = c(t + |y |), the convex
function t ι-* d(c(t), d(t)) is periodic of period |y|. In particular it is bounded and
therefore constant.
For (4), one notes that Min(y) is a convex subspace of X, and hence is itself a
CAT(O) space. In the light of (3), we can apply the Decomposition Theorem (2.14)
to the set Min(y) to obtain a product decomposition Min(y) = Fxl, where each
{y} χ Μ is an axis for y, so in particular y.(y, t) = (y, t + \γ\).
(5) Let a be an isometry of X that commutes with у. It leaves Min(y) invariant
(6.2(2)). Moreover, a takes axes of у to axes of y. Hence (see 1.5.3) a preserves the
product decomposition Min(y) = У χ Μ and splits as (a', a"). Because a" e Isom(lR)
commutes with the translation defined by y, it must be a translation.
Finally we prove (2). If γ"1 is hyperbolic, then Min(ym) splits as У χ Μ where ym
is the identity on Υ and a non-trivial translation on M. As у commutes with ym, by
(5) we see that the restriction of у to Min(yOT) = Υ χ Κ splits as (у', у"), where γ'"'
is the identity of Υ and y" a non-trivial translation of R. But У is a complete CAT(O)
space (because it is closed and convex in X), so the periodic isometry y' must have
a fixed point in Y. The product of any such fixed point with К yields an axis for y,
so by (1) we are done. Π
In relation to part (5) of the preceding proposition, we note the following fact
concerning isometries of product spaces.
6.9 Proposition. Let X be a metric space which splits as a product Χ' χ Χ", and let
У = (/'. У") be an isometry preserving this decomposition. Then, γ is semi-simple
if and only if γ1 and y" are semi-simple. Moreover, Min(y) = Min(y') χ Min(y").
Proof, у is semi-simple if and only if Min(y) φ 0, so it is enough to show Min(y) =
Min(y') χ Min(y").
For any xf, y' e X' and x", у" е X", we have
d{Y'{x!), x') < d(y'(y'\ /) <=* d(y(x', x"), (x1, x")) < d(Y(y', x"), (y1, x"))
d(Y"(x"), x") < d(Y"(y"), У") <=* ά{γ{χ!, χ"), (χ1, χ")) < ά{γψ, у"), (х>, /')).
Thus (from <ί=), we have Min(y) С Min(y') χ Min(y") and (from =>) Min(y) 3
Min(y') χ Min(y"). □

On the General Structure of Groups of Isometries 233
On the General Structure of Groups of Isometries
In this section we gather some general facts about groups acting properly by
isometries. The first result does not involve any hypothesis of curvature.
6.10 Proposition. Suppose that the group Γ acts properly by isometries on the metric
space X. Then:
(1) If a subspace X' ofX is invariant under the actioTTofa subgroup Γ" ο/Γ, then
the action ofY' on X' is proper.
(2) If the action of Γ is cocompact then every element of Γ is a semi-simple isometry
ofX.
(3) Ifthe action ofY is cocompact then the set of translation distances {\γ\ \ у e Γ}
is a discrete subset o/K.
(4) Assume that X splits isometrically as a product Χ' χ Χ" and that each element
γ of Υ splits as (/', γ"). Let N be a normal subgroup of Υ formed by elements
of the form у = (idy, γ"), and assume that there is a compact subset К in X"
such that X" = [JysN y".K. Then the induced action ofY/N on X' is proper.
(In this action the coset (y', γ")Ν acts as γ'.)
Proof. (1) is immediate from the definitions.
(2) Let К be a compact set whose translates by the action of Г cover X. Fix у e Γ
and consider its displacement function dY. Let (xn) be a sequence of points in X such
tbatdY(x„) —> |y|as«—» oo. We choose elements y„ e Tsuchthat}',, := γ„.χη e K.
Note that d(ynYY~l .yn, y„) = d(y.xn,x„) tends to |y|, as η —> oo. Hence, for every
point χ e K, the sequence ά(γηγγ~ι.x, x) remains bounded. Because the action of
Γ is proper, there are an infinite number of integers η such that γηγγ~ι is equal to
the same element У of Г. Passing to a subsequence if necessary, we assume that
У = ΥηΥΥ,Γ1 for every positive integer n. Because К is compact, we may pass to a
further subsequence in order to assume that (y„) converges to some point у е К. Then,
for every positive integer n, the function dY assumes its minimum at y~' .y, because
dY<?yfl-y) = <*(УУп1-У>Уп1-У) = d(Y-y,y) = Ьт<1Су-Уп,Уп) = limd(y.xn,xn) =
\Yl
(3) Let К С X be a compact set whose translates by Γ cover X. In order to obtain a
contradiction, let us suppose that there is a sequence of elements y„ e Γ and a number
а > 0 such that \γ„\ φ \ym\ for all m φ n, and \yn\ —» aasn-> oo. According
to (2), we can choose points xn e X such that d(yn.xn, xn) = \yn\. Replacing yn by
a suitable conjugate if necessary, we may assume that xn e К for all n. Since К is
compact, it is contained in a bounded set, say B(x, r). But then, for all η sufficiently
large, y„ .B(x, r + а + 1) Π B(x, r) φ 0, which contradicts the properness of the action
(see 1.8.3(1)).
(^Letx' e X' and choose a point x= (χ',χ") e X such that x" e K. As the action
of Γ is proper, there exists ε > 0 such that {у e Γ | γ.(Β(χι,ε)χΚ)Γ)(Β(χ',ε)χΚ) φ
0} is finite (see 1.8.3(1)). If у' е Г/Л? is such that γ'.Β{χ!, ε) П ДУ, ε) / 0, then
there exists у = (у', у") in the coset y' such that y".x" e #. But this implies that

234 Chapter II.6 Isometries of CAT(O) Spaces
y.(5(x/, ε) χ Κ) Π (B(xf, ε) χ Κ) φ 0. Thus the number of such γ' is finite and the
action of Γ/Ν on X' is proper. D
6.11 Remark. Parts (2) and (3) of the preceding proposition obviously remain valid
under the weaker hypothesis that Γ is a subgroup of a group that acts properly and
cocompactly by isometries. In section 5 of Chapter ΙΙΙ.Γ we shall see that in the
context of CAT(O) spaces, many groups that cannot act properly and cocompactly
by isometries can nevertheless be embedded in groups that do.
The following result places severe restrictions on the way in which central
extensions can act by isometries on CAT(O) spaces. Some topological consequences
of this will be discussed in Chapter 7. We emphasize that in the following result we
assume nothing beyond what is stated about the group actions involved, in particular
the action of the group Γ is not required to be proper or semi-simple.
6.12 Theorem. Let X be a CAT(O) space and let Γ be a finitely generated group
acting by isometries on X. If Υ contains a central subgroup A = IIх that acts faithfully by
hyperbolic isometries (apart from the identity element), then there exists a subgroup
of finite index Я С Г which contains A as a direct factor.
Proof. Fix a e A, a non-trivial element. According to 6.8(5), the action of Γ leaves
Min(a) = Υ χ К invariant, and the restriction of each γ e Γ to Υ χ К is of the form
(y\ y")> where γ' is an isometry of Υ and γ" is a translation of №.. The map γ м> γ"
defines a homomorphism from Γ to a finitely generated group of translations of K.
Such a group of translations is isomorphic to Zm, for some m, so we have a surjective
homomorphism ψ : Γ —» Zm. The image under ψ of A is non-trivial, because ψ(α)
is non-trivial.
We compose ψ with the projection of Ът onto a suitable direct summand so as
to obtain a homomorphism φ : Γ —» Ζ such that φ(Α) is non-trivial. We then choose
a e A so that φ(α) generates φ(Α). Let Ho = φ~ι(φ(Α)) and note that #o has finite
index in Γ. The map ф(а) м> a splits ф\щ, giving #0 = ker$ χ (a) (sincewis
central) and A = Α' χ (α), where A' = Α Π кетф. By induction on m (the rank of A)
we may assume that A' is a direct factor of a subgroup of finite index H' с кегф.
LetH = H' χ (a). D
6.13 Remark. The first paragraph of the above proof establishes the following: if Γ
is any group of isometries of a CAT(O) space X such that every homomorphism
Γ —» Ж is trivial, then the centre of Г contains no hyperbolic elements.

Clifford Translations and the Euclidean de Rham Factor 235
Clifford Translations and the Euclidean de Rham Factor
A simply connected complete Riemannian manifold of non-positive curvature admits
a canonical splitting Μ = Ν χ Ε" such that N cannot be further decomposed as a
Riemannian product with a Euclidean factor; the factor E" is called the Euclidean
de Rham factor. In this section we construct a similar splitting for arbitrary CAT(O)
spaces.
6.14 Definition. An isometry у of a metric space X is called a Clifford translation
ifdY is a constant function, i.e., Min(y) = X.
The pre-Hilbert space Η constructed in the following theorem is the analogue
for CAT(O) spaces of the Euclidean de Rham factor in Riemannian geometry.
6.15 Theorem. LetX be a CAT(O) space.
(1) If γ is a non-trivial Clifford translation ofX, then X splits as a product X =
Υ χ Κ, andy(y,t) = (y,t+ |y|) for ally e Υ and all t e K.
(2) If X splits as a product Χ' χ Χ", every Clifford translation of X preserves
this splitting and is the product of a Clifford translation ofX' and a Clifford
translation ofX".
(3) The Clifford translations form an abelian subgroup of the group of isometries
ofX; let Η denote this subgroup.
(4) The group Η is naturally a pre-Hilbert space, where the norm of a Clifford
translation γ is equal to its translation length |y|.
(5) IfX is complete, then Η is a Hilbert space.
(6) If the space Η of Clifford translations is complete (in particular if it is finite
dimensional or X is complete), then X admits a splitting Υ χ Η such that, for
every у e Y, the subspace {y} χ Η is Η.у, the orbit of у under the group H.
fhirthermore, every isometry of X preserves this splitting.
Proof. Part (1) is a special case of 6.8(4). To prove (2), we consider the foliation of
X by the axes of a non-trivial Clifford translation y. If these axes project to points in
X' then у is of the form (id, γ"). Otherwise, the orthogonal projection of these axes
give a foliation of X' by geodesic lines that are parallel; if у(x?0, л$) = (ή, χ![) then
x'q and jtj lie on the same line. We obtain a Clifford translation y' of X' by translating
each of these parallel geodesic lines a distance d(x?0, x',) in the appropriate direction.
(The product decomposition theorem (2.14) ensures that this is an isometry.) In the
same way we obtain a Clifford translation y" of X" with y"(*o) = χ"(*Ί')· Now,
(/'> y") is a Clifford translation of X whose action on the y-axis through (x!0, xfj) is
the same as that of y, and therefore by (1) we have у = (у', у").
For (3) we consider two Clifford translations a and β of X. If α is non-trivial, then
X splits as a product Kxl, and a is the product of the identity on Υ and a translation
τα of K. By (2), β is the product of a Clifford translation β' of Υ and a translation

236 Chapter II.6 Isometries of CAT(O) Spaces
Τβ of К. Thus αβ = βα is the Clifford translation (β', τατβ) onX=Y xM.. (Notice
that if β' is non-trivial, then by applying (1) to the action of β' on Υ we get that X
splits as Υ' χ Ε2, where a and β project to the identity on Y' and translations of E2.)
(4) We first define the scalar multiplication giving the vector space structure on
the abelian group Η of Clifford translations. Each non-trivial Clifford translation γ
defines a splitting Υ χ R of X such that y(y, t) = (y, t+\y\). Given any λ e K, the
map (y,t) h+ (y, t + λ | γ |) is again a Clifford translation of X, which we denote λ · γ.
We define λ · γ to be the product of γ by the scalar λ, and claim that this defines a
vector space structure on H. The only non-trivial point to check is that λ · (αβ) is the
composition of λ · a and λ · β, for all α, β e Η and lei. But this follows easily
from the parenthetical remark at the end of the preceding paragraph.
With the above definition of scalar multiplication, it is clear that the map which
assigns to each γ e Η its translation length | γ | is a norm (the triangle inequality
being an immediate consequence of the triangle inequality in X). In order to complete
the proof of (4) we must show that this norm satisfies the parallelogram law. But the
parallelogram law only involves comparing two vectors, and so we may again use
the final sentence of the proof of (3) to reduce to the case X = E2, where the result
is clear.
(5) For any χ e X and a e H, the map а м> a.x is an isometry of Η onto H.x,
because d(a.x, β.χ) = d(x,a~l β.χ) = \a~~l β\. Thus, if an is a Cauchy sequence
in H, then an.x is a Cauchy sequence in X. If we assume that X is complete, then
this sequence must have a limit point, which we denote by a.x. The map χ \-> a.x
is an isometry of X, moreover it is a Clifford translation, because for all χ e X
we have d(x, a.x) = ]imd(x,an(x)) = ]im|an|. And by construction, |α~'θ!π| =
d(a.x, a„.x) —> 0, as η —■► oo. This proves (5).
(6) Suppose that Η is complete. Choose a base point xq e X. Let ρ denote the
projection of X onto the closed convex set H.xq, and let Υ Ъер"' (jco). We claim that the
map (у, а) м> a.y is an isometry from Υ χ Η onto X. This map is surjective because,
given χ e X, its projectionp(x) e H.xq is of the form a.xo; we have p(a~l.x) ~
a~~l .p(x) = xq, hence у := a"1 .x belongs to Υ and χ = a.y. Consider y\, y2 e Υ
and G!i,G!2 e Η and let α = а^'аг; we have to check that а(а\.у\,а2У2)2 ~
d(y\, У2)2 + |a|2. We can assume that a is not the identity, otherwise the assertion* is
obvious. Henceforth we work with the splitting У χ Μ of X associated to a, as in (1).
Without loss of generality, we may assume that xq corresponds to the point (x'o, 0);
thus Υ is contained in У = Υ' χ {0}, and у\,уг correspond to the points (y\, 0)
and (y2,0). Therefore, d{a\.yuсцуг)2 = d(yua.y2)2 = d((yu0), (y2, \a\))2 ~
d(yi,y2)2 + \a\2, as required.
Finally, let γ be an isometry of X. For each χ e X, H.x is the union of all axes of
Clifford translations passing through x. As γ carries an axis of a Clifford translation a
onto an axis of the Clifford translation γαγ~~ι, we have γ {H.x) = Η(γ.χ). Therefore,
by 1.5.3(4), γ preserves the splitting X = Υ χ Η. Ο
The next result indicates how Clifford translations may arise.

The Group of Isometries of a Compact Metric Space of Non-Positive Curvature 237
6.16 Lemma. Let X be a complete CAT(O) space in which every geodesic can be
extended to geodesic line. If a group Γ acts cocompactly by isometries on X and
a e Isom(X) commutes with T, then a is a Clifford translation.
Proof. The displacement function da is Γ-invariant, hence bounded because the
action of Γ is cocompact. The geodesic segment connecting any pair of distinct
points x,y e X can be extended to a geodesic line с : К —> X. The function
11-* d(c(t), a.c(t)) is convex and bounded, hence constant, so da(x) = da(y) for all
x, у e X. О
The Group of Isometries of a Compact Metric Space
of Non-Positive Curvature
Let У be a compact metric space. In 1.8.7 we showed that Isom(J0 equipped with the
metric
d(a, a') = supd(a.y, a'.y)
ysY
is a compact topological group. In this section we show that one can say considerably
more if Υ is non-positively curved.
If Υ is compact and non-positively curved, then by the Cartan-Hadamard theorem
(4.1), its universal covering is a proper CAT(O) space X. And we can think of Υ as
the quotient Γ\Χ of X, where Г is the fundamental group of Υ acting freely and
properly by isometries (as deck transformations) on X. In the following proof we
shall need the fact (proved in 1.8.6) that the group of isometries Isom(J0 of Υ is
naturally isomorphic to the quotient Ν(Γ)/Γ, where N(T) is the normalizer of Γ in
Isom(X).
Recall (5.7) that a complete geodesic space Υ is said to have the geodesic extension
property if every local isometry from a non-trivial compact interval of the real line
into Υ can be extended to a local isometry of the real line into Y. If Υ has this
property then so does any covering space of it. Examples of non-positively curved
spaces which enjoy this property include closed topological manifolds and metric
simplicial complexes with no free faces (5.10).
6.17 Theorem. Let Υ be a compact connected metric space of non-positive curvature,
the quotient of the CAT(O) space X by a group Γ acting properly and freely on X by
isometries. (Thus Γ = π\ Υ and Χ = Υ.) Suppose that Y has the geodesic extension
property. Then:
(1) The group Ьот(У), equipped with the metric described above is a compact
topological group with a finite number of connected components.
(2) The connected component of the identity in Isom(y) is isomorphic to a torus —
the quotient by a lattice Λ of a finite dimensional Euclidean vector space Ho,
which is a subspace of the Clifford translations ofX.

238 Chapter II.6 Isometries of CAT(O) Spaces
(3) In fact, Ho is 'isomorphic to the centralizer С(Г) of Г in Isom(X) and Л =
С(Г) П Г is the centre of Т.
(4) If the centre of Γ = π\Υ is trivial, then Isom(y) is a finite group. In general,
the centre of Γ is a free abelian group whose rank is smaller than or equal to
the maximal dimension of flat subspaces ofX.
Proof. We showed in (6.16) that the centralizer С(Г) of Г in Isom(X) consists entirely
of Clifford translations.
We wish to describe the component of the identity in Isom(y). Because Υ is
compact and the covering map ρ : X —> У is a local isometry, we can choose ε > 0
such that the restriction of ρ to every closed ball of radius ε in X is an isometry onto
its image. Let ν be an isometry of Υ whose distance to the identity (in the metric
defined above) is equal to α < ε. Then, there is a Clifford translation ν of X of
norm a which is a lifting of v: the map ν sends each point χ e X to the unique
point of the ball of radius ε and centre χ whose image under ρ is v(p(x)). The map ν
is clearly length-preserving, and hence is an isometry of X. Moreover it commutes
with every element of Γ, and hence is a Clifford translation. Indeed, for every χ eX,
d(x, v(x)) = d(p(x), v(p(x)) is equal to a.
The group of Clifford translations of X which centralize Γ form a Hubert sub-
space Ho of the Hilbert space Η of all Clifford translations of X (cf. 6.15). As X is
locally compact, Η is finite dimensional. The argument of the previous paragraph
shows that the natural homomorphism/?* from С(Г) = Но to Isom(F) induced by
ρ sends the ball of radius ε centred at 0 e Ho isometrically onto the ball of radius
ε centred at the identity in Isom(y). A first consequence of this observation is that
the identity has a connected open neighbourhood in Isom(y). In light of the fact that
Isom(y) is a compact topological group, this establishes (1). A second consequence
of this observation is that the homomorphism p* is a covering map from Ho onto
the connected component of the identity in Isom(J0; its kernel, С(Г) П Г, must be a
lattice in С(Г) = #0· This establishes parts (2) and (3) of the theorem. (4) follows
immediately from (3) and the fact that Ho is isometrically embedded in X (cf. 6.15).
D
6.18 Remarks
(1) In the preceding theorem, the hypothesis that Υ has the geodesic extension
property is essential. (Consider for example the case of a compact ball in E".)
(2) The preceding proof, like that of (1.8.6), is not valid for proper actions by
groups which have torsion. Let X be a CAT(O) space and suppose that Г с Isom(X)
acts properly and cocompactly. Let Ν(Γ) denote the normalizer of Γ in Isom(X). As
in (1.8.6), there is a natural homomorphism from МТ)/Г to the isometry group of the
compact orbit space Г\Х, but in general this map is neither injective nor surjective
(Exercise: why not?). Note however that the image of Ν(Γ)/Γ in Isom(r\X) is
closed, so with the induced topology N(T)/T becomes a compact topological group
(cf. 1.8.7).
For future reference we note an obvious consequence of (6.17).

A Splitting Theorem 239
6.19 Corollary. Let X be а С AT(0) space with the geodesic extension property and
suppose that Г с Isom(X) acts properly and cocompactly. If Г has a torsion-free
subgroup of finite index with trivial centre, then the centralizer of Г in Isom(X) is
trivial.
Lemma 6.16 can be viewed as a special case of the following lemma which will
be needed in the proof of the Splitting Theorem (6.21) as well various points in later
chapters.
6.20 Lemma. IfX is a complete CAT(O) space with the geodesic extension property,
and Г is a group which acts cocompactly on X by isometries, then X contains no
T-invariant, closed, convex subsets other than X and the empty set.
Proof. Let С be a non-empty Γ-invariant, closed, convex subset of X. Let π denote
the orthogonal projection of X onto C. Since С is Γ-invariant, it contains а Г orbit,
and since the action of Г is cocompact there is a constant К such that every point
of X is within a distance К of С This implies that in fact С itself is the whole of X,
for if there were a point ieX\C then one could extend the geodesic [π(χ), χ] to a
geodesic ray с : [0, oo) —» X such that d(c(t), C) = t for alii (because ί м> d(c(t), С)
is convex and d(c(t), C) = t for small f). D
A Splitting Theorem
Splitting theorems of the following type were proved in the framework of Rieman-
nian manifolds of non-positive curvature by D. Gromoll and J. Wolf [GW71] and
B. Lawson and S. Yau [LaY72] (see also Schroeder [Sch85]). A similar theorem
for CAT(O) spaces was proved by Claire Baribaud in her diplome work [Bar93]. A
splitting theorem of a slightly different nature will be proved in Chapter 9.
6.21 Theorem. Let X be a CAT(O) space with the geodesic extension property.
Suppose that Γ = Γ] χ Γ2 acts properly and cocompactly by isometries on X and
suppose that Υ satisfies one of the following conditions:
(1) Υ has finite centre and Υ \ is torsion free; or
(2) theabelianizationofY\ is finite.
Then, X splits as a product of metric spaces X\ xX2 andY preserves the splitting;
the action of Υ = Γ] χ Γ2 on X = Χι χ Χι is the product action and Y\X =
Γ,\Χ, χ Γ2\Χ2.
6.22 Corollary. Let Υ be a compact geodesic space of non-positive curvature that
has the geodesic extension property. Assume that the fundamental group of Υ splits
as a product Υ = Υι χ Γ2 and that Υ has trivial center. Then Υ splits as a product
Y\ χ Y% such that the fundamental group of Υ ι is Γ„ / = 1,2.

240 Chapter II.6 Isometries of CAT(O) Spaces
We shall deduce Theorem 6.21 from the following more technical result.
6.23 Proposition. Let X be a CAT(O) space with the geodesic extension property.
Suppose thatT = Y\ хТг acts properly and cocompactly onX. Ifthe action ofT\ on
the closed convex hull С of one of its orbits is cocompact, then X splits as a product
of metric spaces Χι χ Χχ and Γ preserves the splitting. Moreover, the subspaces of
the form Χι χ {χι} are precisely the closed convex hulls of the Y\-orbits
Proof. We shall present the proof of the proposition in several steps. Let Σ be the set
of closed, convex, non-empty, Γι-invariant subspaces of X, and let λί be the subset
of Σ consisting of those subspaces which are minimal with respect to inclusion. Note
that since a non-empty intersection of subspaces in Σ is again in Σ, the members of
λί are disjoint. Note also that each member of TV is the closed convex hull of some
r,-orbitC(ri.jc).
Claim 1. λί is non-empty.
Let С be as in the proposition and let К be a compact set such that Y\.K = C.
Every Γι-invariant closed, convex, non-empty subspace of С intersects K, so the
intersection of a decreasing sequence of such subspaces has a non-empty intersection
with K; such an intersection is again in Σ. Therefore we may apply Zorn's lemma to
deduce the existence of a minimal, closed, convex, Γι-invariant subset in С
Given С ι, C2 € λί, let p, : X —> C, denote the projection of X onto C, and let
d = d(C{, C2) := inf{d(xi, xj) \x\ eCi.^e C2}.
Claim 2. There is a unique isometry j of С] х [0, d] onto the convex hull of
C\ U Сг such thatj(x, 0) = χ andj(x, d) = рг(х).
The function dCj : χ м> d(x,pi(x)), which is convex and Γι-invariant, must be
constant on C2, because if there were points x, у e C% such that dct (x) < dct (y) then
{z e C2 I dc^iz) < dc](x)} would be a closed, convex, Γι-invariant proper subspace
of C2, contradicting the minimality of C2. Apply 2.12 (2).
Claim 3. For every χ e X there exists a unique Cx e Μ such that χ e Cx. In fact,
Cx = С(Г[.х), the closed convex hull of the Г1-orbit of x.
The uniqueness of Cx is obvious because distinct elements of Л^ are disjoint. And
if χ e Cx then Γ ι .x с Cx and hence C(T 1 .jc) = Cx, because Cx is minimal.
Let X' = [J{Ci I C, e λί}. To establish the existence of Cx it is enough to show
that X' is a closed, convex, Γ-invariant subspace of X (see 6.20).
We first prove that X' is Γ-invariant. Each С e Μ is Γι-invariant. Given С =
С(Г\.х) e λί and γι e Γ2, we have y2.C = C{y2Y\.x) = C(ri.^jc), a closed,
convex, Γι-invariant subspace of X. And if С" с С(Г[.у2х) were a smaller such
subspace then y2~'С" с С would contradict the minimality of C. Thus yi.C e ЛЛ
Claim 2 implies that X' is convex.
To prove that X' is closed, we consider a point χ e X and a sequence of
points jc„ e X' converging to x. Passing to a subsequence we may assume that
d(xn, x„+i) < 1/2". Let C„ be the element of Л^ containing x,u let pn : X —> C„
be the orthogonal projection, and let P„ be the restriction to C\ of the composition

A Splitting Theorem 241
PnPn-x ■ ■ .p2- Note that for any ζ e C\, the sequence Pn(z) is a Cauchy sequence
because d(Pn(z), Pn+k(z» < E"=,f~' d(Q, Ci+l) < Σ£ί~4*ί,*ί+ι) < V2"~]-
Since X is complete, the sequence of Γι-equivariant maps Pn : C\ —> X converges.
The limit Ρ : Ci -» X is a rrequivariant isometry of C\ onto its image, which is
therefore closed, convex and Γ]-invariant. The minimality of C\ implies that P(C\)
is also minimal, hence P(C\) e N.
It remains to show that χ e P(C\). Let zn e C\ be such that Pn(z,i) = x,i- Then,
d(P(zn), x) < d(P(zn), xn)+d(xn, x) = d(P(zn), Pn{zn))+d{x,u x) < l/2"~l+d(xn, x\
which tends to 0 as η —» oo. Thus χ lies in the closure of P(C\), hence in P(C\).
Claim 4. If C\, C2, C3 e TV, and if p,- : X —» C, denotes projection onto C,, then
Pi =PlP2 0nC3.
If each C, is a single point (the case where Γι is finite) then the claim is trivial. If
not, then each point χ e C3 lies on a non-trivial geodesic path in C3. By hypothesis
this can be extended to a geodesic line c(M) in X. It follows from Claim 2 that c(M)
cannot intersect any member of TV other than C3 and hence, by Claim 3, c(M) С С^.
The present claim then follows by application of (2.15) to the parallel geodesic lines
c(M),pic(M) andp2c(R).
Fix X\ e N. Let ρ : X —> X\ be orthogonal projection. It is Γι-equivariant. Let
Χι denote the metric space (Af, d), where d(C, C") = ίηί{ύ?(χ, χ') | χ e С, х' е С'}.
Given jc e X, let Сл е TV be as in Claim 3.
By restriction to X\ we get an action of Г] on ΧΊ. The action of Γι on Χι is trivial.
There is an isometric action of Γ2 on X2 by Y2-Cx = CnJC. According to Claims 2
and 4, χ ι-* ρ(χ2 -x) defines an action of Γ2 on X\ commuting with the action of Γ ι,
and by Claim 2 this action is by isometries. Thus we obtain an action of Γ on X\ χ Хг
by isometries.
Claim 5. The map X —» X\ χ X2 given byj:n· (p(x), Cx) is a Γ-equivariant
isometry.
The map is Γ-equivariant by construction.
Given x, x1 e X, let p' be projection onto Cy. By applying Claim 2 twice and
then applying Claim 4, we see that X —> X| χ X2 is an isometry:
rf(jc, jc')2 = rf(i, p'(^))2 + d{Cx, Cy)2
= i?(pW,pp'(x'))2+i?(C„Cy)2
= ί?(ρ(χ),ρ(χ'))2+ί?(^,^)2.
It is clear from the construction of X| χ Χ2 that the action of Г on it has the
properties stated in the proposition. D
We now turn to the proof of Theorem 6.21.
Proof of Theorem 6.21(1). We first prove that hypothesis (6.21(1)) is sufficient to
guarantee that the hypothesis of Proposition 6.23 holds. This follows from the next
lemma, which can be found in [GW71] and [ChEb75].

242 Chapter II.6 Isometries of CAT(0) Spaces
6.24 Lemma. Let X be a proper С AT(0) space. Let Г = ΓΊ xY2be a group acting
properly on X by isometries, and suppose that the centre of Υ2 is finite. Let С с X
be a closed, convex, Υ {-invariant subspace and suppose that there exists a compact
subset К С X such that С С Υ.Κ. Then there is a compact subset К' С С such that
C = Ti.K'.
Proof. Assuming that there is no compact subspace K' such that C(Y\.K) = Г] .К',
we shall construct below a sequence of distinct elements β„ e Γ2 such that, for any
β e Γ2, the family of displacement functions ^я да-1 is uniformly bounded on K. As
Γ acts properly on X, this implies that the subset {βηββ„ '}« С Г2 is finite. Hence we
may pass to a subsequence and assume that all of the βηββ„ ' are equal. Applying
this argument as β ranges over a finite generating set for Γ2 (cf. 1.8.10), we obtain a
sequence of distinct elements /},· such that β„ιββ~χ = βηββ^] for every β e Γ2 and
all n, m e N. But then we would have an infinite number of elements β„β^1 in the
centre of Γ2, which is supposed to be finite.
To construct the sequence (/}„), first note that if the desired K' does not exist
then one can find a sequence of points xn in С such that d(xn, Yi.K) —> 00. Choose
βη e Y2 so that βη(χη) e Γ,.ΛΓ. Then ά(β~\Κ),Κ) > ά(β~\Κ), C{YX.K)) >
d(x„, C(Yi.K)) — diam(if). Hence, after passing to a subsequence, we may assume
that all of the elements β„ are distinct.
It remains to check that, for any β e Γ2, the family of displacement functions
^β,,ββ-' is uniformly bounded on K. Observe that the displacement function άβ (which
is convex) is constant on the orbits of Γι because β commutes with Γι; if r is the
maximum of dp on K, then the set of points where άβ is not bigger than r is a
closed, convex, Γι-invariant subspace containing Γι .Κ and hence C. Therefore άβ is
bounded by r on C{Y\.K). As άβηββ-\{βη.χη) = άβ(χη) < r, for any χ € К wehave
άβηββ-\{χ) <r + 2 diam(iQ.
Indeed if a„ e Γ] is such that β„α„.χ„ e K, then
ά(β„ββ-ι.χ,χ)
<ά(β„ββ~].χ, β„ββ~].(β„α„.χ„)) + ά(βηβαη.χη, αηβ„.χ„) + ά(αηβ„.χ„, χ)
= ά(β„ββ~].χ, β„ββ~ι.(β„αη.χ„)) + ά(α„β„β.χ„, α„βη.χ„)+ ά(α„βη.χ„, χ)
< diam(iQ + r + diam(iQ.
D
Applying 6.23, we get a splitting X] xX2oiX preserved by the action of Г, the
action of Γι being trivial on X2. It remains to show that the action of Γ2 on the first
factorX] is also trivial. To see this we apply Corollary 6.18. As Γι is assumed to be
torsion free and as it acts properly on Xb it acts freely on X[. The centre of Γι is
assumed to be finite and torsion free, hence trivial, so by (6.18) the centralizer of Γι
in Isom(Xi) is trivial. As each element of Γ2 commutes with Гь it follows that the
action of Г2 on X[ is trivial. D

A Splitting Theorem 243
Proof of Theorem 6.21(2). We wish to apply (6.23). In order to do so, we shall prove
by induction on the maximum dimension of flats in X that Γ^ acts cocompactly on
the closed convex hull of each Г\ -orbit in X. If the centre of Г is finite then we can
apply (6.24). As we remarked in (6.13), because Γι has finite abelianization it has
no elements of infinite order in its centre, so if the centre of Γ (which is a finitely
generated abelian group) is infinite then there exists an element of infinite order у
in the centre of Гг. Such а у is a Clifford translation (6.16), so we get a Γ-invariant
splitting X = Υ χ К, where у acts trivially on Υ and Γ acts by translations on the
second factor K. Because the abelianization of Γ! is finite, its action on the second
factor is trivial; in particular, every Γι-orbit in X is contained in some slice Υ χ {ή.
Applying our inductive hypothesis to the action of Γ/( γ ) = Γι χ Γ2/( у ) on Y we
deduce that the action of Γ ι on the closed convex hull of each of its orbits in Υ (and
hence X) is cocompact.
In order to complete the proof we must show that the action of Γ2 on X[ arising
from (6.23) is trivial. For this it suffices to show that for every γι e Γ2 there exists a
Г!-orbit Y\.x in X with closed convex hull C{Y\.x) such that, if ρ : X -> C(T\.x) is
the orthogonal projection then ργι restricts to the identity on C(T ι .x). If y2 is elliptic
then its action on the closed convex hull of any orbit in Min(y2) is trivial, so we
choose χ e Min(y2). If γι is hyperbolic, then we consider Мш(уг) with its splitting
У χ Μ as in (6.8(4)). (Recall that the action of y2 on Y' is trivial.) As Γι commutes
with γι, it preserves Мш(уг) and its splitting. As the abelianization of Γι is finite,
its action on the second factor is trivial, so each Γι-orbit in Min(y2) is contained in a
slice Υ' χ {ή. lip is the projection onto such an orbit, then ργι is the identity, because
the action of y2 on Y' is trivial. D
6.25 Exercises
(1) Let Γ be the fundamental group of a closed surface of positive genus. Construct
a proper cocompact action of Γ χ Ζ on И2 χ Μ with the property that Γ does not act
cocompactly on the convex hull of any of its orbits. (Hint: Observe that there exist
non-trivial homomorphisms Γ —» Ζ via which Γ acts on K. Combine such an action
with a cocompact action of Γ on H2 and consider the diagonal action on H2 х К.
Extend this to a cocompact action of Γ χ Ζ.)
(2) Show that the hypotheses of (6.21(1)) can be weakened to: the centre of Γ is
finite and Γ [ modulo its centre has a torsion-free subgroup of finite index.
(3) Let X be a proper CAT(O) space with the geodesic extension property. Let
Γ be a group acting by isometries on X such that the closure in X of the Γ-orbit
of a point of X contains the whole of dX (notation of Chapter 8). Suppose that no
point of dX is fixed by all the elements of Γ. If Γ splits as a product Γι χ Γ2, prove
there is a splitting X = X\ χ X2 such that if у = (γ\, y2) and χ = {χχ,χι), then
γ.(χι, χι) = (yi, χι, γι, χι), (cf. Schroeder [Sch85].)

Chapter II.7 The Flat Torus Theorem
This is the first of a number of chapters in which we study the subgroup structure
of groups Γ that act properly by semi-simple isometries on С АТ(0) spaces X. In this
chapter our focus will be on the abelian subgroups of Г. The Flat Torus Theorem
(7.1) shows that the structure of such subgroups is faithfully reflected in the geometry
of the flat subspaces in X. One important consequence of this fact is the Solvable
Subgroup Theorem (7.8): if Г acts properly and cocompactly by isometries on a
CAT(O) space, then every solvable subgroup of Г is finitely generated and virtually
abelian. In addition to algebraic results of this kind, we shall also present some
topological consequences of the Flat Torus Theorem.
Both the Flat Torus Theorem and the Solvable Subgroup Theorem were
discovered in the setting of smooth manifolds by Gromoll and Wolf [GW71] and,
independently, by Lawson and Yau [LaY72]. Our proofs are quite different to the
original ones.
Throughout this chapter we shall work primarily with proper actions by semi-
simple isometries rather than cocompact actions. Working in this generality has a
number of advantages: besides the obvious benefit of affording more general results,
one can exploit the fact that when restricting an action to a subgroup one again obtains
an action of the same type. (In particular this facilitates induction arguments.)
The Flat Torus Theorem
Recall that, given a group of isometries Г, we write Μϊη(Γ) for the set of points
which are moved the minimal distance \γ| by every γ e Γ.
7.1 Flat Torus Theorem. Let Abe a free abelian group of rank η acting properly
by semi-simple isometries on a CAT(O) space X. Then:
(1) Min(A) = П<*<=л Min(ce) is non-empty and splits as a product Υ χ Ε".
(2) Every element a e A leaves Min(A) invariant and respects the product
decomposition; a acts as the identity on the first factor Υ and as a translation on the
second factor E".
(3) The quotient of each η-flat {y} χ Ε" by the action of A is an n-torus.

The Flat Torus Theorem 245
(4) If an isometry ofX normalizes A, then it leaves Min(A) invariant and preserves
the product decomposition.
(5) If a subgroup Г с Isom(X) normalizes A, then a subgroup of finite index in
Г centralizes A. Moreover, if Υ is finitely generated, then Γ has a subgroup of
finite index that contains A as a direct factor.
Proof. We shall prove parts (1), (2) and (3) by induction on the rank of A. As the
action of A is proper and by semi-simple isometries, each non-trivial element of A
is a hyperbolic isometry. Suppose that A = Z" and choose generators a\,... ,an.
We have seen (6.8) that Min(ai) splits as Ζ χ Ε1 where αϊ acts trivially on the
first factor and acts as a translation of amplitude |αι | on the second factor. Every
α e A commutes with αϊ and therefore preserves the subspace Ζ χ Ε1 with its
decomposition, acting by translation on the factor E1 (see (6.8)).
We claim that the subgroup N С A formed by the elements a e A which act
trivially on the factor Ζ is simply the subgroup of A generated by a\. To see this,
note that N, which is free abelian, acts properly on each line {y} χ Ε1 by translation,
hence it is cyclic; and since αϊ e N is primitive in A, it must generate N.
The free abelian group A0 = A/N is of rank η — 1. Its action on Ζ is proper
(6.10(4)) and by semi-simple isometries (6.9). As Ζ is a convex subspace of X, it
is а СAT(0) space, so by induction Min(A0) С Z splits as У χ Ε"~~', where A0 acts
trivially on Υ and acts by translations on E""1 with quotient an (n — l)-torus. Thus
Min(A) = Υ χ Ε""1 χ Ε1 = Υ χ Ε", and (1), (2) and (3) hold.
If an isometry γ normalizes A, then it obviously preserves Min(A); we claim that
it also preserves its product decomposition. Indeed γ maps Α-orbits onto A-orbits,
and therefore maps the convex hull of each Α-orbit onto the convex hull of an A-orbit;
and it follows from (3) that the convex hull of the Α-orbits of points of Min(A) are
the и-flats {y} xE",ye Y. This proves (4).
It also follows from (3) that for any number r there are only finitely many elements
a e A with translation length \a\ = r. But |yay~'| = |a| for all γ e Isom(X), so
if Г С Isom(X) normalizes A, then the image of the homomorphism Г —» Aut(A)
given by conjugation must be finite (because the number of possible images for each
basis element a,· e A is finite). The kernel of this homomorphism is a subgroup of
finite index in Г and its elements centralize A. If Г is finitely generated then (6.12)
tells us that there exists a further subgroup of finite index which contains A as a direct
factor. This proves (5). D
We generalize the Flat Torus Theorem to the case of virtually abelian groups. (A
group is said to virtually satisfy a property if it has a subgroup of finite index that
satisfies that property.)
7.2 Corollary. Let Τ be a finitely generated group which acts properly by semi-
simple isometries on a complete С AT(0) space X. Suppose that Γ contains a subgroup
of finite index that is free abelian of rank n. Then:
(1) X cqtttains a Y-invariant closed convex subspace isometric to a product Υ χ Ε".

246 Chapter II.7 The Flat Torus Theorem
(2) The action of Τ preserves the product structure on Υ χ Ε", acting as the identity
on the first factor, and acting cocompactly on the second.
(3) Any isometry ofX which normalizes Γ preserves Υ χ Ε" and its splitting.
Proof. Let A0 = Z" be a subgroup of finite index in Γ. Because Γ is finitely generated,
it contains only finitely many subgroups of index |Γ/Α0|. Let A be the intersection
of all of these subgroups. Note that A is a characteristic subgroup of finite index in
Γ, that is, φ(Α) = A for all φ e Aut(T). In particular, if a e Isom(X) normalizes Γ,
then it also normalizes A. As A is of finite index in A0, it is also isomorphic to Z".
According to (7.1), Min(A) splits as Ζ χ Ε", where Ζ is a closed convex subspace
of X; A acts trivially on the first factor, and cocompactly by translations on the
second factor. Because А С Г is normal, Γ acts by isometries of Min(A), preserving
the splitting (7.1(4)). Thus we obtain an induced action of the finite group Γ /A on
the complete CAT(O) space Z. According to (2.8), the fixed point set for this action
is a non-empty, closed, convex subset of Z; call this subset Y.
By construction Υ χ Ε" is Γ-invariant and the action of Γ on the first factor is
trivial. Since the action of A on each flat {y} χ Ε" is cocompact, the action of Γ on
each {y} χ Ε" is also cocompact.
To prove (3), one first notes that Υ χ Ε" is preserved by any isometry of X that
normalizes Γ, because (as we observed above) the normalizer of Γ in Isom(X) is
contained in the normalizer of A. Assertion (3) then follows, as in (7.1), from the
fact that the flats {y} χ Ε" are precisely those subsets of Υ χ Ε" that arise as convex
hulls of Γ-orbits. D
7.3 Remarks
(1) The invariant subspace Υ χ Ε" constructed in the preceding corollary contains
Μϊη(Γ), but in general it is not equal to Μίη(Γ). For example, consider the standard
action of the Klein bottle group Γ = (α, β | α~ιβ~ια = β) on the Euclidean plane:
the action of α is a translation of norm 1 followed by a reflection about an axis of
this translation; β acts as a Clifford translation whose axis is orthogonal to that of a.
In this case, the invariant subspace yielded by Corollary 7.2 is the whole of the
plane, while Μϊη(Γ) is empty.
(2) (Bieberbach Theorem) A finitely generated group Γ acts properly and
cocompactly by isometries on E" if and only if Γ contains a subgroup A = Z" of finite
index. In the light of the preceding corollary and (6.5), in order to prove the 'if
direction in this statement it suffices to get Γ to act properly by isometries on Ed for
some d. To produce such an action, one fixes a proper action of A by translations on
E" and considers the quotient X of Γ χ Ε" by the relation [(γα, χ) ~ (у, а.х) for all
x e E" and a e A]. Let m = \Y/A\. There is a natural identification of Em" with the
set of sections of the projection X —» Г/А, and the natural action of Г on the set of
sections gives a proper action by isometries on E™.
See section 4.2 of [Thu97] for an elegant account of the discrete isometry groups
ofE".

Cocompact Actions and the Solvable Subgroup Theorem 247
Cocompact Actions and the Solvable Subgroup Theorem
The main goal of this paragraph is to prove that if a group Γ acts properly and
cocompactly by isometries on а С AT(0) space then all of its solvable subgroups are
finitely generated and virtually abelian. First we show that abelian subgroups of Г
are finitely generated and satisfy the ascending chain condition. For this we need the
following geometric observation. Recall that a metric space X is said to be cocompact
if there exists a compact subset К С X whose translates by Isom(X) cover X.
7.4 Lemma. If a proper CAT(O) space X is cocompact, then there is a bound on the
dimension of isometrically embedded flat subspaces Em ^-» Χ.
Proof. We fix je0 e X and choose r sufficiently large to ensure that the translates of
the ball B(x0, r) by Isom(X) cover X. As B(x0, 2r) is compact, it can be covered by
a finite number N of balls B; of radius r/2. If there is a flat subspace of dimension
ρ in X, then we can translate it by the action of Isom(X) so as to obtain a convex
subspace Dp С B(xq, 2r) isometric to a Euclidean ball of dimension ρ and radius r.
An orthonormal frame at the centre of Dp gives 2p points on the boundary of Dp,
each pair of which are a distance at least -J2r apart. Each of these points is contained
in a different ball 5, of our chosen covering of B(xq, 2f). Therefore 2p < N. D
If A is an abelian group, then its rank гк^А is, by definition, the dimension of
the Q-vector space A ® Q. In other words, rkijA is the greatest integer η such that A
contains a subgroup isomorphic to Z".
7.5 Theorem (Ascending Chain Condition). Let H\ cff2c·.· be an ascending
chain of virtually abelian subgroups in a group Γ. If Υ acts properly and cocompactly
by isometries on a CAT(O) space, then H„ = Hn+\for sufficiently large n.
7.6 Corollary. If a group Γ acts properly and cocompactly by isometries on а С AT(0)
space, then every abelian subgroup of Τ is finitely generated.
Proof of the corollary. If А С Г is not finitely generated then there exists a sequence
of elements ai,a2,... such that, for all η > 0, the subgroup Hn generated by
{αϊ,..., an} does not contain an+\. Thus we obtain a strictly ascending chain of
subgroups H\ С #2 С · · ·. Π
Proof of the theorem. The proof of the corollary shows that if there were a strictly
ascending chain of virtually abelian subgroups in Γ then there would be a strictly
ascending chain of finitely generated virtually abelian subgroups. Thus we may
assume that the #,· are finitely generated.
Let X be а С AT(0) space on which Г acts properly and cocompactly by isometries.
The existence of a proper cocompact group action implies that X is proper (1.8.4)
and that each element of Г acts on X as a semi-simple isometry (6.10). Lemma 7.4
gives a bound on the rank of flat subspaces in X, so it follows from the Flat Torus

248 Chapter II.7 The Flat Torus Theorem
Theorem (7.1) that there is a bound on the rank of the abelian subgroups in Γ. Thus,
without loss of generality, we may assume that there is an integer η such that each
Hi contains a finite-index subgroup А,- с #, isomorphic to Z".
If η = 0, the groups #,· are finite, and by (2.8) there is a bound on the orders of
finite subgroups in Γ.
Consider the case η > 0. We claim that A\ has finite index in each #,. To see
this, consider the action of Αχ on #, by left translation and the induced action of A\
on the finite set #,/A,. The stabilizers,- с A] of the coset A, is contained in A,. Since
Bi has finite index in A\, it has rank n, therefore it has finite index in A/, and hence
inH;.
To complete the proof we must show that the index of Ai in H, is bounded
independent of i. By the Flat Torus Theorem (7.1), the convex subspace Min(Ai) splits
isometrically as Υ χ Ε", and Αι acts trivially on Υ and cocompactly by translations
on E". According to (7.2), for each i there is a convex subspace Y; of Υ such that #,
preserves F,· χ Ε" with its product structure and acts trivially on the factor Y,.
As the action of Γ on X is proper and cocompact, for each R > 0, there is an
integer Nr such that, for every χ e X, the set (у еГ| γ.χ e B(x, К)} has cardinality
less than Ng. We can choose R big enough so that in the action of A\ on the second
factor of Υ χ Ε", the orbit of any ball of radius R covers E". Then, given h e #, and
χ e Yi χ {0}, we can find an element a e A\ such that d(x, ah(x)) < R. It follows
that the index of Ai in each #, is bounded by NR, which is independent of i. D
7.7 Remarks
(1) If X is a complete simply-connected Riemannian manifold of non-positive
curvature, then every abelian subgroup of Isom(X) either contains an element of
infinite order or else it is finite. Indeed, one can easily show that the set of points
fixed by any group of isometries of X is a complete geodesically convex submanifold,
and this allows one to argue by induction on the dimension of the fixed point sets of
the elements of Γ (see [ChEb75] and [BaGS85]).
A similar induction can be used to show that any abelian subgroup of Isom(X)
which acts properly by semi-simple isometries is finitely generated. The analogous
statements are not true for proper CAT(O) spaces in general (see (7.11) and (7.15)).
(2) In the preceding theorem one can replace the hypothesis that the action of Γ
on X is cocompact by the following assumptions: (i) the action of Γ is proper and
by semi-simple isometries; (ii) there is a bound on the dimension of flat subspaces
in X; (iii) the set of translation numbers {\γ\ \ γ e Γ} is discrete at zero; and (iv)
there is a bound on the order of finite subgroups in Γ.
The examples given in (7.11) and (7.15) show that each of these conditions is
necessary. Torsion-free groups acting by cellular isometries onMK -complexes К with
Shapes(AT) finite satisfy all of these conditions.
Recall that the и-th derived subgroup of a group G is defined recursively by
G'0) = Ga.ndG(n+l) = [G^, G(n)], where the square brackets denote the commutator
subgroup. G is said to be solvable of derived length и if и is the least integer such
thatG*'0 = {1}.

Cocompact Actions and the Solvable Subgroup Theorem 249
7.8 Solvable Subgroup Theorem. If the group Γ acts properly and cocompactly
by isometries on a CAT(O) space X, then every virtually solvable subgroup S с Г is
finitely generated and contains an abelian subgroup of finite index. (And the action
ofSonX is as described in Corollary 7.2.)
Proof. Any countable group can be written as an ascending union of its finitely
generated subgroups, so in the light of the preceding result it is enough to consider
the case where S is finitely generated and solvable. Arguing by induction on the
derived length of S, we may assume that the commutator subgroup 5(1) = [S, S] is
finitely generated and virtually abelian. As in the proof of (7.2), we choose a free
abelian subgroup A that has finite index in 5(1) and is characteristic in 5(1), hence
normal in S.
By applying (7.1(5)) we obtain a subgroup So of finite index in S that contains
A as a direct factor. The commutator subgroup of So has trivial intersection with A,
and hence injects into the finite group S^/A. The following lemma completes the
proof. G
7.9 Lemma. If a finitely generated group Γ has a finite commutator subgroup, then
Γ has an abelian subgroup of finite index.
Proof. The action of Γ by conjugation on [Γ, Γ] gives a homomorphism from Γ to
the finite group AutflT, Γ]). Let Γι С Г denote the kernel of this map and notice
that g"h = hg"[g, h\n for all g,heTx and aline Ζ (because [Γ, Γ] η Γι is central).
Setting η equal to the order of [Γ, Γ], we see that g" is central in Π for every geTi.
But this implies that the finitely generated abelian group Γι/Ζ(Γι) is torsion, hence
finite. Thus Γι (and hence Γ) is virtually abelian. G
7.10 Remarks
(1) The hypotheses of the Solvable Subgroup Theorem can be relaxed as
indicated in (7.7(2)). In particular, if Γ acts freely by semi-simple isometries on an
MK -polyhedral complex К with Shapes(if) finite, then every solvable subgroup of Г
is finitely generated and virtually abelian.
(2) Let Я be a group that acts properly and cocompactly by isometries on а С AT(0)
space and let Г be a non-uniform, irreducible, lattice in a semi-simple Lie group of real
rank at least two that has no compact factors. Such lattices have solvable subgroups
that are not virtually abelian (see [Mar90]), so by the Solvable Subgroup Theorem, the
kernel of any homomorphism φ : Γ -» Η must be infinite. The Kazhdan-Margulis
Finiteness Theorem ([Zim84] 8.1.2) states that every normal subgroup of Γ is either
finite or of finite index. Thus the image of φ must be finite.

250 Chapter II.7 The Flat Torus Theorem
Proper Actions That Are Not Cocompact
Our first goal in this section is to examine how (7.5) can fail when the action is
not cocompact. We begin with a construction which shows that if G is a countable
abelian group all of whose elements have finite order, then G admits a proper action
on a proper CAT(O) space (a metric simplicial tree in fact). We shall then describe
how this construction can be modified to yield proper actions by infinitely generated
abelian groups such as Q. In each case the action has the property that Min(G) is
empty.
7.11 Example: Infinite Torsion Groups. Let G be a group which is the union of
a strictly increasing sequence of finite subgroups {1} = Go С G\ С G% С
Note that G can be taken to be any countably infinite abelian group that has no
elements of infinite order. (The reader may find it helpful to keep in mind the example
Gn = (Z/2Z)".)
Let X be the quotient of G χ [0, oo) by the equivalence relation which identifies
(g, t) with (g\ t') if g~lg' e G, and t = t' > i. Let [g, t] denote the equivalence
class of (g, t). We endow X with the unique length metric such that each of the maps
t м> [g, t] from [0, oo) into X is an isometry. With this metric, X becomes a metric
simplicial tree whose vertices are the points [g, i], where g e G, i e Z.
The action of G on G χ [0, oo) by left translation on the first factor and the
identity on the second factor induces an action of G by isometries on X. This action
is proper. Indeed, if we denote by Ig{i) the interval which is the union of points [g, t]
with i - 3/4 < t < i + 3/4, then g'(Ig(i)) П Ig(i) φ 0 if and only if g' belongs to the
finite subgroup G,·. Each element g e G, is an elliptic isometry, fixing in particular
those points [1, t] with t > i. Finally, we note that the map [g, t] м> t induces an
isometry from the quotient space G\X onto [0, oo).
7.12 Exercises
(1) Let X be a metric simplicial tree with all edges of length 1. Suppose that
each vertex of X has valence at most m. Let Tm be the regular tree in which each
vertex has valence m and each edge has length 1. Prove that there exists an injective
homomorphism φ : Isom(X) —> \som{Tn) and a 0-equivariant isometric embedding
(2) Deduce that there exist proper CAT(0) spaces Υ such that Isom(7) contains
subgroups that act properly and cocompactly on Υ and also contains abelian
subgroups that act properly on Υ by semi-simple isometries but are not finitely generated.
(Hint: The free group of rank m acts freely and cocompactly by isometries on Ггт.
The construction of (7.11) with G„ = (Z/rZ)" produces a tree of bounded valence.)
We shall need the following (easy) fact in the course of the next example.
(3) Consider two actions Φ : G —> Isom(X) and Φ : G —> Isom(y) of an abstract
group G on metric spaces X and Y. Suppose that the induced action of G/ ker Φ on
X is proper and the restricted action Ф|кегФ is proper. Show that the diagonal action
g н> (Ф(#), Ф(#)) of G on Χ χ У is proper.

Proper Actions That Are Not Cocompact 251
7.13 Example: Infinitely Generated Abelian Groups. We describe a proper action
of Q, the additive group of rational numbers, by semi-simple isometries on a proper
2-dimensional CAT(O) space (the product of a simplicial tree and a line in fact).
Q acts by translations on the line E1 in the obvious way (the action is not proper
of course). Q/Z is a countable abelian group with no elements of infinite order,
so by the preceding example it acts properly on a simplicial tree X. According
to the preceding exercise, the diagonal action of Q on Χ χ Ε1 (via the obvious
homomorphism Q —> Q/Z x Q) is a proper action. This action is by semi-simple
isometries. Note thatX χ Ε1 is proper.
In exactly the same way, we can combine the natural action of Q" by translations
on E" with a proper action of the torsion group Q"/Z" on a tree to obtain a proper
action of Q" by semi-simple isometries on a proper С AT(0) space of dimension η + 1
(the product of E" and a tree in fact). The following lemma shows that the dimension
of this example is optimal.
7.14 Lemma. Suppose that the group Γ acts properly by semi-simple isometries on
a proper CAT(O) space X and suppose that Γ contains a normal subgroup A = Z".
IfX does not contain an isometrically embedded copy <?/[0, oo) χ Ε", then A has
finite index in Γ.
Proof. By the Flat Torus Theorem, Min(A) splits as Υ χ Ε" and Γ preserves Min(A)
and its splitting. Let N С Г be the subgroup consisting of elements which act trivially
on Y. Note that N contains A. Since the action of N on each of the flats {y} χ Ε" is
proper and the action of A is cocompact, A must have finite index in N.
Υ (with the metric induced from X) is a proper CAT(O) space, so it is either
bounded or else it contains a geodesic ray; we are assuming that X does not contain
a copy of [0, oo) χ E", so Υ must be bounded. Therefore, since the action of Γ/Ν on
Υ is proper (6.10(4)), Γ/Ν and hence Γ /A must be finite. D
The following exercise gives a more explicit description of a proper action of
Q that can be constructed by the above method, and describes the quotient space
(which is a piecewise Euclidean 2-complex of finite area).
7.15 Exercises
(1) A Non-Positively Curved 2-complex К with π\Κ = Q
The 1-skeleton of К consists of a half-line [0, oo) with vertices vn at the integer
points, together with countably many oriented 1-cells {ε„}η(=Ν, where εη has length
\/n\ and has both of its endpoints attached to v„. For every non-negative integer η we
attach a 2-cell en to K, where e„ is metrized as a rectangle with sides of length \/n\
and 1. We orient the boundary of en and describe the attaching map reading around
the boundary in the positive direction: the side of length \/n\ traces across ε„ in the
direction of its orientation, beginning at vn, then a side of length 1 runs along [0, oo),
joining vn to vn+i, then the side of length \/n\ traces (n+ 1) times around επ+1 in the
direction opposite to its orientation, and finally the remaining side of length 1 runs
back along [0, oo) to vn.

252 Chapter II.7 The Flat Torus Theorem
Note that the 2-cell en has area \/n\, and hence К has total area e = 2 · 71
(a) Prove that n\K = Q.
(b) Prove that when endowed with the induced length metric, the universal cover
of К is a proper CAT(O) space and the action of π\Κ — Q on it by deck
transformations is a proper action by semi-simple isometries.
(c) Prove that the action of Q is precisely that which one obtains by applying the
construction of (7.13) with Q/Z expressed as the union of the finite cyclic
groups Gn = (xn | xnn' = 1), where Gn ^ Gn+i by xn м> *£+}.
(2) In contrast, prove that if a group G acts freely on a non-positively curved
MK-complex К by cellular isometries, and if Shapes(/sT) is finite, then every abelian
subgroup of G is finitely generated. (This is a matter of verifying the remarks that
we made in 7.7(2). You will need to use 6.6(2).)
Polycyclic Groups
One would like to weaken the hypotheses of the Solvable Subgroup Theorem so
as to allow proper actions which are not cocompact. The preceding examples show
that in order to do so one must contend with the possible presence of infinitely
generated abelian groups. On the other hand, in some situations (e.g., when one is
considering actions on Riemannian manifolds) one knows that the groups at hand
do not contain infinitely generated abelian subgroups. If the abelian subgroups of a
solvable group are finitely generated (as the case with finitely generated nilpotent
groups for example) then the group is polycyclic (see [Seg85]). Polycyclic groups
were defined following (1.8.39).
7.16 Theorem. A polycyclic group Г acts properly by semi-simple isometries on a
complete CAT(O) space if and only if Υ is virtually abelian.
Proof. For the "if" assertion, see (7.3(2)). To prove the "only if" assertion we argue
by induction on the Hirsch length of Γ. This reduces us to consideration of the
case Γ = Η ΧφΖ with Η finitely generated and virtually abelian. As in the proof
of (7.2), we can choose a free abelian subgroup А С Η that has finite index and
is characteristic in Η (hence normal in Γ). According to 7.1(5), the abelian group
A x ker φ has finite index in Γ. G
The preceding proof relied on an appeal to 7.1(5) which is underpinned by the
following observation: the translation numbers for the elements of a free abelian
group A acting properly by semi-simple isometries on a CAT(O) space X can be seen
by looking at the action of A on a flat subspace Ε" ^-» X which has quotient an
и-torus. We close this section with a lemma and some exercises that hint at the wider
utility of this observation.

Proper Actions That Are Not Cocompact 253
7.17 Lemma. Let X be a complete С AT(0) space, and let Γ = Z" be a group which
acts properly on X by semi-simple isometries.
(1) The translation length function γ м> \γ\ is the restriction to Γ ofa norm on
Γ <g> M. = R", and this norm satisfies the parallelogram law (i.e., arises from a
scalar product).
(2) If S с Г is a subset with the property that there does not exist any scalar
product on Г <g> К = К" such that (s \ s) = (У | s')for all s, s' e S, then S does
not lie in a single conjugacy class of the full isometry group Isom(X).
(3) Let G be an abstract group with a free abelian subgroup G = Z". IfS с G
is a subset with the property that there does not exist any scalar product on
G <g> К = R" such that (s \ s) = (s' | s1) for all s, s1 e S, and if S lies in a
single conjugacy class of G, then there does not exist a proper action of G by
semi-simple isometries on any complete CAT(O) space.
Proof. According to the Flat Torus Theorem (7.1), Г acts by Clifford translations on
the complete CAT(O) space Μϊη(Γ) — Υ xE". The map of Γ into #Min(n> the group
of Clifford transformations of Μΐη(Γ), extends in the obvious (linear) way to give
an injection Γ ® Κ —> #Μίπ(Γ)· The norm а ы> |α| on #мт(Г) makes it a pre-Hilbert
space (6.15). This proves (1).
Part (2) follows immediately from (1) and the fact that translation length is
preserved under conjugation. (3) follows immediately from (2). D
It is easy to construct sets S as in part (3) of the preceding lemma. Any infinite
subset of Z" has the appropriate property, as do many finite sets, for example any set
which meets a coset of a cyclic subgroup in more than two points.
7.18 Exercises
(1) Let X be a complete CAT(O) space. Prove that if an abelian subgroup Г С
Isom(X) acts properly by semi-simple isometries, then the intersection of Г with
each conjugacy class of Isom(X) is finite.
(2) Let F be the free group on {a, b, c] and let φ be the automorphism of F
given by φ(α) — а, ф(Ь) — ba, ф{с) = ca2. Prove that the group Г — F χφ Ζ,
which has presentation (a, b,c,t \ tat~l — a, tbt~l — ba, tct~l — ca2), cannot act
properly by semi-simple isometries on any complete CAT(O) space. (Hint: Note that
a, t generate a free abelian subgroup of Г and apply part (3) of the above lemma to
the set S = {t, at, a2t}.)
The example in (7.18(2)) is due to S.M. Gersten [Ger94]. He used the above
example to show that the outer automorphism group of any free group of rank at
least 4 cannot act properly and cocompactly by isometries on any complete CAT(O)
space. (A different argument shows that Out(/<3) does not admit such an action either
[BriV95].)

254 Chapter II.7 The Flat Torus Theorem
Actions That Are Not Proper
The following version of the Flat Torus Theorem can be applied to actions which are
not proper.
7.19 Lemma. If Γ is an abelian group acting by isometries on a complete CAT(O)
space, then the set of elements of Υ that act as elliptic isometries is a subgroup.
Proof. Because they commute, if γ, γ' e Γ are semi-simple then Min(y) and Min(y')
intersect (6.2). If γ and γ' are elliptic then γ γ' fixes Min(y) Π Min(y') pointwise.
D
7.20 Theorem. Let Υ be a finitely generated abelian group acting by semi-simple
isometries on a complete CAT(O) space X. (The action of Υ need not be proper.)
(1) Μϊη(Γ) := Пуег Min(y) is non-empty and splits as a product Υ χ Ε".
(2) Every γ e Γ leaves Μΐη(Γ) invariant and respects the product structure; γ
acts as the identity on the first factor Υ and acts by translation on the second
factor E"; moreover, η < rkqY and the action of Υ on E" is cocompact.
(3) The η-flats {y} χ Ε" are precisely those subsets ο/Μϊη(Γ) which arise as the
convex hull of a Y-orbit.
(4) If an isometry a e Isom(X) normalizes Y, then a leaves Μϊη(Γ) invariant and
preserves its splitting Υ χ Ε".
Proof. Let £ denote the subgroup of Γ consisting of those elements which act as
elliptic isometries. An easy induction argument on the number of elements needed
to generate £ shows that Min(£) is non-empty. By (6.2), Min(£) is convex and Γ-
invariant. The action of Υ/6 on Min(£) is by hyperbolic isometries (except for the
identity element). From this point one can proceed by induction on rkqY to prove
parts (1), (2) and (3) of the theorem. The argument is essentially the same as in (7.1),
so we omit the details. Part (4) is also proved as in (7.1). G
7.21 Exercise. Let Γ and X and η be as in the above theorem. Show that if a convex
subspace Ζ С X is Γ-invariant and isometric to Er for some r, then r > n. Show
further that if r = η then Ζ С Μΐη(Γ).
Some Applications to Topology
In this section we gather a number of results which illustrate the great extent to which
the structure of the fundamental group determines the topology of a non-positively
curved space. Most of the results which we present are applications of the Flat Torus
Theorem.

Some Applications to Topology 255
Recall that a strong deformation retraction of topological space Υ onto a subspace
Ζ is a continuous map F : [0, 1] χ Υ —> Υ such that F|(o)Xy = idy, im F|(ijxy = Z,
and F(r, z) = ζ for all ί e [0, 1] and ζ e Z. Recall also that a_/?ar manifold is the
quotient of Euclidean space E" by a group acting freely and properly by isometries.
7.22 Theorem. Let Υ be a compact non-positively curved geodesic space, and
suppose that its fundamental group Γ = π{Υ is virtually solvable. Then, Γ is virtually
abelian, and
(1) there is a strong deformation retraction of Υ onto a compact subspace that is
isometric to aflat manifold;
(2) if Υ has the geodesic extension property, then Υ itself is isometric to a flat
manifold.
Proof. According to (7.8), Γ is virtually abelian. Corollary 7.2 yields a Г-invariant
copy of Euclidean space Ε in the universal cover Υ of Y, such that Γ acts properly
by isometries on this fiat subspace. Since the action of Γ on У is free, Ζ := T\E is a
flat manifold.
Ε С Υ is closed, complete and convex; consider the orthogonal projection π :
Υ —> Ε, as defined in (2.4). Because Ε is Г-invariant, the projection map is Γ-
equivariant. Indeed there is a Γ-equivariant retraction R : [0, 1] χ Υ —» Υ that sends
(t, x) to the point a distance td(x, π(χ)) from χ on the geodesic segment [x, n(x)].
Let ρ : Υ —» Υ denote the universal covering map (so Ζ = p(E)). Because the
retraction R is Γ-equivariant, the map [0, 1] χ Υ -> Ζ given by (t, p{x)) = p(R(t, y))
is well-defined. This is the desired strong deformation retraction of Υ onto Z.
The key point to observe for the second part of the theorem is that under the
additional hypothesis that Υ has the geodesic extension property, the Euclidean sub-
space Ε must actually be the whole of Y. This follows from (6.20) and the fact that
Υ inherits the geodesic extension property from Υ (because this property is local and
Υ is locally isometric to Y). G
Combining Theorem 7.8 with Propositions (5.10) and (5.12) we obtain:
7.23 Corollary. Let Υ be a compact geodesic space of non-positive curvature, and
suppose that π\Υ is virtually solvable.
(1) If Υ is homeomorphic to a topological η-manifold (or indeed an homology
manifold) then the universal cover of Υ is isometric to E".
(2) If Υ is isometric to an ΜK—complex with no free faces, then the complex is
actually aflat manifold.
Our next result is of a more combinatorial nature. Recall that the girth of a graph
is the combinatorial length of the shortest injective loop that it contains.
7.24 Proposition. Let К be a finite, connected, 2-dimensional simplicial complex
with no free faces (i.e., every edge of К lies in the boundary of at least two 2-

256 Chapter II.7 The Flat Torus Theorem
simplices). Suppose that the for each vertex υ e К the girth of the link of υ is at least
6. Suppose further that the fundamental group of К is solvable.
Then, К is homeomorphic to either a torus or a Klein bottle, and exactly six
2-simplices meet at each vertex ofK.
Proof. We metrize К as a piecewise Euclidean complex with each 2-simplex
isometric to an equilateral triangle of side length 1. Metrized thus, К satisfies the link
condition (5.1), and hence is non-positively curved. The preceding result implies that
К is isometric to a flat manifold, hence it is homeomorphic to either a torus or a Klein
bottle. The fact that the metric is non-singular at the vertices means that exactly six
2-simplices meet there. G
One of the aspects of the study of non-positively curved spaces that we shall
not explore in this book is the theory of super-rigidity and the influence that ergodic
theory has had on the study of lattices in Lie groups, a la Margulis. (Basic references
for this material include [Mar90] and [Zim84]. See also [BuM96], [GrS92] and
[GroP91]).
The following consequence of the Flat Torus Theorem provides a very simple
example of a result from this circle of ideas.
7.25 Proposition. Let Η be a group that acts properly by semi-simple isometries on
a CAT(O) space X and let Γ be an irreducible lattice in a semi-simple Lie group of
real rank η > 2 that has no compact factors. IfX does not contain an isometrically
embedded copy o/E", then the image of every homomorphism Γ —> Η is finite.
Proof. The Flat Torus Theorem implies that Η does not contain a subgroup
isomorphic to Z", whereas Γ does. Thus the kernel of any homomorphism φ : Γ —> Η must
be infinite. If a normal subgroup of Γ is not finite, then it is of finite index ([Zim84]
8.1.2), therefore the image of φ must be finite. G
Low-Dimensional Topology
In this section we present some applications of Theorems 6.12 and 7.1 to the study
of manifolds in dimensions two and three. In the course of the discussion we shall
require a number of standard facts about 3-manifolds and surface automorphisms.
We refer to [Bir76] for facts about the mapping class group, to [Rol91] and [Hem76]
for basic facts about 3-manifolds, and to [Sco83] for facts about the geometry of
3-manifolds.
Mapping Class Groups. Let Σ be an oriented surface of finite type, i.e., a compact,
connected, oriented, 2-dimensional manifold with a finite number of points and
open discs deleted. (The surface may be closed, i.e. the set of punctures and deleted
discs may be empty.) A closed curve с : Sl —> Σ is said to be simple if it has no
self-intersections; it is said to be separating if the complement of its image is not
connected.

Some Applications to Topology 257
We consider the group Λ4(Σ) of isotopy classes of those orientation preserving
self-homeomorphisms of Σ which send each puncture to itself and restrict to the
identity on the boundary of Σ. (We are only allowing isotopies through homeomor-
phisms which restrict to the identity on 3Σ.) Associated to each simple closed curve
с in Σ, there is an element of Λ'ί(Σ) defined as follows. One considers a product
neighbourhood of the image of c, identified with (Κ/Ζ) χ [0, 1]. The Dehn twist
associated to с is the map Tc defined on the given product neighbourhood by (using
the above coordinates) (Θ, t) ы> (θ + t, t)\ one extends Tc to be the identity on the
remainder of Σ. The class of Μ(Σ) thus defined is clearly independent of the choice
of product neighbourhood.
We shall restrict our attention to surfaces Σ which are either closed of genus at
least 3, or else have genus at least two and at least one boundary component or two
punctures. With this restriction, we have the following result, which is a consequence
of Theorem 6.12 and an unpublished result of G. Mess28 concerning centralizers in
the mapping class group. This consequence was first observed by Kapovich and Leeb
[KaL95] (from a different point of view).
7.26 Theorem. If Σ is as above, then the mapping class group Μ(Σ) cannot act
properly by semi-simple isometries on any complete CAT(O) space.
Proof. The restrictions we imposed on Σ ensure that there is a separating simple
closed curve с on Σ with the property that the closure of one of the components
of Σ ν с is a sub-surface of genus 2 with no punctures and only one boundary
component, с itself, and the Dehn twist in с is non-trivial in Λ4(Σ). Let Σ2 denote
the sub-surface of genus 2. The subgroup of Λ4(Σ) given by homeomorphisms which
(up to isotopy) are the identity on the complement of Σ2 is naturally isomorphic to
Μ(Σ2), thus it will suffice to prove that Μ(Σ2) cannot act properly by semi-simple
isometries on any CAT(O) space. Notice that the mapping class of the Dehn twist Tc
is central in Μ(Σ2).
Let Σ'2 denote a closed surface of genus 2 with a single puncture p, and let Σ2
denote the corresponding closed (unpunctured) surface. There is an obvious inclusion
of Σ2 into Σ'2 sending с to the boundary of a neighbourhood D of the puncture in
Σ'2, and an even more obvious inclusion of Σ'2 into Σ2. These inclusions induce
surjective maps from Μ(Σ2) to Μ{Σ'2) and from Μ(Σ2) to Μ(Σ2). (For the first
map, one extends homeomorphisms of Σ2 to be the identity on D.) One checks easily
that both maps are surjective homomorphisms, well-defined on isotopy classes. The
kernel of the first homomorphism is the infinite cyclic subgroup generated by the
mapping class of the Dehn twist Tc. The kernel of the second homomorphism is
the subgroup, isomorphic to 7Γι(Σ2), which consists of mapping classes represented
by homeomorphisms which are the identity off a neighbourhood of some essential
closed curve based at p, and which drag ρ around that curve; a more algebraic
way of describing these mapping classes is to say that they are precisely those
elements of Λ4(Σ'2) which induce an inner automorphism of ττι(Σ2, ρ). (See [Bir76]
MSRI preprint #05708, 1990

258 Chapter II.7 The Flat Torus Theorem
for more details.) Thus the kernel К of the composition of the natural homomorphisms
Μ(Σ2) -> Μ(Σ'2) -» M(E2) is a central extension of 7Τι(Σ2) with infinite cyclic
centre.
Using the above geometric description of the mapping classes in the kernel of
Μ(Σ'2) —> Μ(Σ2), one sees that К is the fundamental group of the unit tangent
bundle of Σ2 (where the surface is endowed with a metric of constant negative
curvature)29. In particular, К is (isomorphic to) a cocompact lattice jn^the Lie group
PSL(2, K), the universal covering group of PSL(2, R). Now, PSL(2, K) does not
contain the fundamental group of any closed surface of genus > 2 (see [Sco83]),
so since every finite index subgroup of πχ (Σ,2) is the fundamental group of such a
surface, the central extension 1 —>- (Tc) —>- К —> 7Γι(Σ2) —> 1 cannot be split even
after passing to a subgroup of finite index in 7Γι(Σ2). Thus, by (6.12), К (and hence
Μ(Σ)) cannot act properly by semi-simple isometries on any CAT(O) space. D
3-Manifolds
There has been a good deal of work recently on the question of which 3-manifolds
admitmetrics of non-positive curvature (see [Le95], [Bri98b] and references therein).
We shall not describe that body of work. Instead, we present a single consequence
of our earlier results in order to exemplify the great extent to which the structure of
the fundamental group determines the topology of a 3-manifold, particularly in the
presence of non-positive curvature.
Building on a sequence of works by various authors (including Tukia, Scott and
Mess), Casson and Jungreis [CasJ94] and Gabai [Gab93] independently proved the
following deep theorem: if a 3-manifold Μ is compact, irreducible and π\Μ contains
a normal subgroup isomorphic to Z, then Μ is a Seifert fibre space (i.e., it can be
foliated by circles). We prove a related (but much easier) result.
7.27 Theorem. Let Μ be a closed 3-manifold which is non-positively curved (i.e.,
locally CAT(O)). Ifn\M contains a normal subgroup isomorphic to Z, then:
(1) Μ can be foliated by circles.
(2) A finite-sheeted covering of Μ is homeomorphic to a product Σ χ Sl, where Σ
is a closed surface of positive genus.
(3) Μ supports a Riemannian metric of non-positive sectional curvature.
Proof. Consider the universal covering ρ : Μ —> Μ, equipped with the length metric
making ρ a local isometry. The action of Γ := τΐ\Μ on Μ by deck transformations is
an action by hyperbolic isometries. Suppose that γ e Γ generates an infinite cyclic
normal subgroup С с Г. Because С is normal, Min(y) is Γ-invariant, and hence is
the whole of the universal cover (see (6.20) or (6.16)). The splitting Μ = Υ χ Μ.
given by (6.8) is preserved by the action of Γ; the group С acts by translations on
the second factor and acts trivially on the first factor. Hence the foliation of Μ by
This is Mess's unpublished observation.

Some Applications to Topology 259
the lines {y} χ К descends to a foliation of Μ by circles. Since У χ Μ is a simply
connected 3-manifold, У is a simply connected 2-manifold (cf. [Bing59]). Since Υ
is contractible (it is a complete CAT(O) space) it must be homeomorphic to K2.
Let Г" С Г be a subgroup of finite index that contains С as a direct factor (6.12),
say Г' = С χ Κ. The action of К = Г'/ С on Υ is proper and cocompact (6.10). Since
К is a subgroup of Г (which acts freely on the CAT(O) space M) it is torsion-free
(2.8). Thus the action of К on Υ is free, K\Y is a closed surface Σ, and the quotient
of Μ by the action of Г' = С χ Κ is homeomorphic to Σ χ Sl.
If Σ has genus at least two, then it follows from Kerckhoff's solution to the
Nielsen Realization problem [Ker83] that there is a metric of constant negative
curvature on Υ that is Г/С invariant; the resulting product metric on Σ χ Μ descends
to the desired Riemannian metric of non-positive curvature on M. If Σ has genus 1
then Γ is virtually abelian and (7.2) implies that Μ admits a flat metric. G

Chapter II.8 The Boundary at Infinity of а С AT(0) Space
In this chapter we study the geometry at infinity of С AT(0) spaces. If X is a simply
connected complete Riemannian и-manifold of non-positive curvature, then the
exponential map from each point χ e X is a diffeomorphism onto X. At an intuitive
level, one might describe this by saying that, as in our own space, the field of vision
of an observer at any point in X extends indefinitely through spheres of increasing
radius. One obtains a natural compactification X of X by attaching to X the inverse limit
of these spheres. X is homeomorphic to a closed и-ball; the ideal points dX = X \ X
correspond to geodesic rays issuing from an arbitrary basepoint in X and are referred
to as points at infinity.
We shall generalize this construction to the case of complete CAT(O) spaces X.
We shall give two constructions of dX, the first follows the visual description given
above (and is due to Eberlein and O'Neill [EbON73]) and the second (described
by Gromov in [BaGS85]) arises from a natural embedding of X into the space of
continuous functions on X. These constructions are equivalent (Theorem 8.13). In
general dX is not a sphere. If X is not locally compact then in general X and dX will
not even be compact. We call X the bordification of X.
Isometries of X extend uniquely to homeomorphisms of X (see (8.9)). In the last
section of this chapter we shall characterize parabolic isometries of complete С AT(0)
spaces in terms of their fixed points at infinity.
Asymptotic Rays and the Boundary dX
8.1 Definition. Let X be a metric space. Two geodesic rays c, c' : [0, oo) —> X are
said to be asymptotic if there exists a constant К such that d(c(t), c'(t)) < К for all
t > 0. The set dX of boundary points of X (which we shall also call the points at
infinity) is the set of equivalence classes of geodesic rays — two geodesic rays being
equivalent if and only if they are asymptotic. The union X U dX will be denoted X.
The equivalence class of a geodesic ray с will be denoted c(oo). A typical point of
dX will often be denoted ξ.
Notice that the images of two asymptotic geodesic rays under any isometry γ of
X are again asymptotic geodesic rays, and hence γ extends to give a bijection of X,
which we shall continue to denote by γ.

Asymptotic Rays and the Boundary dX 261
8.2 Proposition. If X is a complete CAT(O) space and с : [0, oo) —> Xisa geodesic
ray issuing from x, then for every point x1 e X there is a unique geodesic ray c' which
issues from x! and is asymptotic to с
The uniqueness assertion in this proposition follows immediately from the
convexity of the distance function in CAT(O) spaces (2.2). For if two asymptotic rays
c' and c" issue from the same point, then d(c'(t), c"{t)) is a bounded, non-negative,
convex function defined for all t > 0; it vanishes at 0 and hence is identically zero.
In order to prove the asserted existence of c' we shall need the following lemma.
8.3 Lemma. Given ε > 0, a > 0 and s > 0, there exists a constant Τ = Τ(ε, a, s) >
0 such that the following is true: if χ and xf are points of a CAT(O) space X with
d(x, xf) = a, if с is a geodesic ray issuing from x, and ifa, is the geodesic joining xf
to c(t) with σ,(0) = χ1, then d(a,(s), at+t'(s)) < ε for all t > Τ and all f > 0.
x' x'
\\°t+t(s)
x=c(0) c(t) c{t+t')
Fig. 8.1 The construction of asymptotic rays.
Proof. By the triangle inequality,
\t-a\< d(x, c(t)) <t + a,
\t + f - a\ < d(x, c(t + t')) <t+t' + a.
Let a be the comparison angle Z^(c(t), c(t + t')). By the law of cosines,
, ч d(x', c{t)f + d{xf, c(t + O)2 - f2
cos(a) =
2d(x',c(t))d(x',c(t + t'))
(t-a)2 + (t + f - a)2 - f2
2{t + a)(t + t' + a)
_(t- a)(t +f -a)
~ (t + a)(t + f + a)'
Thus cos a —> 1 as t -^ oo, and hence a —> 0, where the convergence is uniform in
f >0.
On the other hand, if s < imn{d(x/, c(t)), d{x!, c(t + f))} then by the CAT(0)
inequality in the comparison triangle for A(x, c(t), c(t + f)),
d(a,(s), σ,+fis)) < d(a,(s), at+t-(s)) = 2isin(a/2).
c(t+t')

262 Chapter Π.8 The Boundary at Infinity of a CAT(O) Space
Thus for sufficiently large t we have d(at(s), at+t'(s)) < ε.
Ώ
Proof of (8.2). It follows from the preceding lemma that for every s > 0 the sequence
(σ„(ί)) (which is defined for sufficiently large ri) is Cauchy, and hence converges in
X to a point which we shall call c'(s). As the pointwise limit of geodesies, s н> c'(s)
must be a geodesic ray — this is the desired ray c' issuing from xf. One can see that
c' is asymptotic to с by checking that d(c(s), c'(s)) < d(x, x1) for all s > 0. Indeed
by considering the comparison triangle A(x, x1, c(n)) one sees that
d(a„(s), c(s)) < d(a„(s), φ)) < d(x, x),
for all η > 0, where the first inequality comes from the CAT(O) condition and the
second from elementary Euclidean geometry. G
Let с : [0, oo) —> E" be a geodesic ray. The asymptotic ray which issues from
x e E" and is asymptotic to с is the unique ray parallel to c. The following exercises
describe the asymptotic rays in W.
8.4 Exercises
(1) The Poincare metric on the open ball В" С Μ" was defined in (1.6.7). The
image of each geodesic ray с : [0, oo) —> B" is an arc of a circle that is orthogonal
to the unit sphere about the origin in E". The closure of this arc has one endpoint on
the sphere and c(i) converges to this point as t —> oo.
Show that geodesic rays in this model of hyperbolic space are asymptotic if and
only if the closures of their images intersect the sphere at the same point.
/\ A A
Fig. 8.2 Asymptotic rays in the Poincare model
(2) B" can also be viewed as the set of points in the Klein model for hyperbolic
space. In this model the images of geodesic rays are half-open affine intervals whose
closures have one point on the bounding sphere (1.6.2). Show that, as in (1), geodesic
rays in the Klein model are asymptotic if and only if the closures of their images
intersect the bounding sphere at the same point.
(3) The geodesic lines in the hyperboloid model of hyperbolic space W С Ε"·1
are intersections with 2-dimensional vector subspaces in E"·1. Show that if two

The Cone Topology on X = X U dX 263
geodesic rays in Ш" are asymptotic, then the intersection of the corresponding 2-
dimensional subspaces of E"·' is a line in the light cone.
(4) Give an example of a non-complete convex subspace of the Euclidean plane
for which the conclusion of (8.2) is not valid.
The Cone Topology οηΙ = ΐυθΧ
Throughout this section we assume that X is a complete CAT(O) space. Let X =
X U dX. We wish to define a topology on X such that the induced topology on X is
the original metric topology.
8.5 Definition of X as an Inverse Limit. We fix a point x0 e X and consider the
system of closed balls B(x0, r) centred at xq. The projection pr of X onto B(x0, r)
is well-defined because B{x0, r) is a complete convex subset of X (see 2.4). If χ $.
B(x0, r) then pr{x) is the point of [xq, x] a distance r from x0. For r1 > r, we have
pr o/v = Pr- And for r e [0, /] and allx' eX with d(x/, xq) = /, the path г м> /?л(х/)
is the unique geodesic segment joining xq to xf.
The projections рг\щх ^ ■ B(x0, /) —» B(x0, r), where / > r, form an inverse
system, and we consider hjn B(x0, r) with the inverse limit topology [Spa66]. A point
in this space is a map с : [0, oo) -> X such that if r1 > r then рг{с{/)) = c(r). Such
maps are of two types: either c(/) / c(r) for all/ φ r (in which case с is a geodesic
ray issuing from x0), or else there is a minimum ro > 0 such that c(r) = c(r0) for all
r > r0 (in which case the restriction of с to [0, r0] is the geodesic segment joining x0
to c(r0), and the restriction of с to [r0, oo) is a constant map). Thus ЦтВ(хо, г) may
be viewed as a subspace of the set of maps [0, oo) —> X. (The inverse limit topology
coincides with the topology of uniform convergence on compact subsets.)
There is a natural bijection ф{хо): X —> hjn5(jco, r), which associates to ξ e ЭХ
the geodesic ray that issues from jco in the class of ξ, and which associates to χ e X
the map cx : [0, oo) -> X whose restriction to [0, d{xo, x)] is the geodesic segment
joining xq to χ and whose restriction to [d(xo, x), oo) is the constant map at x. Let
T(jeo) be the topology on X for which φ(χο) is a homeomorphism. The inclusion
Χ <^-> X gives a homeomorphism from X onto a dense open set of X.
Notice that for every χ e X and ξ e ЭХ there is a natural path with compact
image [χ, ξ] joining χ to ξ in X; this is the union of ξ and the image of [0, oo) by the
geodesic ray с with c(0) = χ and c(oo) = ξ. Notice also that a sequence of points
(xn) in X с X converges to a point ξ e ЭХ if and only if the geodesies joining xq to
x„ converge (uniformly on compact subsets) to the geodesic ray that issues from xq
and belongs to the class of ξ.
8.6 Definition of the Cone Topology. Let X be a complete CAT(0) space. The cone
topology on X is the topology T(jc0) defined above. (This is independent of the choice
of basepoint jcq e X (see 8.8).)

264 Chapter II.8 The Boundary at Infinity of a CAT(O) Space
The topology on dX induced by the cone topology on X will also be referred to
as the cone topology. When equipped with this topology, dX is sometimes called the
visual boundary of X. However, we shall normally refer to it simply as the boundary.
Note that dX is a closed subspace of X and that X is compact if X is proper.
In order to work effectively with the cone topology, we need to understand an
explicit neighbourhood basis for it. A basic neighbourhood of a point at infinity has
the following form: given a geodesic ray с and positive numbers r > 0, ε > 0, let
U(c, r, ε) = {χ e X \ d(x, c(0)) > r, d(pr(x), c{r)) < ε},
where pr is the natural projection of X = ЦтВ(с(0), г) onto B(c(0), r).
8.7 Exercise. Prove that one obtains a basis for the topology T(xq) on X by taking
the set of all open balls B(x, г) С X, together with the collection of all sets of the
form U(c, r, ε), where с is a geodesic ray with c(0) = xq.
8.8 Proposition. For all xo, x'0 eX, the topologies T(xq) and Т(д^) are the same.
Proof. It is enough to prove that the projection pro : X —> B(x0, ro) is continuous
when X is endowed with the topology T(xJ,). The continuity at points у е X is
obvious. Given ξ e dX, let cq and c'0 be the geodesic rays issuing from jeo and xfQ
respectively such that c0(oo) = c'0(oo) = ξ. Let χ = Co(r0) = ρΓ0(ξ) and let ε
be an arbitrary positive number. Let R be a number bigger than ro + d(xo, x!0) and
also bigger than the number Τ(ε/3, d(x0, x!0), r0) given by (8.3). We shall argue that
pro U(c'0, R, ε/З) с B(x, ε). To see this, given у е X, let cy be the geodesic segment
or the geodesic ray joining x*0 to y. We claim that if у e U(c', R, ε/З) then
d(pro(y), x) < d(pro(y), Pro(cy(R))) + d(pro(cy(R)), pro(c'0(R))) + d(pro(c'0(R)), x)
< ε/3 + ε/3 + ε/3.
The bound on the second term comes from the definition of U(c', R, ε/З), which
implies that d(cy(R), c'(R)) < ε/З. The desired bounds on the first and third terms
are obtained by applying (8.3) with xq in the role of x1, and xfQ in the role of x; the
role of с is played in the first case by cy and in the second case by c'0. G
8.9 Corollary. Let γ be an isometry of a complete CAT(O) space X. The natural
extension of γ toX is a homeomorphism.
Proof. The action of γ on dX is induced by its action on geodesic rays: t ы> c(t) is
sent to t м> γ.φ). An equivalent description of the action of γ onX = XU dX can be
obtained by fixing x0 e X and noting that γ conjugates the projection X —> B(x0, r)
to the projection X —> Β(γ(χ0), r), and therefore induces a homeomorphism γ from
l^mB(xQ, r) to ЦтВ(у0, г). Modulo the natural identifications ф(хо) and φ(γ(χο))
described in (8.5), this homeomorphism γ : X —> X is the natural extension of γ
described above. G

The Cone Topology on X = X U ΘΧ 265
Fig. 8.3 The cone topology in independent of base point
8.10 Remark. If X is a proper CAT(O) space, then the natural map с н> end(c)
from the set of geodesic rays to Endi(X) (as defined in 1.8.27) induces a continuous
surjection dX —»■ Endi(X). If the asymptotic classes of с and d are in the same path
component of dX, then end(c) = end(c')·
8.11 Examples of ЭХ
(1) If X is a complete и-dimensional Riemannian manifold of non-positive
sectional curvature, then dX is homeomorphic to S"~', the (n — 1 )-sphere. Indeed, given
a base point χ e X, one obtains a homeomorphism by considering the map T\ —»■ dX
which associates to each unit vector и tangent to X at χ the class of the geodesic ray
с which issues from χ with velocity vector u.
(2) As a special case of (1), consider the Poincare ball model (1.6.8) for real
hyperbolic и-space. In (8.4) we noted that two geodesic rays с and d in H" in this
model are asymptotic if and only if c(t) and d(t) converge to the same point of W
as t —»■ oo. Thus the visual boundary 3H" is naturally identified with the sphere
of Euclidean radius 1 centred at 0 e W, and И is homeomorphic to the unit ball
5(0, l)inE".
We noted in (8.4) that geodesic rays in the Klein model for H" also define the
same point in the bounding sphere. It follows that the natural transformation h^lhP
between the models (see 1.6.2 and 1.6.7) extends continuously to the bounding sphere
as the identity map.
(3) In [DaJ91a] M. Davis and T. Janusciewicz used Gromov's hyperbolization
technique to produce interesting examples of contractible topological manifolds X

266 Chapter II. 8 The Boundary at Infinity of a CAT(O) Space
in all dimensions η > 5. Each of their examples supports a complete CAT(O) metric
d such that (X, d) admits a group of isometries which acts freely and cocompactly.
However, in contrast to (1), one can construct these examples so as to ensure that dX
is not a sphere (see (5.23)). If X is a complete CAT(O) 3-manifold, then one can use
a theorem of Rolfsen ([R0I68]) to show that dX is homeomorphic to a 2-sphere and
X is a 3-ball.
(4) If Xq is a closed convex subspace of a complete CAT(O) space X, then dXo is
a closed subspace of dX. For instance, consider the Klein model of hyperbolic space
and let F be any closed set in ЭН" identified to the unit sphere in E", as in example
(2). Consider the convex hull F of F, that is, the intersection with B" of the closed
affine half-spaces containing F. (This is the same as the intersection of the closed
hyperbolic half-spaces whose boundary contains F.) The boundary at infinity of F
is F.
(5) Let X be an K-tree (1.15(5)) and fix a basepoint χ e X. The boundary of X
is homeomorphic to the projective limit as r —»■ 00 of the spheres Sr(x). Each such
sphere is totally disconnected, hence dX is also totally disconnected. If X is locally
compact, then each sphere in X contains only a finite number of points, hence dX
is a compact space. If X is an infinite simplicial K-tree in which every vertex has
valence at least three, then dX is a Cantor set (see (1.8.31)).
(6) Let X = X\ χ Χ2 be the product of two complete CAT(O) spaces X\ and
X2. If |i € dX\ and %i e ЗХ2 are represented by the geodesic rays c\ and cj. and if
θ e [О,π/2], we shall denote by (cos0)fi + (sin0)f2 the point of dX represented
by the geodesic ray t ы> (o(icos0)), C2(isin0)). The boundary dX is naturally
homeomorphic to the spherical join dX\ * 8X2, that is, the quotient of the product
3Xi x [0,7Г/2] χ ЗХ2 by the equivalence relation which identifies (f 1,6), £2) to
(Ιί, θ, ξ{) if and only if [Θ = 0 and fc = l^ or № = π/2 and li = !i']·
(7) If in the previous example we take X\ to be a metric simplicial tree in which
all the vertices have the same valence > 3 and we take X2 = K, then the boundary
of X\ x X2 is the suspension of a Cantor set — in particular it is connected but not
locally connected30. In contrast, Swarup [Sw96], building on work of Bowditch and
others, has shown that if X is compact and has curvature < к < 0 and if X is not
quasi-isometric to K, then the boundary of its universal covering is connected if and
only if it is locally connected.
(8) The Menger and Sierpinski curves can be obtained as the visual boundaries
of certain CAT(O) polyhedral complexes: see (12.34(4)) and [Ben92].
Further examples are given in "CAT(O) groups with non-locally connected boundary", by
M. Mihalik and K. Ruane, to appear in Proc. London Math. Soc.

Horofunctions and Busemann Functions 267
Horofunctions and Busemann Functions
Let X be a complete CAT(O) space. We have explained how one can attach to X a
boundary at infinity dX, whose points correspond to classes of asymptotic geodesic
rays in X. This boundary was attached to X, intuitively speaking, by fixing a point
xo e X and attaching to each geodesic ray issuing from xq an endpoint at infinity.
Since thehomeomorphism type of the spaceX = XUdX obtained by this construction
is independent of the choice of xq, one expects that there might be a more functorial
way of constructing the boundary. Our first goal in this section is to describe such a
construction.
Following Gromov [BaGS87], we shall describe a natural embedding of an
arbitrary metric space X into a certain function space. This embedding has the property
that when X is complete and CAT(O), the closure of X is naturally homeomorphic
to X. In this construction, the ideal points of X appear as equivalence classes of
Busemann functions (see 8.16).
8.12 The Space X. Let X be any metric space. We denote by C(X) the space of
continuous functions on X equipped with the topology of uniform convergence on
bounded subsets (so in particular if X is proper then this is the more familiar topology
of uniform convergence on compact subsets). Let C*(X) denote the quotient of C(X)
by the 1-dimensional subspace of constant functions, and let/ denote the image in
C*(X) of/ e C(X). Notice that/„ -» / in C*(X) if and only if there exist constants
i!„il such that/, + an —>- / uniformly on bounded subsets; equivalently, given
a base point xq e X, the sequence of functions ζ м> f„(z) — /n(*o) converges to the
function ζ м> f(z) —/(го) uniformly on all balls B(xq, r). In fact, given a base point
jco € X, the space C*(X) is homeomorphic to the subspace of C(X) of continuous
functions/ on X such that/(xo) = 0.
It is also useful to observe that, given x, у e X, the value of f(x) —f(y) is an
invariant of/.
There is a natural embedding ι : X ^ C*(X) obtained by associating to each
x e X the equivalence class dx of the function dx : у ы> d(x, у). We shall denote
by X the closure of t(X) in C*(X). The ultimate goal of this section is to prove the
following theorem.
8.13 Theorem. LetX be a complete CAT(O) space. The natural inclusion ι : X —►
С* (X) extends uniquely to a homeomorphism X —»■ X.
It will be convenient to identify X with i(X), and we shall do so freely, unless
there is a danger of ambiguity.
8.14 Definition, h e C(X) is said to be a horofunction (centred at h) if h e Χ χ Χ.
The sublevel sets h~l(—oo, г] С X are called (closed) horoballs, and the sets h~~l(r)
are called horospheres centred at h.

268 Chapter Π.8 The Boundary at Infinity of a CAT(O) Space
8.15 Exercises
(1) Let X be a metric space. Prove that the map ι : X —»■ С*(Х) is a homeomor-
phism onto its image.
(2) Prove that if X is proper then X and X \ X are compact. (Hint: Given a
sequence/„ in C*(X) and *o € X, there exist functions/„ €/„ such that/„(jto) = 0.)
(3) Show that if h e X then h(x)-h(y) < d(x, y) for all x, у e X. Show that if X is
CAT(O) then hoc : [a, b] —> №. is convex for every geodesic segment с : [a, b] —> X.
The following general observation explains the connection between horofunc-
tions and sequences of points tending to infinity.
8.16 Lemma. Let X be a complete metric space. If the sequence dXn converges to
h e Χ ν X, then only finitely many of the points xn lie in any bounded subset ofX.
Proof. Fix xq e X and h e h with h(xo) = 0. If there is an infinite number of
points x„ contained in some ball B{xq, r), passing to a subsequence we can assume
that d(xo, x„) converges asn^ oo. By hypothesis the sequence of functions χ м>
d{x„,x) — d(x„, xo) converges uniformly to h on B(xq, r). Therefore, given ε > 0
we can find N > 0 such that \h(x) — d{x„,x) + d(xn,xo)\ < ε for all η > N
and χ e B(xq, r), and \d{xn, xq) — d(xn>, xq)\ < ε, for all n, n' > N. In particular
\h(xn) — d(xn, xn) + d(xn, xq)\ < ε and \h(xn) — d{xn, xn·) + d(xn,, xq)\ < ε. Hence
d(xn, xni) < 2ε. Thus the sequence (xn) is Cauchy and converges to some point zel
Therefore h = dz, contrary to hypothesis. G
Typical examples of horofunctions are the Busemann functions bc associated to
geodesic rays c. We shall prove that in complete CAT(O) spaces every horofunction
is a Busemann function.
8.17 Definition of a Busemann Function. Let X be a metric space and let с :
[0, oo) —»■ X be a geodesic ray. The function bc : X —> M. defined by
bc{x) = lim (d(x, c{t)) - t)
t—>oo
is called the Busemann function associated to the geodesic ray c. (On occasion bc
may be denoted bx^, where χ = c(0) and ξ = c(oo).)
The first part of the following lemma implies that the limit in the above definition
does indeed exist. Examples of Busemann functions in CAT(O) spaces are given at
the end of this section.
8.18 Lemma. Let X be a metric space and let с : [0, oo) —»■ X be a geodesic ray.
(1) For each χ e X, the function [0, oo) —»■ Μ given by t м> d{x, c(t)) — t is
non-increasing and is bounded below by —d(x, c(0)).
(2) IfX is a CAT(0) space, then as t —> oo the functions dc(t){x) = d(x, c(t)) — t
converge to bc uniformly on bounded subsets.

Horofunctions and Busemann Functions 269
Proof. By the triangle inequality, if 0 < f < ithen d(x, c(t)) < d(x, c(f))+t—f, and
hence d(x, c(t)) -1 < d(x, c(f)) - f. But we also have t - f < d(x, c(0) + d(x, c(t)),
so taking f = 0 we get -d(x, c(0)) < d(x, c(t)) - t. This proves (1).
Part (2) is an immediate consequence of (8.3). D
Our next goal is to establish a natural correspondence between horofunctions and
equivalence classes of Busemann functions.
8.19 Proposition. Let X be a complete CAT(0) space and let xn be a sequence of
points in X. Then dXu converges to a point ofX^X if and only if xn converges in
X to a point of dX, that is, for fixed xo the sequence of geodesic segments [xq, xn]
converges to a geodesic ray [xq, ξ].
We defer the proof of this proposition for a moment and move directly to:
8.20 Corollary (The Ideal Points X \ X). IfX is a complete CAT(O) space, then the
Busemann functions associated to asymptotic rays in X are equal up to addition of
a constant, and for every xq e X the map i: dX —>- X \ X defined by ί{ξ) = bXo^ is
a bijection.
Proof. We first prove that i is surjective. Given h e X \ X choose x„ e X with
HXii -+ Ji. By the proposition, there exists ξ € dX such that xn -»■ ξ in X. Let
с : [0, oo) -»■ X be the geodesic ray with c(0) = xo and c(oo) = ξ, and let y2n = xn
and yi„+\ = c(n) for all n. Then y„ —>- ξ, so by the proposition dyii converges to a
point of Χ ν X, and since X is Hausdorff this point must be h. Thus dC(,,) —> h, so by
definition (8.17) h = bxo^.
Given any x/0 e X, in the preceding paragraph one can replace с by d : [0, oo) —►
X, where c'(0) = x/0 and c'(oo) = ξ to deduce that fry ^ = h = Ъха^. Thus the
Busemann functions of asymptotic rays differ only by an additive constant.
Given distinct points ξ, ξ' e ЭХ, let с and с' be the geodesic rays in X with
c(0) = c'(0) = xo, c(oo) = f and c'(oo) = £'. The sequence z„ defined by zin = c{n)
and Zin+\ = d(n) does not converge in X, so by the proposition, the sequence dZii
does not converge in X, hence bXo^ φ bXo^>. Thus i is injective. G
In order to prove (8.19), we shall need the following lemma.
8.21 Lemma. Let X be a CAT(O) space.
(1) Given a ball B(xq, p)ands > 0, there exists r > 0 such that, for all ζ e В(х$, р)
and all χ e X with d(x, xq) > r,
d(z, y) + d(x, y) - d(x, ζ) < ε,
where у is the point of [xq, x] a distance rfrom xq.
(2) Consider xq, y, χ e X with у e [xq, x] and d(x$, y) = ρ > 0. Then,

270 Chapter Π.8 The Boundary at Infinity of а С AT(0) Space
At ^ At ^d(y^
d(x, z) - d(x, y) > —-
2p
for all ζ such that d(xo, z) = P-
Proof. (1) We first note that if a segment \p, q] of length t is contained in an infinite
strip in E2 whose boundary is the union of two parallel lines a distance ρ apart, and
if f is the length of its orthogonal projection onto the side of the strip, then
U-t'\<p2/t.
Indeed, if q' is the orthogonal projection of q onto the straight line through ρ parallel
to the sides of the strip and q" is the orthogonal projection of q' on \p, q], then
f = d{p, q') and \t — f\ < d(q, q")\ obviously d(q, q')/t = d(q, q")/d(q, q') and
d(q, q') < p, so we have \t — f\ < p2 /t.
Choose r big enough so that p2/(r — p) < ε/2. If Ζ = Ε2 and ζ' is the orthogonal
projection of ζ onto the line through χ and xq, then using the above inequality twice
we get: d(z, x) - d(t', χ) < ε/2 and d(z, y) - d(z', y) < ε/2. But dtf, x) = d{z', y) +
d{y, x), hence d(z, y) + d(y, x) < d(z, χ) + ε. In the general case, one applies the
CAT(O) inequality to the triangle A(jcq, z, x) and considers the comparison point for
у e [xo, x].
(2) We first check the inequality in the case X = E2. We have d(z, x) >
d(x, y) + d(y, z'), where z' is the projection of ζ on the line through xo and x. But
d(z', y)/d{y, z) = d(y, z)/2p, whence the desired inequality forX = E2. The
inequality in the general case is obtained by applying the CAT(O) inequality to the triangle
A(jcq, z, x) and considering the comparison point fory. G
Proof of (8.19). Choose a basepoint xq e X and let dx be the function on X defined
by dx(z) = d(x, z) - d(x, xq).
Assume first that xn converges to ξ e dX. We want to prove that the sequence dXli
converges uniformly on any ball B(xq , p). Given ε > 0, choose r > 0 as in (8.21 (1)).
Choose N big enough so that d(x„, xq) > r for η > N. lfy„ is the point of [xo, xn) at
distance r from xq, then by making two applications of (8.21(1)) we see that for all
ζ e B(x0, p)
\~dx,Xz) - ~dXin{z)\ = \d(xn, z) - d(xn, yn) - d{xm, z) + d(xm, ym)\
< \d(y„, z) - d(ym, z)\ + 2ε
<ά^η^ιη) + 2ε.
By hypothesis d(y„, ym) —»■ 0 as m, η —»■ oo. Thus we have uniform convergence on
the ball B(x0, p).
Conversely, assume (dX/i) converges uniformly on bounded sets but (xn) is
unbounded (cf. 8.16). We need to prove that the geodesic segments [xq, xn] converge to
a geodesic ray issuing from xq. Given ρ > 0 and ε > 0, we fix an integer N such that
d{x„, xq) > ρ and \d(xn, z) — d(xn, xo) — d{xm, z)+d(xm, xq)\ < ε forallm, η > Nana
Ζ € 5(*o> p)· Let Уп be the point on the geodesic segment [xq, x„] at distance ρ from

Horofunctions and Busemann Functions 271
xq. In the above inequality, replacing ζ by yn, we get \d(xm, y„) — d(xm, ym)\ < ε,
because d(xo,yn) = d(xo,ym) = p. (8.21(2)) implies that for n,m>Nwe have
d(y„, ym)2/2p < d(xm, yn) — d(xm, ym) < ε, hence y„ is a Cauchy sequence. G
Proof of Theorem 8.13. Because X is dense in X, there is at most one continuous
extension to X of ( : X ^ X. We showed in (8.20) that the map i : dX -»■ X \ X
given by г(с(оо)) = bc is a bijection, so it only remains to show that if we extend
ι by i then the resulting bijection i : X —*■ X and its inverse are continuous on the
complement of X.
We choose a base point xq in X and identify С*(Х) to the subspace Cq(X) of C(X)
consisting of those functions/ such that/(xo) = 0; thus X is identified to a subspace
of C(X). With this identification, the map i: X —> X associates to χ e X the function
dx(z) '■= d(x, z) — d(x, xq), and associates to f e dX the Busemann function bc of
the geodesic ray с with c(0) = xq and c(oo) = ξ.
A fundamental system of neighbourhoods for a point χ e X is given by the balls
B(x, r). A fundamental system of neighbourhoods for bc in X is the collection of sets
V(bc, p,B) = {feX\ \f(z) - bc(z)\ <S,Vze B(x0, p)},
where ρ > 0 and ε > 0. A fundamental system of neighbourhoods for c(oo) in X
was described in (8.6):
U(c, r, 5) = {x e X | ύ?(χ, *ο) > r, d(pr(x), c(r)) < 5}
where r > 0, 5 > 0, and pr : X —»■ 5(jcq, r) is the map whose restriction to X is the
projection on B(xq, r) and which maps c(oo) e 3X onto c(r).
By letting ρ, ε, r, δ take all positive rational values, we see that each point of X
and X has a countable fundamental system of neighbourhoods, (i.e. as topological
spaces X and X are first countable). It is therefore sufficient to prove that both i and
i~~l carry convergent sequences to convergent sequences. In other words, writing 5„
to denote du or bxou (according to whether и e X or и е ЭХ), what we must show is
that a sequence xn in X converges to ξ e ЭХ if and only if δΧιι —>- bxo^ uniformly on
bounded subsets. But this was done in (8.18) and (8.19). G
Characterizations of Horofunctions
Let В be the set of functions h on X satisfying the following three conditions:
(i) h is convex;
(ii) \h(x) -h(y)\< d(x, y) for all x, у е X;
(iii) For any xq e X and r > 0, the function h attains its minimum on the sphere
Sr(xQ) at a unique point у and h(y) = H{xq) — r.
Notice that if h e В then И е В whenever h = h .
8.22 Proposition. Let X be a complete CAT(0) space. For functions h: X —>- K, the
following conditions are equivalent:

272 Chapter Π.8 The Boundary at Infinity of a CAT(O) Space
(1) h is a horofunction;
(2) h e B;
(3) h is convex, for every (it the set h~l(—oo, t] is non-empty, and for each
χ e X the map cx : [0, oo) —»■ X associating to t > 0 the projection of χ onto
h~~x(—oo, h(x) — t] is a geodesic ray.
Proof. (1) => (2): Let h be a horofunction. After modification by an additive
constant we may assume that h is the Busemann function bc associated to a geodesic
ray issuing from x. Let x„ be a sequence of points tending to c(oo) along the geodesic
ray c, and let dXn be the function defined by dXn{z) = ύ?(χ„,ζ) — d{x„, xo). Then
fec = Ηπι,,^οοίΐ^. The convexity of the metric on X (2.2) ensures that each dXn
satisfies conditions (i) and (iii) in the definition of B, and (ii) is simply the triangle
inequality for dXn. Each of these is a closed condition, so the limit bc also satisfies
them.
(2) => (3): Consider a function h e B, a point χ e X, and a number ί > 0. Let
cx(t) be the unique point of S,(x) such that h(cx(t)) = h(x) — ί (condition (iii)). This
point is the projection of χ onto the closed convex subset A, = h~~x((—oo, h(x) — t]).
Let r : [0, i] —»■ X be the geodesic segment joining jc to c^(i)· Because /г is convex,
we have that h(x) — h(r(s)) < s for all s e [0, t]. Condition (iii) applied to the sphere
Ss(x) implies that cx(s) = r(s), hence t ы> cx(t) is a geodesic ray.
(3) => (1): Let b be the Busemann function associated to the geodesic ray cx.
For convenience (replacing h by h — h(x) if necessary) we assume that h(x) = 0. We
want to prove that b = h. For r > 0 let πΓ be the projection of X onto the closed
convex set Ar = h~l(—oo, —r]. Note that ύ?(χ, πν(χ)) = г.
Given yel, for each (sufficiently large) r > 0, we have A(y) = d(y, Ttriy)) — r.
Consider the Euclidean quadrilateral obtained by joining the comparison triangles
A(x, y, nr(x)) and A(y, nr{y), nr(x)) along the edge [nr(x), y], so that nr(y) and χ lie
on opposite sides of the line through у and nr(x) (see Figure 8.3). (The degenerate
case nr(y) = nr(x) is easily dealt with.)
Let (pr be the angle of the triangle A(x, y, nr(x)) at the vertex nr(x) and let
ocr, βι-, Ψγ be the angles of the triangle A(y, πΓ(χ), nr(y)) at the vertices nr(x), nr(y)
and у respectively. Because d{nr{x), π,-iy)) < d(x, y) (cf. 2.4), and d(x, nr(x)) and
d(y, ^riy)) both tend to oo as r —»■ oo, the angles φτ and ^rr tend to 0 as r —>■ oo.
From (1.7) and (2.4) we have
<pr + ar > Z„rM(x, y) + Z„r(x)(y, nr(y)) > Z^(.v)(x, 7rr(y)) > 7Γ/2.
As ;вг > 7Г/2, we have limr^oo ;βΛ = lim^oo ar = я/2. And therefore b(y) =
limr^oo(d(y, nr(x)) -r) = limr^TO d(y, nr(y)) -r = My). D
8.23 Exercises
(1) Prove that the Busemann functions associated to asymptotic rays in a CAT(0)
space X differ only by an additive constant even if X is not complete. (Hint: Use
(2.15).)

Horofunctions and Busemann Functions 273
Fig. 8.4 Characterizing horofunctions
(2) Let с be a geodesic ray in E". Prove that hc(x) = — r < 0 if and only if c{r)
is the image of χ under orthogonal projection to im(c).
(3) Prove that in W the assertion of (2) is false.
(4) Let X be a complete CAT(O) space and let с : [0, oo) —»■ X be a geodesic ray.
Fix a geodesic с : К —>- Ε2 in the Euclidean plane. For every χ e X and every и > 0
consider a comparison triangle Δ(χ, c(0), с(и)) with c(0) = c(0) and с(и) = с(и).
Let c(rn) be the foot of the perpendicular from χ to c(K). Prove that the sequence r„
converges to a point r such that &с(дс) = — r.
8.24 Examples of Busemann Functions
(1) Consider W equipped with the Euclidean metric, and let с : [0, oo) —»■ W
be a geodesic ray of the form c(t) = a + te, where e is a vector of unit length and
a e M" is fixed. The associated Busemann function bc is defined by
bc(x) = (a — χ | e)
where (· | ·) is the Euclidean scalar product. Its horospheres (i.e., the sets where bc
is constant) are precisely those hyperplanes which are orthogonal to the direction
determined by e.
(2) Consider the Poincare disc model for real hyperbolic space W. The Poisson
kernel is the function Ρ : W χ ЭИ" -» К defined by P(x, ξ) = ψ^, where | · |
denotes the Euclidean norm. The Busemann function bx£ associated to the geodesic
ray с with c(0) = χ and c(oo) = ξ is
ЬхЛ = -1о£Р(х,1).
(3) Let X = Χι χ X2 be the product of two CAT(O) spaces X\ and X2. Let c\
and C2 be geodesic rays in X\ and X2 respectively, and let |i = ci(oo) e dXi and
fc = сг(оо) е ЗХ2. If θ e [0, π/2], then c(i) = (ci(icos0), C2(isin#)) is a geodesic

274 Chapter П. 8 The Boundary at Infinity of a CAT(O) Space
Fig. 8.5 Horospheres in the Poincare mode)
ray in X and c(oo) is denoted (cos6)^\ + (sin0)^2 (see (8.9(5)). If b\, bi, b are the
Busemann functions associated to c\, сг, с respectively, then
b = (cos6)bx + (sine)b2.
Indeed, since lim^ooidfo c{t)) + t)/2t = 1,
b(x) = lim (d(x, c(t)) -t)= lim —(d(x, c(t))2 - i2)
i->00 t-*0O 2ί
= lim — \d(xx, Ci(icos6>))2 - ^cos2^ + d(x2, c2(isin6l))2 - ^sin2^
ί->00 2ί
= (cos Θ) bx(x\) + (s'me)b2(x2)-
(4) Let X be the tree constructed in (7.11). The boundary dX consists of exactly
one point and (up to sign) the natural projection onto [0, oo) is a Busemann function.
Further examples of Busemann functions and horospheres are described in
(10.69).
Parabolic Isometries
Recall from (6.1) that an isometry у of a metric space X is said to be parabolic if
d(y.xo, xo) > mf{d(y.x, χ) \ χ e X] for every xq e X. In this section we consider the
action of such isometries on CAT(O) spaces X, and more particularly on X.
To say that an isometry у of a CAT(O) space X is parabolic means precisely that
if one takes a sequence of points xn e X such that the sequence of numbers dy (xn)
approximates |y|, then the sequence xn does not converge in X. But if X is proper,

Parabolic Isometries 275
then such a sequence must have a convergent subsequence in X, which is compact.
Thus one expects to find fixed points of parabolic isometries at infinity.
8.25 Proposition. Let γ be an isometry of a proper С AT(0) space X. If γ is parabolic,
then it fixes at least one point и е дХ and leaves invariant all of the horoballs centred
atu.
Proof. Consider the displacement function dy of γ. It is a γ -invariant convex function
which does not attain its infimum in X. Thus the proposition is a consequence of the
following lemma.
8.26 Lemma. Let X be a proper С AT(0) space and letf : X —>- Ж be a continuous
convex function which does not attain its infimum uelU {—oo}. If γ is an isometry
of X such thatf(y.x) = J?(x) forall χ e X, then γ fixes at least one point и е дХ, and
also fixes all of the horofunctions centred at u.
8.27 Remark. Notice that, in general, when an isometry γ fixes a point и е дХ, the
action of γ may permute the set of horofunctions centred at u. On the other hand,
if γ leaves invariant a horofunction centred at и then it must fix и and all of the
horofunctions centred at u.
Proof of Lemma 8.26 For every t > a, we let A, = {x e X \ f(x) < t]. This is a
closed, convex, non-empty subspace of X. Obviously А, с Af if t < f, and by our
hypothesis on/ we have dt>aA, = 0. Since X is proper, for every closed ball В in
X we have Β Π Α, = 0 when t is close enough to a.
Let π, be the projection of X onto A, and let/, be the function χ м> d(x, π,{χ)).
Choose a base point jeq € X, and let h,(x) = f(x) —f(xo). We claim that we can
find a sequence of numbers t„ tending to a such that the corresponding sequence of
functions hhi : χ ы> ft„(x) — /r„(*o) converges uniformly on compact sets to some
function h. Indeed, the family of functions h, : χ м> f(x) — fi(xo) is uniformly
bounded on balls and equicontinuous because \h,(x) — h,(y)\ < d(x, y) (8.22), and
hence, since X is proper, we can apply the Arzela-Ascoli theorem (1.3.9) to obtain
the desired convergent sequence of functions htii. By Corollary 2.5, the limit h of the
sequence of functions (htu) satisfies the list of conditions preceding (8.22) and hence
h is a horofunction. Since each of the functions h,i: is invariant by y, so too is h. G
The following example shows that (8.25) does not remain true without the
hypothesis that X is proper.
8.28 Example. Let Η be the Hilbert space £2(Z), as defined in Chapter 1.4. (We
write points in Η as χ = (xn).) The boundary at infinity dH can be identified with
the set of equivalence classes of Η \ {0} modulo the relation generated by scalar
multiplication by positive numbers: χ ~ у if there exists λ > 0 such that xn = Xyn
for all η e Z.

276 Chapter II.8 The Boundary at Infinity of а С AT(0) Space
Translations of Η act trivially on dH. One example of an isometry that does not
act trivially on dH is the shift map σ. By definition, σ.χ is the sequence whose и-th
entry is xn+i. The only element of Η fixed by σ is 0, because if x„ = xn+\ φ 0 for
all η e Z, then (jc„)„ez is not an element of H. Let у be the isometry χ м> σ.χ + 5,
where 5 is the sequence whose only non-zero entry is So = 1. Arguing as above, one
sees that у does not fix any point in H, so it is not elliptic. But у does not fix any
point in dH either, so it can't be hyperbolic because a hyperbolic isometry must fix at
least two points at infinity, namely the endpoints of its axes. Hence у is a parabolic
isometry that does not fix any point of dH.
8.29 Exercise. Calculate the translation length | γ | of the isometry у described in the
preceding example, and prove directly that dy does not attain this value.
8.30 Example. Let X = Χι χ Χ2 be the product of two complete CAT(O) spaces. Let
у be an isometry of X that splits as у = (уь γι). If у, fixes ξι e ЭХ,, for i = 1, 2,
then у fixes the whole segment (cos0)|i + (sin#)f2 e ЗХ for θ e [0,7г/2] (in
the notation of 8.11(6)). If γ\ is parabolic and preserves the horofunctions of X\
centered at |i, then у is parabolic (6.9), but it preserves the horofunction centred at
(cos θ)ξι + (sin θ)ξ2 if and only if θ = 0 or if γι preserves the horofunctions centred

Chapter II.9 The Tits Metric and Visibility Spaces
Let X be a complete С AT(0) space. In the preceding chapter we constructed a
boundary at infinity dX and studied the cone topology on it. (This topology makes dX
compact if X is proper.) ЭН" and ЭЕ" are homeomorphic in the cone topology, but
at an intuitive level they appear quite different when viewed from within the space.
Consider how the apparent distance between two points at infinity changes as one
moves around in E" and Й": in E" the angle subtended at the eye of an observer
by the geodesic rays going to two fixed points at infinity does not depend on where
the observer is standing; in W the angle depends very much on where the observer
is standing, and by standing in the right place he can make the angle π. Thus by
recording the view of dX from various points inside X one obtains a metric structure
that discriminates between the boundaries of E" and Ш". In this chapter we shall
consider the same metric structure in the context of complete CAT(O) spaces.
More precisely, we shall define an angular metric on the boundary of a complete
CAT(O) space X. The associated length metric is called the Tits metric, and the
corresponding length space is called the Tits boundary ofX, written djX- The natural
map djX —*■ dX is continuous (9.7), but in general it is not a homeomorphism.
37-И" is a discrete space. In contrast, 3rE" is isometric to S""1 and the natural map
Э7-Е" —>- ЭЕ" is a homeomorphism.
This contrast in behaviour is indicative of the fact that the Tits boundary encodes
the geometry of flats in a complete CAT(O) space X (see (9.21)). Moreover, if X has
the geodesic extension property, then the existence of product compositions for X is
determined by dTX (see (9.24)).
Ш" is a visibility space (9.28), which means that every pair of distinct points
in ЗИ" can be connected by a geodesic line in W. The concept of visibility is
closely related to Gromov's 5-hyperbolic condition (see III.H). In the final section
of this chapter we shall prove that a complete cocompact CAT(O) space X contains
an isometrically embedded copy of E2 if and only if X is not a visibility space (The
Hat Plane Theorem (9.33)).

278 Chapter II.9 The Tits Metric and Visibility Spaces
Angles in X
Most of the results in this chapter concern the behaviour of angles in the bordification
of a complete CAT(O) space.
9.1 Definition. Let X be a complete CAT(O) space. Given x, у e X and ξ, η e ЭХ, we
shall use the symbol Zx (ξ, η) to denote the angle at χ between the unique geodesic rays
which issue from χ and lie in the classes ξ and η respectively, and we write Zx(y, ξ)
to denote the angle at χ between the geodesic segment [x, y] and the geodesic ray
that issues from χ and is in the class ξ.
The continuity properties of angles in X are much the same as those of angles in
X (see (3.3)).
9.2 Proposition (Continuity Properties of Angles). Let X be a complete metric
CAT(O) space.
(1) For fixed ρ e X, the function (x, x1) ы> Zp(x, x1), which takes values in [0, π],
is continuous at all points (x, x1) e Χ χ X with χφρ and χ1 φ p.
(2) The function (p, x, x1) ы> Zp(x, x1) is upper semicontinuous at points (p, x, xf) e
Χ χ Χ χ X with χφρ and χ1 φρ.
Proof. Suppose xn —>- χ e X, xfn —»■ xf e X and p„ —»■ ρ e X, where χφρ and xf φ
ρ'. Let с and с' be the geodesic paths or rays joining/? to χ and xf respectively, and let c„
and c'n be the geodesic paths or rays joining/?,, to x„ and xfn respectively. We fix t > 0 so
that for all sufficiently large n the points с (ί) and c'n (t) are defined and different from
x„ and x!n. By definition ZPa(c„(t), c'„(t)) = ZPn{xn,xfn) and Zp(c(t), c(t)) = Zp(x, x).
And from the convexity of the metric on X (2.2) and the definition of the topology
on X we have c„(t) —»■ c(t) and c'„(t) —»■ c'(i)·
Thus, by replacing x„ and jcJ, by c„(t) and c^(i). we can reduce the present
proposition to the corresponding properties for angles in X (see (3.3)). G
With the notion of asymptotic rays in hand, one can consider triangles which
have vertices in X rather than just X. Many basic facts about triangles in CAT(O)
spaces can be extended to this context. The following proposition provides a useful
illustration of this fact (cf. (2.9)).
9.3 Proposition (Triangles with One Vertex at Infinity). Let A be a triangle in
a CAT(O) space X with one vertex at infinity; thus Δ consists of two asymptotic
rays, с and c' say, together with the geodesic segment joining c(0) to c'(0). Let
x = c(0), x1 = c'(0) and ξ = c(oo) = c'(oo). Let γ = Ζχ(χ/, ξ) and γ' = Ζχ·(χ, ξ).
Then:
(1) γ + γ'<π;
(2) γ + γ' = π if and only if the convex hull of A is isometric to the convex hull
of a triangle in E2 with one vertex at infinity and interior angles γ and γ'.

The Angular Metric 279
^ (i) (ii)
Fig. 9.1 (i) A triangle in E2 with one vertex at infinity, (ii) A triangle in H2 with two vertices
at infinity
Proof. Let Δ, denote the geodesic triangle in X with vertices x, x1 and c'{t). Let γ,
and y/ be the angles at the vertices χ and χ'. Let Υ and Y' be the corresponding angles
in a comparison triangle Δ,. Let a, denote the angle at χ between с and the geodesic
segment joining χ to c'(t).
With this notation, γ < γ, + α,, γ, < γ, γ' = у/ < Υ' and Υ + Υ' < π. Hence,
У + у' < Yt + <*t + у/ < Yt + Yt + a, <π +at.
But it follows from (8.3) that lim^oo a, = 0, hence у + у' <π.
We also have that lim,^^ γ, = γ because, by the triangle inequality for angles
(1.1.14), \y,-at\ <y <y,+at. _
Suppose now that у + y' = π. We claim that y' = y/ = y/ for all t > 0.
Since y/ > y/ = y' and Y' is a non-decreasing function of i, it suffices to show that
lim^oo Y' < γ'. But у + y/ < yt +Y' <Y + Y' < я, so passing to the limit as
t —>- oo, we get π = γ + γ' < γ + lim,^^ Υ' <π. Thus Υ' = у/, and (2.9) implies
that the triangles Δ, are flat for all t > 0.
Let Δ be a Euclidean triangle with one vertex at infinity, consisting of two
asymptotic geodesic rays с and c' issuing from points χ and x1 which are a distance
d(x, x1) apart, such that the angles of Δ at 3c and x1 are у and y' respectively. For
all t > 0, the Euclidean triangle Δ, = A(x, x', c'(0) is_jf comparison triangle for
Δ,. As t varies, the isometries from the convex hull of Δ, onto the convex hull of
Δ,, establishedjn the previous paragraph fit together, to give an isometry from the
convex hull of Δ onto the convex hull of Δ. G
The Angular Metric
Intuitively speaking, in order to obtain a true measure of the separation of points at
infinity in a CAT(O) space, one should view them from all points of the space. This
motivates the following definition.

280 Chapter Π.9 The Tits Metric and Visibility Spaces
9.4 Definition. Let X be a complete CAT(O) space. The angle Ζ(ξ,η) between
ξ, η e ЭХ is defined to be:
Z(|, 7j) = supZ^(|,7j).
xeX
9.5 Proposition (Properties of the Angular Metric). Let X be a complete CAT(O)
space.
(1) The function (ξ, ξ') ы> Ζ(ξ, ξ') defines a metric on дХ called the angular
metric. The extension to дХ of any isometry ofX is an isometry ofdX with the
angular metric. (We show in (9.9) that this metric is complete.)
(2) The function (ξ, ξ') ы> Ζ(ξ, ξ') is lower semicontinuous with respect to the
cone topology: for every ε > 0, there exist neighbourhoods Uof% and U' of
ξ' such that Ζ(η, η') > Ζ(ξ, ξ') - ε for all η e U and η' e U'.
(3) IfXis proper and cocompact then for all ξ, ξ' e дХ there exist у е X and
η, η' e дХ such that Ζ?(η, η') = Ζ(|, ξ').
Proof. (1) The triangle inequality for Ζ follows from the triangle inequality for
Alexandrov angles (1.1.13). We have to check that if ξ φ ξ' then Z(|, ξ') > 0. Let с
and с' be geodesic rays issuing from the same point χ e X and such that c(oo) = |
and c'(oo) = |'. We choose t > 0 large enough to ensure that d(c(t), c'(t)) > 0. Let
c" be the geodesic ray issuing from c(t) that is in the class ξ'. If ZC(f)(|, £') were zero,
then ZC(t)(x, ξ') would be π, and the concatenation of [x, c(t)] and c"|[,,«,) would be
a geodesic ray distinct from c', contradicting the fact that d is the only geodesic ray
in the class ξ' that issues from x.
(2) For fixed ρ e X, the function (ξ, ξ') м> Ζρ(ξ, ξ') is continuous (9.2). The
supremum of a bounded family of continuous functions is lower semicontinuous.
(3) Choose a sequence of points xn in X such that lim^oo ΖΧα(ξ, ξ') = Ζ(ξ, ξ').
As X is assumed to be cocompact, there exists a compact subset С С X and a sequence
of isometries y„ e Isom(X) such that y„.jc„ = yn e C. Clearly Zyn(yn.%, γη.ξ'„) =
■£t„(t· I') ar,d АУл-!> Ул·!') = Z(|, I')· Passing to a subsequence we may assume
that yn converges to a point у e C. Furthermore, since we are assuming that X is
proper, дХ is compact, so by passing to a further subsequence we may assume that
γ„.ξ and γ„.ξ' converge in ЭХ, to points η and η' say. The lower semicontinuity of
^-i.Yn-%, Yn-H') yields the third inequality given below, and the upper semicontinuity
of Ζγη(γ„.ξ, Yn-ξ') (see (9.2)) yields the first (the remaining inequality and the two
equalities hold by definition):
Ζ(ξ, ξ') = lim ZySYn-ξ, Yn-ξ') < Z,(»7, η') < Ζ(η, η')
п-юо
<ΙιπιΜΖ(γη.ξ,γη.ξ') = Ζ(ξ, ξ').
Equality holds throughout.
D

The Angular Metric 281
9.6 Examples
(1) If X = E" then Zp(|, ξ') = Z9(|, £') for all p, q e E" and all ξ, ξ' e ЭХ,
hence Zp(|, f') = Z(|, f')· I£ follows that the map which associates to each ξ e dX
the point of intersection of [0, f ] with the unit sphere about 0 € E" gives an isometry
from3E"toSn-1.
(2) If X is real hyperbolic space W, then Z(|, f') = я for any distinct points
|, f' € ЭН", because there is geodesic line с : К —»■ И" such c(oo) = ξ and
c(—oo) = I'. We shall see later that the same is true for any complete CAT(— 1)
space X(cf. 9.13).
(3) If X is the Euclidean cone CqY on a metric space Y, then ЭХ with its angular
metric is isometric to Υ with the truncated metric max{d, π]. (This generalizes (1).)
Our next goal is to prove:
9.7 Proposition. Let X be a complete CAT(O) space.
(1) The identity map from ЭХ equipped with the angular metric to ЭХ with the cone
topology is continuous. (In general it is not a homeomorphism, cf. 9.6(2)).
(2) ЭХ with the angular metric is a complete metric space.
We postpone the proof of this proposition for a moment, because it will be
clearer if we first articulate a reformulation of the definition of Z. In fact we take
this opportunity to list several such reformulations. (The proof of their equivalence
intertwines, so it is natural to prove them all at once.) The list of corollaries given
below indicates the utility of these different descriptions of Z. Recall that for x, у € X
distinct from/? e X, the symbol Zp(jc, y) denotes the angle at the vertex corresponding
to ρ in a comparison triangle A(p, x, у) с Е2.
9.8 Proposition. Let X be a complete С AT(0) space with basepointxo. Let ξ, ξ' e ЭХ
and let c, d be geodesic rays with c(0) = d(0) = xq, c(oo) = ξ and ί/(οο) = ξ'.
Then:
(1) Z(f, ξ') = linv^oo Z4(c(f), c'(f)) = sup{Z,0(c(i), с'(0) 11, f > 0}.
(2) The function t ы> ZC(,)(|, ξ') is non-decreasing and
Z(f, Г) = Hm Ζφ)(ξ, Г).
(3) The function
(t, f) m> [π - Zc(()(jco, c'(i')) - ^ю<Л, c(i)]
is non-decreasing in both t and f, and its limit as t, f —> oo is Z(|, £')·
(4)
2sin(Z(!,!')/2) = lim -d{c{t),c'{f)).

282 Chapter Π.9 The Tits Metric and Visibility Spaces
Proof. We begin with the characterization of Ζ(ξ,ξ') given in (1) and explain why
this implies (4). Whereas the Alexandrov angle between the rays с and d is defined
to be
^o(M0 = /»noZ|o(c(i),c'(i')),
we now wish to consider the limit as t, f —> oo,
ΖΧο(ξ,ξ'):= Jim Z,0(c(i),c'(i))-
This limit exists because (t, f) м> Zxo(c(t), d(f)) is a non-decreasing function of
both t and f (see (3.1)) and it is bounded above by π. And 2 sin(ZXo(c(t), c'{t))/2) =
d{c{t), d(t))/t, so we have
2 sin(Z,0(f, f')/2) = urn - d(c(t), d{t)).
t-*oo t
This formula shows that (4) is a consequence of (1). It also shows that ΖΧο(ξ, ξ') is
independent of the choice of basepoint x0, for if с and c' are geodesic rays issuing
from some other point in X and c(oo) = ξ and c'(oo) = ξ', then there is a constant #
such that d(c(t), c{t)) + d(d(t), c'{t)) < К for all t > 0, so by the triangle inequality
Нт^схМф), c'(t))/t = Цт^ооДОф, с'(0)Л·
_ Let Z(£, £')_:= A,(f - Г)· Note that Z(f, £') < Д£, £'). because ΖΛο(£, ξ') <
Ζ^ξ, ξ') and Ζχο(ξ, f') does not depend on *o· Thus, in order to prove (1) (and
hence (4)) it suffices to show that Ζ(ξ, ξ') < Ζ(ξ, ξ'). The proof of this inequality
intertwines naturally with the proofs of (2) and (3).
We use the following notation (see figure 9.2):
a, = Ζφ)(χ0, c'(t% a', =Zd(l>)(x0, c(t)), γ, = Ζφ)(ξ, c'(i'))
у', = Ζ^η(ξ', φ)), φ, =Zc(/)(f, Г), Ψ, = ^c(,)(f - c'(i'))-
The function t м> φ, is non-decreasing because for i > t we have
φ, + Ζφ)(χ0, ξ') < π < Ζφ)(χ0, ξ') + φ,,
where the first inequality comes from (9.3) and the second is the triangle inequality
for angles together with the fact that the angle at c(s) between the incoming and
outgoing germs of the geodesic с is it. This proves the first assertion in (2). The
proof that the function described in (3) is non-decreasing is similar.
In the same way as we obtained the displayed inequality above, we get γ, <
φ, + ψ,, α, + γ, > π, at> + γ,> > π and ψ, + γ,, < π, hence
Φ, > (π- - α, - α',).
And if we let a, = Ζφ){χΰ, d(f)) and a', = ZdV){xo, c(t)), then by the CAT(O)
inequality we get
(л-a,- a',) > (π - a, - a',) = 7Xo(c(t), c'(f)).

The Angular Metric 283
£
at\Ut
xo
Fig. 9.2 Characterizations of Ζ(ξ, ξ')
Therefore,
7(ξ, Г) > AH, Η') > Φ, > (* - α, - α',) > 7Χ0(φ), c(t')).
By definition lim,/-^ Zxa{c{f), с'(0) = Ζ(ξ, ξ'), so passing to the limit as t, f -»
oo, all of the above inequalities become equalities. This proves (1), (2), (3) and (4).
D
Proof of (9.7). We first prove (1). Fix x0 e X and ξ e dX and let с be the geodesic
ray with c(oo) = ξ that issues from xq. Consider a basic neighbourhood U(c, r, ε)
of ξ in the cone topology (see (8.7)). Let 5 = arcsin(e/(2r)). Given |' e dX with
Ζ(ξ, ξ') < δ, let с' be the geodesic ray with c'(0) = xo and c'(oo) = ξ'. From part (1)
of the above proposition we have ZXo(c(r), c'{r)) < 8 and hence d(c(r), c'(r)) < ε.
Thus Ζ(ξ, ξ') < δ implies ξ' e U(c, r, ε).
The same argument shows that if ξη is a Cauchy sequence for the angular metric
on dX, and c„ is the geodesic ray with c„(0) — xq and c„(oo) = ξη, then for every
r > 0 the sequence c„(r) is Cauchy and hence convergent in X. Let с : [0, oo) —»
X be the limit ray. Given ε > 0, since |„ is Cauchy, we can apply part (1) of
the preceding proposition to deduce that for sufficiently large m and η we have
sup{ZX0(cm(t), сп(0) | t, f > 0} < ε. Fixing η and letting m->oowe get:
Ζ(ξ, ξη) = sup{7^0(c(i), cntf)) | t, t' > 0} < ε.
Thus !„-►£. D
9.9 Corollary (Flat Sectors). Let X be a complete CAT(O) space. If for some point
xo e X we have Ζχο(ξ, ξ') = Ζ(ξ, ξ') < it, then the convex hull of the geodesic rays
с and d issuing from x0 with c(oo) = ξ and (/(oo) = |' is isometric to a sector in
the Euclidean plane E2 bounded by two rays which meet at an angle Ζ(ξ, ξ').
Proof. The function t м> Ζφ)(ξ, ξ') is a non-decreasing function of t, its limit as
t -> 0 is ΖΧο(ξ, ξ') and by part (2) of the proposition, its limit as t -> oo is Z(f, f ')·
If Z.t0(£, ξ') = Z(f, f') then this function must be constant, so we may apply the

284 Chapter Π.9 The Tits Metric and Visibility Spaces
Flat Triangle Lemma (2.9). (The hypothesis Z(f, f) < π is needed to ensure that
the comparison triangles A(x0, c(t), d{t)) are non-degenerate.) G
9.10 Corollary. Let X be a complete CAT(O) space and let с, с' be geodesic rays
issuing from xq in the direction of ξ, ξ' e ЭХ respectively. For any real numbers
a,a'>0 we have
lim -d(c(at), c'(a't)) = (a2 + a'2 - lad cos Z(|, ξ'))ι/2.
/->oo t
In other words, this limit exists and is equal to the length of the third side of a
Euclidean triangle whose other sides have lengths a and a' and meet at an angle
Proof. The Euclidean law of cosines gives
d(c(at), c\a't))2 = a2? + a'V - laa'i1 cos (Zio(c(ai). c'{dt))).
To obtain the desired formula, we divide by t2, let t —» oo and appeal to (9.8(1)). G
9.11 Corollary. Let X\ and X2 be two complete CAT(O) spaces. Then д(Х\ х Хг)
with the angular metric Ζ is isometric to the spherical join ЬХ\ * ЭХ2 of(dX\, Z) and
(8X2, Z). More specifically (in the notation of 1.5), given ξ = (cos θ ξι + sin θ ξ2)
and ξ' = (cos θ' К + sin θ' ξ2) in д(Х{ χ Χ2), we have
cos(Z(|, I')) = cos0 cos0'cos(Z(|b !{)) + sin0 sin0'cos(Z(f2, li»·
Proof. Choose basepoints x\ e X\ and x2 e X2 and let c\, сг, с\, c'2 be geodesic
rays issuing from these points in the classes l1.f2.l1M2 respectively. c(t) —
(ci(icos0), C2(isin0)) and c'(t) = (c',(icos0'), c2(isin0')) are geodesic rays in
Χι χ X2 with c(oo) = I and c'(oo) = f and c(0) = c'(0) = (χι, x2)- The
previous corollary implies:
(2-2cos(Z(f,r))
= lim (d(c(t), с (ή)It)2
= lim(rf(ci(icos0), c'At cos θ'))/ή2 + lim (d(c2(t sin θ), c'M sin θ'))/t)2
= cos2 0 + cos2 0' - 2cos0 cos0' cos Z(|i, £,')
+ sin2 0 + sin2 0' - 2 sin0 sin0' cos Ζ(|2, Φ
= 2 - 2 cos0 cos 0' cosZ(|b ξ[) - 2 sin0 sin 0' cos Z(|2,ξ2),
as required. Q
Later (9.24) we shall prove that the above result admits a converse in the case
where geodesies can be extended indefinitely. In other words, if X has the geodesic

The Boundary (ΘΧ, Ζ) is a CAT(l) Space 285
extension property then one can detect whether or not X is a product by examining
the geometry of (ЭХ, Ζ).
9.12 Examples
(1) If X is a complete CAT(— 1) then Ζ(ξ, ψ) = π for all pairs of distinct points
ξ, ξ' e ЭХ. To see this, we fix xq e X such that ΖΧο(ξ, ξ') > 0 and consider с, с'
with c(0) = c'(0) — xq and c(oo) = f, c'(oo) = f'. Let m, denote the midpoint of
[c(t), c'(t)]. For a comparison triangle A(x0, c(t), c'(t)) in H2 we have
4*W0, ?W) > ^(c(f), c'(0) - A„(f, Г) > 0.
It follows that d(xo, m,) is bounded above by some constant A. Hence,
2i = d(x0, c{t)) + d(x0, c'{t)) < 2d(x0, m,) + d(mt, c{t)) + d(mt, c'(t))
= 2d(x0,m,) + d(c(t),c'(t)) < 2A + 2t.
Dividing by t and let t -» oo, we get that lim,^oo d{c{t), c'{f))/t ■= 2, and conclude
from (9.8(4)) that Ζ(ξ, ξ') = π.
(2) Let Υ — R χ X. By (9.11), the angular metric on Э Υ makes it isometric to the
suspension of (X, Z). The case X — И2 is described in (4). Consider the case where
X is a regular metric tree. In this case Э Υ with the cone topology is metrizable as
the suspension of the standard Cantor set, and (Э Υ, Ζ) is the length space associated
to this metric — the metric topology and length-space topology are different in this
case.
(3) If Υ = Ш2 χ И2, then (dY, Z) is the spherical join of two uncountable spaces
in each of which all distinct points are a distance π apart.
(4) If Υ = Ε" χ И2, then (dY, Z) is the geodesic space obtained by taking
uncountably many copies of S"_1 and identifying their north poles and south poles.,
(Э Y, Z) would remain unchanged if we were to replace H2 by the universal cover
of any compact space X of strictly negative curvature with the geodesic extension
property (other than a circle).
The Boundary (ΘΧ, Ζ) is a CAT(l) Space
The following fundamental property of the angular metrics was proved by Gromov
in the case of Hadamard manifolds [BaGS87].
9.13 Theorem. IfX is a complete CAT(O) space, then ЭХ with the angular metric is
a complete CAT(l) space.
First we prove that ЭХ is a 7r-geodesic space. This follows from:

286 Chapter Π.9 The Tits Metric and Visibility Spaces
9.14 Proposition. Let X be a complete CAT(O) space. Given%, η e bXwithZ{%, η) <
it, there exists μ e dX such that Z{%, μ) = Ζ(η, μ) = Ζ(ξ, η)/2.
We follow a proof of В. Kleiner and B. Leeb ([K1L97]); this requires two lemmas.
9.15 Lemma. Let X be а С AT(0) space. Fixx,y eX with d(x, y) = 2r and let m be
the midpoint of the geodesic segment [x,y]. Ifm! is a point such that d(m' ,x) < r( 1 +ε)
and d(m', y) < r{\ + ε), then d(m, m!) < r(s2 + 2ε)ι/2.
Proof. At least one of the angles Zm(m', x) and Zm(m', y) is greater than or equal to
7Г/2, let us say Zm(m', χ) > π/2. In the comparison triangle A(m, т', x), the angle
at m is also greater than or equal to π/2 (1.7), hence τ^(1 + ε)2 > d(m',x)2 >
r2 + d(m, m')2. Therefore d(m, m')2 < r2^2 + 2ε). Ώ
9.16 Lemma. Let X be a complete CAT(O) space with basepoint xq. Let x„ and y„
be sequences of points in X converging to points ξ and η ofdX in the cone topology.
Then,
liminfZ^0(x„,}'n) > Ζ(ξ, η).
n—*co
Proof. By hypothesis the sequences of geodesic segments [xq, x„] and [xq, y„]
converged the geodesic rays [xo, ξ] and [xo, η] respectively. Define^ e [xq, ξ] to be the
point furthest from xq such that d{x'n, [xo, ξ]) < l.Letx^'be the orthogonal projection
of xfn to [xo, xn]. Let y'n and y"n be defined similarly. Because x„ —» ξ and y„ —> η,
we have d(x0, x!n), dfo, y'n) -> oo. It follows_from 9.8(1) that limn-*cx, Z^x1^ y'n) —
Ζ(ξ, η). But lim,MOO ZXa{x!n, y'n) = lim,MOO Ζ^(χ"η, yl) and by the CAT(O) inequality
Ао(*л> Уп) > Ao«, y'n)- Therefore liminf,^^^, yn) > Ζ(ξ, η). Π
Proof of 9.14. Choose a basepoint xq e X and let t м> xt and t н> у, be geodesic
rays issuing from xq in the classes ξ and η respectively. Let m, be the midpoint of
the geodesic segment [x,, yt]. We wish to show that the geodesic segments [xo, m,]
converge to a geodesic ray [xo, μ] as t tends to oo. This will prove the
proposition, because liminf, ZXo(x,, mt) > Ζ(ξ, μ) and liminf, ZXo(y,, mt) > Ζ(η, μ) by
(9.16), and lim, Zxo(x,, y,) ■= Ζ(ξ, η) by (9.8(1)), so letting t -> oo in Zxo(xt, m,) =
^(y„ m,) < (\/2)ZX0(x„yt), we get Ζ(ξ, μ) = Ζ(η, μ) < (1/2)Ζ(ξ, η), and the
triangle inequality implies that in fact we have equality.
It remains to show that the geodesic segments [xo, mt] converge to a geodesic ray
as t —> oo. Since X is complete, it is sufficient to prove that for all R > 0 and ε > 0
the geodesic segments [*o, mt] and [xo, ms] intersect the sphere of radius R about x0 in
points a distance at most ε apart whenever t and s are sufficiently large. Note first that
d(xo, mt) -» oo as t -> oo, because by (9.8(4)) lim,^oo d(xt, m,)/t ■= sin(Z(f, η)/2),
which by hypothesis is less than 1, and by the triangle inequality t < d(x,, m,) +
d(xo, m,), so dividing by t and letting t —> oo we have lim,^oo d(x0, mt)/t > 1 —
sin(Z(f, η)/2) > 0.

The Boundary φΧ, Ζ) is a CAT(l) Space 287
Given s > t we write (t/s) ms to denote the point on the geodesic segment [xo, ms]
that is distance (t/s) d(xo, ms) fromxo· Applying the CAT(O) inequality to the triangle
A(x0,xs,ms) we get
t t
d(x„ (t/s) ms) < - d(xs, ms) - — d(xs, ys),
s 2s
and from Δ(χο, ys, ms) we get
t t
d(yt, (t/s) ms) < - d(ys, ms) - — d(xs, ys).
s 2s
Therefore
d(x,, y,) < d(x„ (t/s) ms) + d(y,, (t/s) ms) < (t/s) d(xs, ys),
for all s > t. If we divide all of these inequalities by d(x,, y,) and pass to the limit
as t —> oo, they all become equalities because s > t and \ims-J>0Od(xs,ys)/s —
lim^oo d(xt, y,)/t = sin(Z(f, η)/2). Thus
d(x„(t/s)ms)
lim — — =1 and
j,/-*c» d(xt, m,)
d(y,,(t/s)ms)
llm —τ, τ- =L
s,t->-oo d{y,,m,)
Using (9.15) we conclude that lim,,*-^ d(mJl^i"'s) = 0. But we proved in the
second paragraph that lim,^oo d(x0, m,)/t > 0, hence
d(m„(t/s)ms)
hm = 0
i,/->oo d(xQ, m,)
and therefore
lim ZXo(m„(t/s)ms) = 0.
t,s-+co
Given a positive number R and t > R we write p, for the point of [xq, m,] a
distance R from xq. The CAT(O) inequality for Δ(χ0, m,, (t/s) ms) gives
d(p,, ps) < 2Rsin((l/2)7Xo(mt, (t/s)ms),
which we have shown converges to 0 as s, t —» oo. Thus the geodesic segments
[xq, m,] converge to a geodesic ray [xq, μ]. D
9.17 Remark. There is an easier proof of (9.14) in the case where X is proper: as in
the second paragraph of the proof given above one sees that the set of midpoints m,
is unbounded, then using the properness of X one can extract a sequence of points
mt(n) from this set so that [xo, mi(n)] converges to a geodesic ray. The argument of the
first paragraph can then be applied.

288 Chapter II.9 The Tits Metric and Visibility Spaces
Proof of Theorem 9.13. We showed in (9.7) that (ЭХ, Ζ) is a complete metric space,
so the existence of midpoints (9.14) implies that it is a 7r-geodesic space. It remains
to show that triangles satisfy the CAT(l) condition.
Consider ξ0, ξι,ξ2, μ e ЭХ with Ζ(ξ0, ξ{) + Ζ(ξ{,_ξ2) + Α&. £ο) < 2ττ and
Ζ(ξ0,μ) = Ζ(£ι,μ) = (1/2)Ζ(ξ0,ξι), let Atf0,JbJ2) C S2 be a comparison
triangle and let μ be the midpoint of the segment [ξ0, ξ ι ]. We must show that
Ζ(ξ2, μ) < d(f2,JC).
Let co,c\,C2,c' be geodesic rays issuing from a fixed point ξο e X with
c0(oo) = |o,ci(oo) = |i,c2(oo) = |2.c'(oo) = μ, and let α = ΑΙο,Ιι) and
α = cos(a/2) > 0. Let m, be the midpoint of the segment [co(i)> ci(i)]· We claim
that lim,^oo d(m„ c'{at))/t = 0. For i — 0, 1, by (9.8(4)) we have
lim - d(a(t), mt) — lim — d(co(t), c\(t)) — sin(a/2),
/->oo ί ί->οο 2ί
and by (9.10)
1
lim - d(d(t), с (at)) = sm(a/2).
Hence lim^oc Д^» - 1, so by (9.15)
(1) lim d(mt, c'(at))/t = 0.
Consider S2 as the unit sphere in K3. Let m denote the midpoint of the Euclidean
segment [|o. Ill- The spherical distance d(%2,m) is equal to the vertex angle at
0 € R3 in the Euclidean triangle Δ(0, ξ2, Щ. The law of cosines applied to this
triangle yields:
(2) ||F2 - mf = a2 + 1 - 2a cos(rf(F2, μ)).
We wish to compare this with the fact that (cf. (9.10))
(3) lim -r d(c2(t), с {at))2 = a2 + 1 - 2a cos(Z(|2, μ))·
By the CAT(0) inequality,
d(c2(t), m,)2 < -d(c2(t), c0(t))2 + - d(c2(t), cj(i))2 - - d(c0(t), cx{t))2.
Combining this with (1), we get:
lim - d(c2(t), c'(at))2
< lim —r d(c2(t), c0(t))2 + lim —r d(c2(t), c,(i))2 - Hm —r d(c0(t), cx{t))2
t-+oo Ltl t-+oo 2tL '->°° At1
= \\\h - Foil2 + \\\h - F, II2 - ^llFi - Foil2

The Tits Metric 289
This inequality allows us to compare (2) and (3) and hence deduce that
Ζ(ξ2,μ)<ά(ξ2,μ)-
Ώ
The Tits Metric
For many purposes it is convenient to replace the angular metric Ζ on the boundary
ЭХ of a complete CAT(O) space by the corresponding length metric. This metric is
named in honour of Jacques Tits.
9.18 Definition. Let X be a CAT(O) space. The Tits metric on dX is the length metric
associated to the angular metric; it is denoted dT. The length space (ЭХ, dT) is called
the Tits boundary of X, and is denoted Э7Х.
9.19 Remarks
(1) If ξ and η are points of dX which cannot be joined by a path which is rectifiable
in the angular metric, then άτ(ξ, η) = oo. In particular, if X is a CAT(—1) space,
then the Tits distance of any two distinct points in dX is infinite (see 9.12(1)).
(2) It follows from Theorem 9.14 that two points of dX which are a distance less
than π apart are joined by a (unique) geodesic in (ЭХ, Ζ), so if Ζ(ξ, η) < π then
άτ(ξ, η) — Ζ(ξ, η). In particular the identity map from (ЭХ, Ζ) to (ЭХ, dT) is a local
isometry and (ЭХ, dT) is a CAT(l) metric space. If X is proper, then one can say
more:
9.20 Theorem. If X is a complete CAT(O) space, then dXj is a complete CAT(l)
space. Moreover, ifX is proper then any two points ξο,ξ\ e ЭХ such that άτ(ξο, ξι) <
oo are joined by a geodesic segment in Э7Х.
Proof. Asdy(fo, fi) < 00, there is a sequence of continuous paths p„ : [0, 1] —> ЭХ,
parameterized proportional to arc length, which join |0 to |i and whose Tits length
tends to άτ(ξο, ξι)- We fix N so that for all n sufficiently large dT(p„(k/N), p„((k +
1)/Л0) < тг/2 for к =. 0,..., Ν — 1. As ЭХ with the cone topology is compact,
we may pass to a subsequence and assume that each of the sequences pn(k/N)
converges in the visual topology, to a point μ!ζ say. As dT(pn(k/N),p„((k + \)/N)) =
Z(p„(k/N,p„((k + \)/N), using the upper semicontinuity of the angular
metric (9.2), we have Ζ(μ*, μ*+ι) < lim,,-^ Z(p„(k/N),p„((k + l)/N) which is
limn-+oo dT(pn(k/N), pn((k + l)/N)) < π/2. Let ck be a geodesic segment (which
exists by 9.14) joining μλ Юд^+ь for A: = 0,... ,N — 1, and let с be the path joining
ξο to ξ ι which is the concatenation of cq, ..., c^-i. Then,

290 Chapter II.9 The Tits Metric and Visibility Spaces
N-l
dT(lo, ξι) < кт{с) = J2ldT(ck)
N-l
- У2 Z^b ^k+^ - lim ldr(Pn) < ΊΤ{ξΰ, ξΐ),
k=0
showing that с is a geodesic segment joining ξο to f ι. G
9.21 Proposition. Let X be a proper CAT(O) space and let ξο and ξι be distinct points
of ax.
(1) If άτ(ξ0, ξι) > it, then there is α geodesic с : К —> X with c(oo) = ξ0 and
c(oo) = ξι.
(2) If there is no geodesic с : К —> Χ mi/г c(oo) = fo аи^ c(—oo) = |i, ί/геи
^r(|o. ξι) — ^(ξο> ξι)αηα* there is a geodesic segment in (ЭХ, dT) joining ξ0 to
ξι-
(3) If с : К —> X is α geodesic, then dT(c(—oo), c(oo)) > я, wii/г equality if and
only ifc(R) bounds aflat half-plane.
(4) If the diameter of the Tits boundary djX is π, then every geodesic line in X
bounds aflat half-plane.
We need a lemma.
9.22 Lemma. Let Xbea complete CAT(O) space, and let (x„) and {x!n) be sequences
in X converging to distinct points ξ and ξ'ofдX. Suppose that each of the geodesic
segments [xn, x?n] meets a fixed compact set К с X. Then, there exists a geodesic
с : Μ —> Xsuchthatc{oo) = ξ andc{—oo) — ξ'. Moreover, the image of с intersects
K.
Proof. For every и e N we fix a point pn e [xn, xfn] Π Κ. By passing to a subsequence
if necessary, we may assume that the sequence pn converges to a point ρ e K. Let
cp and c' be the geodesic rays issuing from ρ which are asymptotic to ξ and ξ'
respectively. We claim that the map с : К -> X defined by c(t) = cp(t) for t > 0
and c(i) = c'A—t) for ί < 0 is a geodesic. Indeed, given у e cp(M+) and / e c'p(R+)
there exist sequences of points yn e [p„, x„] and )/ e [p„, xfn] such that y„■—> у and
)/ -> У as η -> oo; since <5?Суп,У„) = d{yn,pn) + d(pn,y'n), in the limit we have
d(y,y') = d(y,p) + d(p,y'). D
Proof of 9.21. (1) follows immediately from (2).
For (2), we consider geodesic rays cq and ci issuing from the same point and such
that co(oo) = ξο and ci (oo) = ξ ι. Let m, be the midpoint of the segment [co(t), ci(t)].
If there is no geodesic joining ξο to ξ ι then, since X is proper, u?(co(0), m,) —» oo
by (9.22). As in 9.17 one sees that there is a point μ e ЭХ such that Z(fo, μ) =
Ζ(μ, f ι) = 1/2 Ζ(ξο, ξι) < π β. As there are geodesic segments in ЭХ joining ξ0 to

How the Tits^vletric Determines Splittings 291
μ and μ to ξι, we have άτ(ξο, ξι) < Z(f0, μ) + Ζ (μ, ξι) = Z(f0, |i)· The reverse
inequality is obvious, hence άτ(ξ0, ξι) = Z(fo, |i)·
(3) Let |o = c(-oo) and |i = c(oo). We have άτ{ξο, ξι) > Z(fo, ti) = π.
Suppose rfydo. |i) = 7Γ- According to (9.20), there exists η e ЭХ such that άτ(ξο, η) —
Ίτ{ξι, Л) — л/2, and hence by (1) Z(fo, Л) — Z(f ι. У) — π/2. Let с' be the geodesic
ray issuing from χ = c(0) and such that c'(oo) = η. As Ζ*(|;, tj) < Z(f,-, tj) — π/2
and я = Ζχ(ξ0, ξι) < Zx(|0, J?) + Zx(tj, |i) < Z(|0, η) + Δ{η, ξχ) = π, we have
equality everywhere. In particular, Z^(|,-, tj) = Z(|/, tj) = 7Γ/2 for i = 0,1. and
(9.9) implies that the convex hull of c'([0, oo)) U c(R) is isometric to a flat half-plane
bounded by c(R). D
9.23 Examples
(1) Let X = {(x, у) е Μ2 : χ > 0 or у > 0} with the length metric induced by the
Euclidean metric. Then (ЭХ, dT) is isometric to the interval [0, 37Г/2].
(2) Let X = {(x, y) e R2 : \y\ < x2} with the length metric induced by the
Euclidean metric. Then (ЭХ, dj) is isometric to the disjoint union of two intervals
[—7Г/2, π/2]. The two half-parabolae {(x,y) \ у — x*,x > 0} and {(x,y) | —у =
χ2, χ > 0} are the images of two geodesic rays in X that define points of (ЭХ, dT)
that are a distance π apart but which cannot be joined by a geodesic in ЭХ.
How the Tits Metric Determines Splittings
It follows from (9.11) that if a complete CAT(0) space X splits as a product Χι χ Χι,
then its Tits boundary splits as a spherical join дтХ — Э7-Х1 * Э7Х2. In this section we
shall show that, conversely, if X has the geodesic extension property then spherical
join decompositions of the Tits boundary Э7-Х give rise to product decompositions
of X itself. This theorem is due to V. Schroeder in the case where X is a Hadamard
manifold (see [BaGS87], appendix 4).
9.24 Theorem. IfX is a complete CAT(0) space in which all geodesic segments can
be extended to geodesic lines, and if Э7Х is isometric to the spherical join of two
non-empty spaces A1 andAi, then X splits as a product Χι χ Χι, where ЭХ, = Αχ for
i =1,2.
In order to clarify the exposition, we re-express the theorem in a more technical
form that indicates the key features of spherical joins.
9.25 Proposition. Let X be as above and suppose that A \, Αχ С Э^Х are disjoint,
non-empty subspaces such that
(1) if ξι еЛ| and ξ2 e Аг, then Ζ(|ι, Ι2) — π/2, and
(2) for every ξ e Э7-Х, there exists ξ ι e A1 and ξι e Л 2 such that ξ belongs to the
geodesic segment [ξι, ξ2] С Э^Х

292 Chapter II.9 The Tits Metric and Visibility Spaces
Then X splits as a product Χι χ X2 such that dTXi = Aifor i — 1,2.
9.26 Remarks. Under the hypotheses of the proposition:
(1) The Tits metric and the angular metric on dX coincide.
(2) If ξ e Ax and f' e dX are such that Z(f, ξ') = π, then ξ' e Ax.
In order to prove (9.25) we shall need the following lemma.
9.27 Lemma. Let Χ, Α ι andAj. be as in the statement of the proposition and consider
two geodesic rays с\,сг : [0, oo) —» X with c\(0) = сг(0) = xq. 7/'c,-(oo) = f,- e A,-,
fori = 1, 2, then ΖΧο(ξι, ξ2) = π /2 and the convex hull of с ι([0, оо)исг([0, oo)) is
isometric to a quadrant in the Euclidean plane.
Proof. By hypothesis, the geodesic ray c\ is the restriction of a geodesic line c\ :
К —» X. Let ξt = ci(—oo). As we remarked above, ξt e Ль Hence
7Г = ΖΧ0(ξχ, ?,) < Z,„(?b fc) + /,„(?,, fe) < Αξί, ξΐ) + Afl. fe) = Я,
so Z^di, Ι2) = ^(fe. f 1) = τ/2, and we can apply Corollary 9.9. D
Proof of Proposition 9.25. We present the proof in four stages. Fix xq e X and let
Xi (jco) be the union of the geodesic rays joining xq to the points of A1.
Claim 1: Xi(jco) is a closed convex subspace. Any geodesic line which cuts
X^jco) in two distinct points is entirely contained in Xi(*(,). And if x'0 e X\(xq) then
ВД)= *,(;*,).
Our proof relies on the following alternative description of Xi (xq). Consider a
geodesic ray с : [0, oo) -> X with c(0) = *o and c(oo) e Л2, and let bc be the
associated Busemann function. We claim that Xi(*o) = {~\cb~x(0), where с runs
over all such geodesic rays. The key observation is that if c\ is a geodesic ray with
ct(0) = xq and ci(oo) e Ab then the convex hull of c([0, 00)) and Ci([0, 00)) is a
subspace isometric to a Euclidean quadrant (by the lemma) and the intersection of
b~\0) with this subspace is ci([0, 00)) (see 8.24(1)). Since every point of Xi(jc0)
lies on some such geodesic ray c\, we deduce Xi(*o) С b~l(0). Conversely, given
χ e Χ χ X] (xq), we consider a geodesic ray cx issuing from xq and passing through x.
By hypothesis, there exist ξι e A\ and ξ2 e Л2 such that cx(oo) e [|i, £2]· Let c, be
the geodesic rays issuing from xq with c,(oo) = |,·. Since χ is not in X! (x0), the ray Ci
does not pass through x, and so by considering the restriction of bC2 to the Euclidean
quadrant yielded by the lemma, we see that bCl{x) φ 0. ThusXi(jc0) = f}cb~l(0).
It follows from this equality that for all x/0 e Xi(jc0) and с with c(0) = x0 and
c(oo) e Аъ the Busemann function of the ray d issuing fromxO with c'(oo) = c(oo)
is the same as bc. Thus Xi (^o) = Пс К' (0) = ^i (-^o)·
Given any geodesic ray с with c(0) = jco and c(oo) e Л2, we can extend it to a
geodesic line with c(—00) e Л2, as we noted previously. Let с : [0, oo) —» X be the
geodesic ray t м> c(-i). Note that ^'(0) П b=l(0) = b;l((-oo, 0]) П ^'([0, 00))

How the Tits Metric Determines Splittings 293
which is a closed convex set. It follows that Xi(xq), as the intersection of closed
convex sets, is closed and convex.
In the argument given above we described the geodesic ray cx joining jco to
an arbitrary χ e X as lying in a subspace isometric to a Euclidean quadrant. The
intersection of this subspace with Xi(*o) is precisely the image of c\ (one of the
axes). It follows that if χ φ Χ\(χο) then cx meets X\(xq) at only one point. This
completes the proof of the first claim.
Givenx,y e X, define α4 := ά(Χχ (χ), Χχ(у)), and \etpyx : X\ (χ) —» Χχ(у) denote
the restriction of the orthogonal projection X —> X\(y).
Claim 2: The convex hull of X\(x) U X\(y) is isometric to Χχ(χ) χ [0, a^]. And
for all ij.zeXwe have pu = pzypyx.
Given x1 &X\ (x), we extend the geodesic segment [x, x1] to a geodesic line с with
c(0) = x. It follows from Claim 1 that c(K) С X\(x) and c(oo) e A\. Consider the
ray c' issuing from у with c'(oo) = c(oo). The function t м> d(c(t), c'(t)) is convex
and bounded on [0, oo), hence it is non-increasing and d(xf, X\ (y)) < d(x, y). Casting
c(t) in the role of χ1, it follows that t м> d(c(t), X\(y)) is bounded on the whole of
K. But since Xi(y) is convex, this function is convex (2.5), hence constant. Thus
d{x,Xx(y))=d{x!,Xiiy)).
Because the convex hull of X\ (x) U Χχ (у) is isometric to Χι (χ) χ [0, axy] by the
Sandwich Lemma 2.12(2), the mappyx sends each geodesic line in Χι (x) isometrically
onto a geodesic line in X\ (y). And since every point of X\ (x) lies on a geodesic line,
the second assertion in Claim 2 is an immediate consequence of Lemma 2.15.
Claim 3: Let Хг(*о) be the union of the geodesic rays joining jco to the points of Л2
and let/7] : X —» Xi(*o) be orthogonal projection. We claim that Хг{хо) = p\x{xo).
Given χ e X distinct from x^ let cx : [0, 00) —>- X be a geodesic ray issuing from
*o and passing through x. By hypothesis, there exist f, e A, such that с*(оо) е [|i,fe].
Let Cj be the geodesic ray with c,(0) = xq and c,(oo) = £,·. According to the lemma,
the convex hull of the union of these geodesic rays is a subspace of X isometric to
a quadrant in a Euclidean plane. This subspace contains the image of cx and in this
subspace it is obvious that the orthogonal projection of χ onto the image of C\ is xq if
and only if χ lies in the image of Ci\ thus if p(x) = xq then χ e ХгС^о)· Conversely, if
χ e Хг{хо) then in the above construction we may choose c\ so that it passes through
any prescribed point x1 e X\ (xq), and hence d(x, x1) > d{x, xq) (with equality only if
X1 = Xq).
Claim 4; For i = 1, 2, letp,- : X —> X,(^o) be orthogonal projection. We claim
that the map/ = {p\, рг): X —> Χι (^0) x Хг(*о) is an isometry, where Xt (x0) χ Хг(^о)
is endowed with the product metric.
Let x, у e X. It follows from the first assertion in Claim 2 that d(x, y)2 =
d(Xi(x), Xi(y))2 + dix^xyiy))1, and from the second assertion that d{x,pxy{y)) =
d{p\{x),p\{y)). Claim 3 implies that рг{х) — X\{x)V\Xi{xu) (if we exchange the roles
of the subscripts 1 and 2), and we get a similar expression foxpi(y). And Claim 2
(with 1 and 2 exchanged) implies that d(p2(x), Piiy)) — d{X\ (x), Xt(y)).

294 Chapter II.9 The Tits Metric and Visibility Spaces
Therefore d(x, yf = d{px(x), /?, (y))2 + d(p2(x), рг(у))2- It is clear that 3X,(jc0) =
Ait i=l,2. D
Visibility Spaces
The notion of visibility for simply connected manifolds of non-positive curvature
was introduced by Eberlein and O'Neill [EbON73] as a generalization of strictly
negative curvature. This condition can be interpreted in a number of different ways
(see [BaGS87, p.54]) most of which can be generalized to CAT(O) spaces.
9.28 Definition. A CAT(O) space is said to be a visibility space if for every pair of
distinct points ξ and η of the visual boundary dX there is a geodesic line с : К —> Χ
such that c(oo) = ξ and c(-oo) = η.
Note that the geodesic с is in general not unique. For instance, X could be
a Euclidean strip [0, 1] χ К, or X could be obtained by gluing such a strip to a
hyperbolic half-plane H^ using an isometry of {0} χ Κ onto the geodesic bounding
Щ.
9.29 Definition. A CAT(O) space X is said to be locally visible if for every ρ e X
and ε > 0 there exists R(p, ε) > 0 such that if a geodesic segment [x, y] lies entirely
outside the ball of radius R(p, ε) about ρ then Zp(x, y) < ε.
9.30 Definition. A CAT(O) space X is said to be uniformly visible if for every ε > 0
there exists /?(ε) > 0 such that, given ρ e X, if a geodesic segment [x, y] lies entirely
outside the ball of radius /?(ε) about ρ then Zp(x, y) < ε.
9.31 Remarks
(1) Real hyperbolic space H" is uniformly visible and hence so is any CAT(—1)
space (because the distance from ρ to [x, y] does not decrease when one takes a
comparison triangle A(x, y, p) in H2 and the comparison angle zip is at least as large
as Zp(x, y)).
(2) There exist complete non-proper CAT(O) spaces which are locally visible but
which are not visibility spaces and vice versa. An example of a complete visibility
space which is not locally visible can be obtained by taking any complete visibility
space, fixing a basepoint and attaching to that point the comer of a Euclidean square
of side n for every positive integer n. An example of a complete locally visible space
which is not a visibility space is {(x,y) | 1 > у > 1—(l + |x|)-1} with the path metric
induced from the Euclidean plane. Examples of CAT(O) spaces which are visible but
not uniformly visible can be obtained by considering negatively curved 1-connected
manifolds whose curvature is not bounded above by a negative constant. The interior
of an ideal triangle in the hyperbolic plane provides an example of an incomplete
uniformly visible space that is not a visibility space.

Visibility Spaces 295
(3) In contrast to the remark preceding (9.30), note that by the flat strip theorem,
there is a unique geodesic line joining any two points at infinity in any proper
CAT(-l) space X.
One consequence of this is that if we fix ρ e X and ξ, f' e ЭХ, then there is a
well-defined ideal triangle with vertices ρ, ξ, ξ'. (Exercise: If θ is the angle between
the geodesic rays issuing from/? with endpoints ξ and ξ', and D is the distance from
ρ to the geodesic line with endpoints ξ and ξ', then tan(#/4) < e~D.)
9.32 Proposition. IfX is a proper CAT(O) space, then the following are equivalent:
(1) X is a visibility space;
(2) X is locally visible.
If in addition X is cocompact, then (1) and (2) are equivalent to:
(3) X is uniformly visible.
Proof. (2) =Φ· (1): Let ξ and f' be distinct points of dX and let x„ and xfn be
sequences of points in X converging to f and f respectively. If ρ e X is such that
Zp(|, ξ') > ε > 0, then for n big enough we have /Lp{x„, xfn) > ε. By (2), there exists
R > 0 such that [x„, xfn] meets the compact ball B(p, R). It then follows from Lemma
9.22 that there is a geodesic joining ξ to ξ'.
(1) =Φ· (2): We argue the contrapositive. If X is not locally visible, then
there exists a point ρ e X, a number ε > 0, and sequences (xn) and (xj;) such
that dip, [x„, x'J) is unbounded but Zp(x„, x'n) > ε. We can assume, by passing to
subsequences if necessary, that the sequences xn and x!n converge to points ξ and ξ'
in ЭХ. These points are necessarily distinct because Ζρ(ξ, f') > ε.
Letpn be the image of ρ under projection onto the closed convex set [x„, x'J. By
passing to a subsequence if necessary, we may assume that the sequence pn converges
to a point η e ЭХ. Reversing the roles off and ξ' if necessary, we assume that η φ ξ.
If n is big enough then pn φ xn and hence the angle Zp„(p, xn) is well defined.
According to (2.4) this angle is not smaller than π/2, hence the comparison angle
Zpn(p, xn) is also no smaller than π/2, so Zp(p„, x„) < π/2. But then, by the lower
semicontinuity of the comparison angle (9.16) we have Ζ(ξ, η) < π/2. Hence there
is no geodesic in X with endpoints ξ and η.
Finally we prove that (2) =Φ· (3) (the converse is trivial). We argue the
contrapositive. Suppose that X is cocompact, covered by the translates of a compact
subset K. If X is not uniformly visible then there exists ε > 0 and sequences of
points pn, x„, y„ in X such that ZPn(xn, y„) > ε for all n, and d(pn, [x,„ y„]) —> oo as
n —> oo. Translating by suitable elements of Isom(X) we may assume that pn e К
for all n, hence d(K, [xn, ynJ) —> oo as n —> oo. Since К and X are compact, we may
pass to subsequences so thatp„ —» ρ e К and x„ —> ξ e ЭХ, y„ —» η e ЭХ. By the
upper-semicontinuity of angles (9.2), we have Ζρ(ξ, η) > ε. Hence by the continuity
of angles at a fixed p, we have Zp(xn, y„) > ε/2 for all n sufficiently large. Since
dip, [x„, y„]) > d(K, [x„, yn]) —> oo as n —» oo, this implies that X is not locally
visible at p. G

296 Chapter II.9 The Tits Metric and Visibility Spaces
The following theorem was proved by Eberlein [Eb73] in the case of Hadamard
manifolds. The version given here was stated by Gromov [Gro87] and a detailed
proof was given by Bridson [Bri95]. A generalization to convex metric spaces was
proved by Bowditch [Bow95b]. The implications of the stark dichotomy exposed by
this theorem will be considered in Part III.
9.33 Theorem. A proper cocompact CAT(O) space Xisa visibility space if and only
if it does not contain a 2-flat (i.e., a subspace isometric to E2).
Proof. It is clear that if X contains a 2-flat then it cannot be a visibility space. On the
other hand, if X is not a visibility space then according to (9.21(2)) there exist two
distinct points ξ, ξ' e дХ such that Ζ(ξ, ξ') < π/2. But then, by Proposition 9.5(3),
there exist у e X and η, η' e ЭХ such that ^(η, η') = Ζ(ξ, ξ')· It follows from (9.9)
that the convex hull of the geodesic rays issuing from у in the classes η and η' is
isometric to a sector in the Euclidean plane; in particular X contains arbitrarily large
flat discs. The following general lemma completes the proof of the theorem. G
9.34 Lemma. Let Υ be a separable metric space with basepoint yo and let X be a
proper cocompact metric space. If for all n e N there exists an isometric embedding
Фп ■ B(yo, n) <^-> X then there exists an isometric embedding φ : Υ <^-> Χ.
Proof. Let yo, 3Ί, · · · be a countable dense subset of Y. Because X is cocompact, we
can replace фп by a suitable choice of gnφn with gn e Isom(X), and hence assume
that the sequence (ф„(уо))п is contained in a compact (hence bounded) subset of X.
Since X is proper, it follows that for all i the sequence {φη{γί))„ is contained in a
compact subset of X. Hence there exists xq & X and a sequence of integers («o(/))y
such that $п0(/)Суо) —> xo as j —> oo. And proceeding by recursion on r we can find
a sequence of elements xn e X and infinite sets of integers [nr(j)}j С (и,—ι (/)}; such
that фПгЦ)(Ук) —> xk as j —> oo for all к < r. The diagonal sequence m = nr(r)
satisfies фт(Ук) —> хъ as m —> oo for all к > 0. Since the фт are isometries, so too
is the map у к м> Xk- The set {yk}k is dense in Y, so yk м> Xk has a unique extension,
which is the desired φ. G
We close this section by giving a few other characterizations of proper visibility
spaces. In Chapter III.Η we shall consider a much more far-reaching reformulation
of uniform visibility, Gromov's 5-hyperbolic condition.
9.35 Proposition. Let X be a proper CAT(O) space. The following are equivalent:
(1) X is a visibility space.
(2) If h, h' are horofunctions centred at different points ξ, ξ' e ЭХ, then h + h'
assumes its infimum.
(3) Let h be a horofunction centred at ξ e ЭХ and let с : [0, oo] —> X be a geodesic
ray with c(oo) φ ξ. Then h о c(t) tends to infinity when t tends to infinity.

Visibility Spaces 297
(4) The intersection of any two horoballs centered at different points of dX is
bounded.
We first need a lemma.
9.36 Lemma. Let с : К —> X be a geodesic line in a CAT(O) space X, and let bc be
the Busemann function associated to the geodesic ray c|[o,oo]. Let ρ be the projection
onto the closed convex subspace c(R). Then for all χ e X we have
bc(p(x)) < bc(x).
Proof. Let s = bc(x). For any t > s, we have bc{c{—t)) = t > s = bc(c(—s)), and the
projection of c(—i) to thehoroball b~x ((oo, s]) is c(—s). Thus ZC(-S)(c(—t), χ) > π/2,
and d(x, c(—t)) > d(x, c(—s)). Therefore p(x) e c([—s, oo)). G
Proof of Proposition 9.35.
(1) => (2). Let с : К -> X be a geodesic such that c(oo) = f and c(-oo) = f',
and let bc (resp. b_c) be the Busemann function associated to the ray c|[o,oo) (resp.
t м> c(—i))· These functions are equal up to constants to h and h' respectively, so
it is sufficient to show that bc + b-c assumes its infimum. Note bc(c(t)) = —t and
b-c(c(t)) = t, and hence bc + b-c is identically zero on c(K). Let ρ be the projection
of X to c(K) and χ e X. By the lemma, bc{x) > bc(p(x)) and Ь_с(л:) > b-c(p(x)) =
-fec(pW)· Thus ^c(jc) + b-c(x) > 0.
(2) =Φ· (1). Let xeXbea point such that h(x) + h'(x) is minimum. Let с (resp.
c') be the geodesic ray issuing from χ such that c(oo) = f (resp. c'(oo) = ξ')· As
A'eB (cf. 8.22), for ί > 0, we have
h'(c(t)) - h'(x) < t,
and we have equality if and only if the concatenation of с (reversed) and c' is a
geodesic line. As h(x) - t + h'(c(t)) = h(c(t)) + h'(c(t)) > h(x) + h'(x), we have
h'(c(t)) > h'(x) + t,
hence the desired equality.
(1) =» (3). Let d : R -+ X be a geodesic line with c'(oo) = c(oo) := ξ'
and c'(-oo) = |. We have A(c'(0) = Кс'Ф)) + t- If i1 is the projection on c'(K), by
the above lemma we have h(c(t)) > h(p(c(t)). As |A(c(i)) - h{c'{t))\ < d(c(t), c'{t)),
which remains bounded as t —> oo, we have h(p(c(t)) —> oo.
(3) ==Φ· (4). Let /г and /г' be horofunctions centred at different points ξ and £',
and let Я and #' be horoballs centred at ξ and £'. If Η Π Я' is not bounded, there is
an unbounded sequence (x„) in Я Π Η'. As X is proper by hypothesis, we can assume
that (x„) converges to η e ЭХ. Letx0 e Я П Я'. As Я П Я' is convex, the geodesic
segments [*о>*н] converge to a geodesic ray с in Я П Я' with c(oo) = ??. But the
functions h(c(t)) and h'(c(t)) are both bounded, hence ?? = ξ = ξ', a contradiction.

298 Chapter II.9 The Tits Metric and Visibility Spaces
(4) =Φ· (1). Let ξ and ξ' be distinct points of dX and let bc and bd be
Busemann functions associated to geodesic rays issuing from χ e X and such
that c(oo) = ξ and c'(oo) = ξ'. By hypothesis the intersection С of the closed
horoballs Я := ^'((-oo, 0]) and Я' := ^'((-oo, 0]) is bounded. For a > 0, let
H(a) := b~]((-oo, -a]). If a is bigger than the diameter of C, then Я' П H(a) = 0;
indeed for у e X such that ЬсСу) < —a, we have α < \bc{y) — fecWI 5 d(x, y), hence
Я' does not intersect H{a).
Letm := inffa > 0 | Я'ПЯ(а) ψ 0}. As we assumed X to be proper, Я'ПЯ(т) /
0. Given у е Я' П H(m), let с and с7 be geodesic rays issuing from у and such that
c(oo) = ξ and c'(oo) = ξ'. We have frc(c(i)) = -t -m and b^(c'(i) = -t. We claim
that the union of the images of с and c' is a geodesic line joining ξ to f'. It suffices
to check that, for t > 0, we have d(c(t), c'(i)) > 2i. Let г be a point on the geodesic
segment [c(i), c'(t)] such that bc(z) = -m. Then £ν(ζ) > 0, and hence
d(c(f), c{t)) = d(c(t), z) + d(z, ~c'{t)) > \bMt)) - bc(z)\ + \bd{z) ~ ^(c (01 > 2f.
D

Chapter 11.10 Symmetric Spaces
A symmetric space is a connected Riemannian manifold Μ where for each point
ρ e Μ there is an isometry σρ of Μ such that σρ(ρ) = ρ and the differential of
σρ at ρ is multiplication by —1. Symmetric spaces were introduced by Elie Cartan
in 1926 [Car26] and are generally regarded as being among the most fundamental
and beautiful objects in mathematics; they play a fundamental role in the theory of
semi-simple Lie groups and enjoy many remarkable properties. A comprehensive
treatment of symmetric spaces is beyond the scope of this book, but we feel that
there is considerable benefit in describing certain key examples from scratch (without
assuming any background in differential geometry or the theory of Lie groups), in
keeping with the spirit of the book. Simple examples of symmetric spaces include
the model spaces M" that we studied in Part I.
In this chapter we shall be concerned with symmetric spaces that are simply
connected and non-positively curved. If such a space has a trivial Euclidean de
Rham factor, then it is said to be of non-compact type, and in this case the connected
component of the identity in the group of isometries of such a space is a semi-
simple Lie group G with trivial centre and no compact factors. In fact, there is a
precise correspondence between symmetric spaces of non-compact type and such
Lie groups: given G one takes a maximal compact subgroup K, forms the quotient
Μ = G/K and endows it with a G-invariant Riemannian metric— this is a symmetric
space of non-compact type and the connected component of the identity in Isom(M)
is G (see for instance [Eb96, 2.1.1]).
The first class of examples on which we shall focus are the hyperbolic spaces
KH", where К is the real, complex or quaternionic numbers. We shall give a unified
treatment of these spaces based on our earlier treatment of the real case (Chapter 1.2).
We shall define the metric on KH" directly, prove that it is a CAT(-l) space, and
then discuss its isometries, Busemann functions and horocyclic coordinates. Besides
one exceptional example in dimension 8 (based on the Cayley numbers), the spaces
KH" account for all symmetric spaces of rank one, where rank is defined as follows.
The rank of a simply connected symmetric space of non-positive curvature is
the maximum dimension of flats in M, i.e. subspaces isometric to Euclidean spaces.
The results in Chapters 6 and 7 give some indication of the important role that
flat subspaces play in CAT(O) spaces; their importance in symmetric spaces is even
more pronounced. In particular, there is a clear distinction between the geometry of
symmetric spaces of rank one and those of higher rank. Aspects of this distinction
will emerge in the course of this chapter. We should also mention one particularly

300 Chapter II. 10 Symmetric Spaces
striking aspect of the geometry of symmetric spaces that is not discussed here: if the
rank of Μ is at least 2, then Μ and lattices31 in Isom(M) are remarkably rigid (see
[Mar90], [Zim84] and (1.8.41)).
We shall exemplify the theory of higher-rank symmetric spaces by focussing
on P(n, K), the space of symmetric, positive-definite, real (n, «)-matrices. P(n, Щ
has a Euclidean de Rham factor of dimension 1 (namely the positive multiples of
the identity matrix); the complementary factor P{n, K)t is the subspace of P(n, K)
consisting of matrices of determinant one. P(n, K)i is irreducible (i.e. cannot be
decomposed as a non-trivial product). Following the outline of the treatment given
by Mostow [Mos73], we shall prove that P(n, K) is CAT(O) and we shall describe
the geometry of its maximal flats (which are of dimension ή). We shall also describe
the Busemann functions and horospheres in P(n, K). The reason that we chose to
explain the example of P(n, K) in some detail is that it plays the following universal
role in the theory of symmetric spaces: any simply connected symmetric space of
non-positive curvature is, up to rescaling of the irreducible factors in its product
decomposition, isometric to a totally geodesic submanifold of P(n, M.) for some η
(see [Mos73, paragraph 3] and [Eb96, 2.6.5]). Examples of such totally geodesic
submanifolds are given in the section entitled "Reductive Subgroups".
In the final section of this chapter we shall discuss the Tits boundary of symmetric
spaces (which is interesting only in the case where the rank is > 2). This leads us to
a discussion of a remarkable class of polyhedral complexes called buildings. These
complexes, which were discovered by Jacques Tits, provide important examples of
CAT(O) and CAT(l) spaces. We shall not replicate the literature on buildings, but in
an appendix to this chapter we shall present their definition and explain how it leads
to curvature bounds.
For a concise and elegant introduction to symmetric spaces from the point of
view of differential geometry, we recommend the excellent book of Milnor [ΜΪ163].
For the general theory of symmetric spaces, see [Wo67], [Hel78], [Kar65],[Im79],
[Eb96]. In this chapter we have been guided by the treatment of Mostow [Mos73];
we also benefitted from the lecture notes of Eberlein [Eb96], as well as conversations
with Marc Burger.
Real, Complex and Quaternionic Hyperbolic n-Spaces
In this section we shall construct the complex and quaternionic hyperbolic и-spaces
in a manner closely analogous to the way in which we defined real hyperbolic space
in (1.2). We shall give as unified a treatment as possible, writing К to denote either
К, С or the quaternions, and writing KH" to denote the K-hyperbolic 32 space of
31 Discrete subgroups such that T\M has finite volume. For existence, see [Bo63].
32 Throughout this book the symbol H" is reserved for real hyperbolic rc-space, but in this
chapter it will be convenient to write MH" instead of H" so as to give a unified treatment of
the real, complex and quaternionic cases. For the same reason, we change the notation of
(1.6) by writing M"'' instead of E"·' and RP" instead of P".

Real, Complex and Quaternionic Hyperbolic rc-Spaces 301
dimension n. As in the real case, we shall describe the set of points of KH" in terms
of K"'1, the vector space K"+1 equipped with a form of type (и, 1). We shall then
define the distance function by means of an explicit formula on this set of points
(10.5) and give an explicit description of hyperbolic segments and angles (10.7). Our
task, then, will be to show that this 'distance function' is indeed a metric and that our
explicit descriptions of hyperbolic segments and angles are precisely the geodesic
segments and Alexandrov angles associated to this metric. Proceeding as in (II.2),
we shall deduce these facts from an appropriate form of the law of cosines (10.8).
This form of the law of cosines also leads to a short proof of the fact that KH" is a
CAT(-l) space (10.10).
For an alternative introduction to complex hyperbolic space, see [Ep87], and for
the general case of symmetric spaces of rank one see [Mos73, paragraph 19].
Notation. Throughout this section К will denote either the field R of real numbers,
the field С of complex numbers, or the non-commutative field of quaternions. (Recall
that the quaternions are a 4-dimensional algebra over К with basis (1, i,j, k}, where
1 is central, ij = k,jk = i, ki =j and i2 =j2=k2 = —l.)
If χ e C, then we write χ to denote the complex conjugate of x. Conjugation on
M. is trivial. For quaternions, one defines the conjugate of λ = uq + αλ i + a^j + a^k to
be λ = αο — a\ i — a^j — a^k. The norm |λ| of λ e К is the non-negative real number
νλλ. The real part of χ is the real number 9ΐλ = ^=.
In all of the vector spaces which we shall consider, the multiplication by scalars
will be on the right.
10.1 Definition of K"·1 and the Form Q(x, y) = (x\y). We denote by K"·1 the EC-
vector space K"+1 endowed with the form Q(x, y) = (x\y) of type (n, 1) defined
by:
П
(х\У) '■= ^ХкУк -хп+1Уп+и
k=l
where x— {x\,..., x„+\) and у = (yi, ■.., y„+i). If (x\y) = 0 then χ and у are said
to be orthogonal. The orthogonal complement of χ e K"·1, written r1, is {u e K"+1 |
(x\u) = 0}.
10.2 Remark. Note that (y\x) = (x\y) and hence (x\x) is a real number. If (x\x) < 0,
then the restriction of Q to r1 is positive definite, and if (x\x) > 0 then the restriction
of Q to r1 is of type (n — 1, 1) (see, e.g., [Bour59, paragraph 7]).
10.3 Lemma (The Reverse Schwartz Inequality). If (x\x) < 0 and (y\y) < 0, then
(x\y)(y\x) > (x\x)(y\y),
with equality if and only if χ and у are linearly dependent over K.

302 Chapter II. 10 Symmetric Spaces
Proof. If χ and у are linearly dependent then we obviously have equality. The
restriction of Q to x1- is positive definite and (y\y) < 0, so (x\y) φ 0. Let! = — (x\x)(x\y)~l.
Then χ + yk e χ-1, and if χ and у are linearly independent then x + yk φ 0, therefore
(x + yk\x + yk) = (x + yk\yk) > 0. By expanding this inequality we get
0 < -(φ) + (x\x)2 (y\y) {ylx)-1 (х\уГ1.
After dividing by (x\x) < 0, this can be rearranged to give the inequality in the
statement of the lemma. Π
In Part I we described real hyperbolic space as one sheet of the sphere of radius
— 1 in W·'. This sheet maps bijectively onto its image under the natural map ρ :
К"л -» KP"; by endowing the image of the sheet with the metric that makes the
restriction of ρ an isometry one obtains an alternative model for real hyperbolic
space. This second (projective) model lends itself readily to the present more general
context.
10.4 Notation. Let KP" be the и-dimensional projective space over K, i.e. the quotient
of K"+1 χ {0} by the equivalence relation which identifies χ = {χλ,..., χη+\) with
xk = {x\k,..., x„+\k) for all λ e Κ* = Κ \ {0}. The class of χ is denoted [x] and
(jci ,..., xn+\) are called homogeneous coordinates for [x].
10.5 The Hyperbolic и-Space KH" over K. We define KH" to be the set of points
[x] e KP" with (x\x) < 0. The distance d([x], \y]) between two points of KH" is
defined by
coshzd([x], \y]) = ,,,.,,·
M*> (y\y)
The preceding lemma shows that the right hand side of this formula is bigger than
1 unless [x] = [y], so the formula makes sense and the distance between each pair of
distinct points is non-zero. Thus, in order to show that d is a metric, it only remains
to verify the triangle inequality. As in the case Κ = Μ (see 1.2.6), we shall deduce
the triangle inequality from an appropriate form of the law of cosines. Following the
strategy of (1.2.7), this requires that we first define a primitive notion of hyperbolic
segment and hyperbolic angle. This task is complicated in the present setting by the
fact that if Κ φ Κ then we do not have a canonical way to lift KH" into K"·', and as
a result we cannot describe tangent vectors in as concrete a manner as we did in the
real case. We circumvent this difficulty as follows. Given χ e K"·' with (x\x) < 0 we
regard x1- as a model for the tangent space Γ^ΚΗ" of [x] e KH". More precisely:
10.6 Tangent vectors. We identify xL with ГМКН" using the differential of the
natural projection K"·1 \ {0} -> KP".
If и e r1 is identified with U e Г^КН" then we say that и is the tangent vector

Real, Complex and Quaternionic Hyperbolic rc-Spaces 303
at χ representing U. (If λ /0 then uk is the tangent vector zixk representing U, and
since χ1- = (χλ)-1, one must be careful to specify the choice of χ as well as и when
describing U e ГМКН".)
We note in passing that if К is the quaternions, Г^КН" is just a real vector
space, whereas if К = С then Г^КН" is naturally a complex vector space. In any
case, the symmetric positive-definite K-bilinear form associating to tangent vectors
u, ν at χ the real number —ΐϋ(ιι\υ)/{χ\χ) is compatible with the above identifications
and therefore defines a scalar product33 on Γ^ΚΗ". In this way KH" is naturally a
Riemannian manifold. The interested reader can check that the metric associated to
this Riemannian structure is the metric defined in (10.5) (cf. 1.6.17).
10.7 Lines, Segments and Angles in KH". Suppose that (x\x) = —1 and that
и e Xх, with (u\u) = 1, represents U e Г^КН". Then the hyperbolic geodesic
issuing from [x] in the direction of U is defined to be the map К —> KH" given by
t t-» c{t) = [jccoshi + usinht]. Given a > 0, we write [c(0), c(a)] to denote the
image under с of the interval [0, a], and refer to it as a hyperbolic segment joining
[x] = c(0) to c(a). We call и the initial vector of[c(0), c(a)] atx.
We claim that, given any two points [χ] φ \y] in KH", there exists a unique
hyperbolic geodesic segment joining [x] to \y]. To construct this segment one takes
(x\x) = -1 and (y\y) = -1 then multiplies у by the unique λ e К with |λ| = 1
such that (x\yk) is real and negative 34. One then defines a = arccosh (—(x\yk)) and
и = yX~^ha ■ The initial segment [c(0), c(a)] of 1ь» c(t) = [x cosh t + и sinh t] is a
hyperbolic segment joining [x] to \y]. It remains to show that this segment is unique
— we leave this as an exercise for the reader.
Given [x] e KH" and non-zero tangent vectors и, υ atx, we define the hyperbolic
angle between и and υ by
,H, ч ®Ш
cosZ"(u, v) :=
-J{u\u)y/{v\v)
Since Z."(u, υ) = Zf {uk, υλ) for all λ e K, this formula gives a well-defined
notion of angle between vectors U, V e Γ^ΚΗ". The hyperbolic angle between two
hyperbolic segments issuing from [x] e KH" is defined to be the angle between their
initial vectors. A hyperbolic triangle in KH" consists of a choice of three distinct
points (its vertices) A, B,C e KH", and the three hyperbolic segments joining them
(its sides). The vertex angle at С is defined to be the hyperbolic angle between the
segments [C, A] and [С, В].
10.8 Proposition (The Law of Cosines in KH"). Let A be a hyperbolic triangle in
KH" with vertices A,B,C, where С = [χ] and (x\x) = -1. Let a = d(B, C), b =
d(C, A) and с = d(A, B). Let u, ν e xL be the initial vectors at χ of the hyperbolic
segments [C, A] and [С, В]. Then
33 If K = C, there is also a well-defined hermitian form on ΓΜΚΗ" given by — (u\v)/(x\x).
34 The group of elements (numbers) of norm 1 in К acts simply transitively (by right
multiplication) on each sphere {μ : |μ| = г > 0} and each such sphere contains a unique real
negative number.

304 Chapter 11.10 Symmetric Spaces
coshc = | cosha coshb — sinha sinhb (и|и)|.
Proof. We have A = [x cosh a + и sinh a] and В = [χ cosh b + ν sinh b]. Hence
cosh с = | (x cosh a + и sinh a \ χ cosh b + υ sinh b) \
= |cosha coshb — sinha sinhb(и|и)|,
as required. D
10.9 Corollary (The Triangle Inequality and Geodesies). For allA,B,Ce KH",
d(A, B) < d(A, C) + d(C, B),
with equality if and only if С lies on the hyperbolic segment joining A to B. Thus
KH" is a uniquely geodesic metric space: the unique geodesic segment joining A to
В is the hyperbolic segment [A, B].
Proof Let α = d(C, B), b = d(A, С), с = d(A, B). We assume that А, В and С are
distinct (the case where they are not is trivial). Consider a hyperbolic triangle Δ with
vertices A, В, С With the notations of the preceding proposition we have:
cosh с = | cosh α cosh b — sinh α sinh b {u\v)\
< cosha cosh b+ sinh a sinh έ> | (м| υ> |
< cosh a cosh b + sinh a sinh b = cosh (a + b),
with equality if and only if (u\v) = —l.lf(u\v) = —1, then since (u \ u) = (v\v) = 1,
we have Q(u + v) = (u + v\u+v) = 0. The restriction of Q to xL is positive definite,
thus и = — ν and С belongs to the geodesic segment [A, B]. D
The Curvature of KH"
10.10 Theorem. KH" is a CAT(-1) space.
Proof. We shall prove that KH" is a CAT(—1) space by applying (1.8) with the
hyperbolic angle ZH in the role of A.
Let Δ be as in (10.8) and note that in the light of (10.9) we now know that Δ
is a geodesic triangle. Let γ denote the hyperbolic angle at the vertex С in Δ, i.e.
γ = arccos9l(i<|i>).
We consider a geodesic triangle A(A,B, C) in the real hyperbolic plane with
d(B, Q = a and d(A, C) = b, and with vertex angle γ at C. Let с = d(A, B). We
must show that с < с We claim that
cosh с = cosha cosh b — sinha sinh b 91 (и|υ)
< | cosha coshb — sinha sinhb {u\v)\ = coshc.

The Curvature of ЮН" 305
The equality in the first line is the law of cosines in H2, the equality in the second
line is the law of cosines in KH", and the inequality relating the two lines is a special
case of the simple observation that \λ + μq\ > \λ+ μ^\ for all q e К and λ, μ e К
(with equality if and only if q e К or μ = 0).
Thus с <c, and we can apply remark (1.8). □
10.11 Exercise. The Alexandrov angle in KH" is equal to the hyperbolic angle (10.7).
We note one further consequence of the law of cosines. This describes the triangles
in KH" that are isometric to triangles in the model spaces M". Each of these special
triangles actually lies in a complete isometrically embedded copy of the appropriate
model space (cf. Theorem 10.16).
10.12 Proposition. Let Δ с KH" be the geodesic triangle considered in (10.8) and
suppose that the vertex angle Z-c{A, B) is not equal to 0 or π. Then:
(1) The convex hull ο/Δ is isometric to the convex hull of its comparison triangle
in M2_l if and only if (u\v) is real.
(2) The convex hull of A is isometric to the convex hull of its comparison triangle
in M2_ 4 if and only if и and υ are linearly dependent over К
Proof. In the notation of the proof of (10.10), we have с = с if and only if (u\v) is
a real number. If the convex hull of Δ is isometric to that of a geodesic triangle in
Μ2, then obviously с = с Conversely, if с = с then according to (2.10) the convex
hull of Δ is isometric to the convex hull of its comparison triangle in Μ2,.
In order to prove the second assertion we compare the given triangle Δ(Α, В, С)
in KH" with a triangle Δ = (А, В, С) in the plane M24 of constant curvature —4.
We choose d(C, A) = a, d(C, B) = b and Ze(A, B) = y, where γ = ZC(A, B) =
arccosi)i(i<|i>). Let с = d(A, B).
The law of cosines in Μ2 4 states that
cosh 2c = cosh 2a cosh2£> — sinh 2a sinh2bcosy.
Squaring this expression and taking account of the identities cosh 2d = 2 cosh2 d — 1
and sinh 2d = 2 sinh uicosh d, we get:
cosh2 с = cosh2 a cosh2 b + sinh2 a sinh2 b — 2 cosh a cosh b sinh a sinh b cos γ.
On the other hand, the law of cosines in KH" (10.8) gives
cosh2 с =
cosh2 a cosh2 b + \ (u \ ν) \2 sinh2 a sinh2 b — 2 cosh a sinh a cosh b sinh b cos γ.
As |(ϊ<|υ)| < 1, comparing the above expressions we see that с > с with equality if
and only if |(и|и)| = 1.
Since (u\u) = 1, if υ = uk then λ = (u\v). And since the span of {u, v}
is contained in jr1, where the form Q is positive definite, υ = uk if and only if

306 Chapter II. 10 Symmetric Spaces
(υ — uk | υ — uk) = 0. Expanding (υ — uk \ υ — и!) and setting λ = (u\v), we get
(v\v) + |(и|и)|2(и|и) — 2|(и|1>)|2. And since (v\v) = (u\u) = 1, this expression is 0
if and only if | (u\v) \ = 1. Thus с = с if and only if и and υ are linearly dependent
over K.
In order to complete the proof, as in (1), we would now like to appeal to the
analogue 2.10 of the Flat Triangle Lemma. However, since KH" is а С AT(— 1) rather
than а С AT(—4) space, we cannot appeal to (2.10) directly. Instead, we claim that if и
and ν are linearly dependent, then Δ is contained in an isometrically embedded copy
of M™4, where m = dim^ K. To see this, we identify the one-dimensional K-vector
subspace of ΤΧΚ"·} spanned by и with the tangent space at a point о е M™4 by means
of an isometry φ : T0M™4 —»■ Km. The conclusion of the preceding paragraph, с = с,
shows that the map φ : M™4 -> KH" sending the point a distance t e [0, oo) along
the geodesic in M™4 with initial vector ν to the point a distance t e [0, oo) along the
geodesic in K"·1 with initial vector φ(ν) is an isometry onto its image. D
10.13 RemarL·
(1) Consider (10.12) in the case η = 1. Given any [x] e KH1, since лг1 is 1-
dimensional, any two vectors u, ν e xL must be dependent. Thus KH1 is isometric
to M24 if К = С and to M\ if К is the quaternions.
(2) The argument in the last paragraph of the preceding proof shows that the K-
linear map K1,' —> K"'' sending (1, 0) to χ and (0, 1) to и е x1- induces an isometric
embedding; : KH1 -> KH".
(3) The proof of (10.12) shows that the curvature of KH" is bounded below by
—4 and above by — 1.
The Curvature of Distinguished Subspaces of KH"
Continuing the theme of (10.12) and (10.13), our next goal is to characterize those
subsets of KH" that are isometric to Mk = Μ* , and Mk_4. This requires the following
definitions.
10.14 Definitions. Regard K"·' as an R-vector space in the natural way. An K-vector
subspace V С К"'1 is said to be totally real (with respect to Q) if (u\v) el for all
u, υ e V.
Let ρ : К"·1 \ {0} -» KP" be the natural projection. If У с К"·1 is a totally real
subspace of (real) dimension (к + 1), and if there exists χ e V such that (jc|jc) < 0,
thenp(V ν {0}) Π KH" is called a totally real subspace of dimension к in KH".
If W С К"'' is a K-vector subspace of dimension two that contains a vector υ
such that (υ\υ) = -1, then^W \ {0}) П KH" is called a K-affine line. (K-affine
subspaces of dimension к are defined similarly.)
In the course of the next proof we shall need the result of the following simple
exercise in linear algebra.

The Group of Isometries of ЮН" 307
10.15 Exercise. Let V С К"'1 be a totally real subspace of dimension к + 1 and let
x e V be such that Q(x, x) = — 1. Prove that one can find a real basis ux,..., щ, и^+\
for V such that uk+\ = x, (и,-|и(·) = 1 for i = 1,..., к and {щ\щ) = 0 for all i φ j.
(Hint: Argue that V = (χ) φ (r1 Π V), as a real vector space, and use the fact that
the restriction of Q to the second summand is positive definite.)
10.16 Theorem.
(1) The totally real subspaces of dimension к in KH" are precisely those subsets
which are isometric to the real hyperbolic space KH* = Шк.
(2) For Κ φ % the K-affine lines in KH" are precisely those subsets which are
isometric to the model space M"^4 of constant curvature —4, where mo = dimRK·
Moreover, KH" does not contain any subsets isometric to M™4 for m > mo.
Proof. In order to prove (1), given a totally real subspace V С К"'1 of dimension
к + 1 containing a vector υ with (υ, υ) = — 1, we choose a basis щ, ..., и^+ι as in
exercise (10.15). The K-linear map M"·1 -> K"·1 sending the standard basis of M"·1
to щ,..., uk+i induces an isometry of KH* onto p(V χ {0}) Π ΚΗ".
For the converse, given an isometric embedding j : Ш1к —*■ KH", we choose
k + 1 points Ao, Α ι, ..., Ak € 1RH* in general position, write χ to denote the element
of K"·' with Q{x, x) =. -1 and [x] = j(Ao), and then for i = 1,..., η we define
щ e x1- to be the initial vector of the geodesic segment j([Aο, Aj]). Proposition 10.12
implies that the K-subspace V С К"·' generated by x, щ,..., ик is totally real (since
(x\u;) = 0 and (щ\ц) e R for all i,j) and thatp(V χ {0}) П KH" =j(RHk).
The preimage under /j : K"·1 -> KP" of a K-affine line in KP" is a two
dimensional K-subspace for which we can choose a basis {x, u] with Q(x, x) =
-1, β(κ, и) = 1, and(jc|i<) = 0. TheK-linear map K1·1 ->■ K"·1 sending (1,0) to jc
and (0, 1) to и induces an isometric embeddingy : KH1 —> KH" whose image is the
given affine line. Thus the first assertion in (2) follows from (10.13). The assertion
in the second sentence is an immediate consequence of (10.12). D
The Group of Isometries of KH"
We describe the isometry group of KH" by means of a sequence of exercises.
Consider the group GL(n + 1, K) of invertible (n + 1, η + l)-matrices with
coefficients in K. There is a natural left action of this on K"·1 by K-linear
automorphisms: the matrix Л = (a,j) sends χ = (χχ, ... ,x„+\) € К"·1 to Α.χ =
(Σ/=ι °ιλ·>···>Σ/=ι a"+ujxj)·
10.17 The group 0K(Q). Let 0K(Q) denote35 the subgroup of GL(n + 1, K) that
preserves the form Q defining K"'\ i.e. A e Ok(Q) if and only if (A.x\A.y) = (x\y)
foralljc,yeK"+1.
Classically, 0R(Q) is denoted 0(n, 1) and Oc(Q) is denoted U(n, 1).

308 Chapter П. 10 Symmetric Spaces
Note that the induced action of Οκ (β) on KP" preserves the subset KH"
consisting of points [x] with (x\x) < 0. Note also that the action of Ok(Q) on KH" is by
isometries (see 10.5).
The following description of the stabilizer of the point of KH" with homogeneous
coordinates (0, ..., 0, 1) is simply a matter of linear algebra.
10.18 Exercises
(1) The stabilizer in Ok(Q) of the point po with homogeneous coordinates
(0, ..., 0, 1) acts transitively on the sphere of unit tangent vectors in ΓΡοΚΗ".
(2) The stabilizer of po is isomorphic to the group 0(n) χ 0(1) if Κ = К, to
U(n) χ i/(l) if К = C, and to Sp(n) χ Sp(l) if К = И. (Recall that 0(n), (resp.
U(n), Sp(n)) is the subgroup of GL(n, K) leaving invariant the standard scalar product
(x\y) = ΣχΊτ, on K" in the case К = R (resp. С, Н).)
(Hint: If a matrix fixes po then its last column and last row have zeroes everywhere
except their last entry. Let Po = (0, ..., 0, 1) e K"·1. In the product decomposition
of the stabilizer of po, the first factor is the stabilizer of Po and the second factor is
the pointwise stabilizer of Pq. )
(3) In exercise (2), describe the action of the second factor of the stabilizer of po
onKP".
With regard to part (1) of the preceding exercise, we remark that (10.12) shows
that, in contrast to the real case, if Κ φ Κ then the stabilizer of po does not act
transitively on the set of pairs of orthogonal vectors in ГРоКН". On the other hand:
10.19 Exercise. O^(Q) acts transitively on the set of totally real subspaces in KH"
of each dimension k < n. It also acts transitively on the set of K-affine subspaces of
each dimension к <п.
Ok(Q) also contains the one parameter subgroup
(cosh t 0 sinh t \
0 /„_, 0 ,
sinh t 0 cosh 11
where /„_ [ is the identity matrix of size η — 1.
10.20 Exercise
(1) Prove that A (t) is a hyperbolic isometry whose axis passes through po and has
initial vector (1, 0,..., 0) at jco. The translation length of A (t) is t.
(2) Deduce (using (1) and 10.18(1)) that the action of 0K(Q) on KH" is transitive.
(Compare with 10.28.)
(3) A e GL(K, η + 1) belongs to 0κ(0 if and only if the columns αϊ,..., α„ of
A, viewed as vectors in K"+1, satisfy (απ+) | a„+l) = —1, (α,·|α;> = lfori= Ι,.,.,η
and (а;\су) = 0 if i /y.

The Boundary at Infinity and Horospheres in ЮН" 309
(4) Let J be the (n + 1, η + l)-matrix that differs form / only in that its last
diagonal entry is — 1. Show that A e #κ(0 if and only if AJA = J. Deduce that if
A e 0K(Q) then A e 0K(Q).
The kernel of the action of Ok(Q) on KH" is the subgroup formed by the scalar
matrices λ/, where λ e lis central and of norm 1. We write POk(Q) to denote
the quotient of Ok(Q) by this subgroup зб. If К is Μ or the quaternions, then this
subgroup is simply ±/, while in the case К = С it is isomorphic to (7(1).
It follows from Exercise 10.18 that the subgroup of PO^(Q) fixing po is
isomorphic to 0(n), U(n), or (Sp(n) χ Sp(l))/{/, -/}, according to whether К is К, С or
the quaternions.
10.21 Exercise
(1) POc(Q) is isomorphic to the quotient of SU(n, l)37 by the cyclic subgroup
generated by el7lil{n+x)l.
(2) In addition to POc(Q), the group of isometries of CH" also contains the
involution given in homogeneous coordinates by (z\,..., z„+\) ы- (zi,..., zn+\).
(3) The fixed point set of the involution in (2) is isometric to KH". (It is a totally
real subspace of maximal dimension.)
(4) Prove that POc(Q) and the involution in (2) generate Isom(CH").
(5) Prove that if К is the quaternions and η > 1, then POk(Q) = Isom(KH").
The Boundary at Infinity and Horospheres in KH"
10.22 The Boundary at Infinity. In Chapter 8 we saw that if X is a complete
non-positively curved Riemannian manifold, then dX is homeomorphic to §""', the
(n — 1 )-sphere. Thus 3KH" is a sphere of dimension η — 1, 2и — 1 or An — 1, according
to whether К is К, С or the quaternions. There is a natural way to realize 3KH" as a
subset of KP": it is the subspace defined by the homogeneous polynomial equation
Q{x, x) = 0. More precisely, KH" С KP" is given by the equation Q(x, x) < 0
and the inclusion map KH" <^-> KP" extends uniquely to a homeomorphism from
KH" U ЭКН" onto the subspace defined by Q(x, x) < 0.
It follows from (10.18) that the stabilizer of each point of KH" acts transitively
on ЭКН".
10.23 Horospheres in Hyperbolic Spaces. In Chapter 8 we saw that if X is a
CAT(0) space, χ e X and xn e X is a sequence such that xn —> ξ е ЭХ, then the
horosphere centred at ξ that passes through χ is the limit of the spheres S(xn, rn)
36 Classically, one writes PO(n, 1) (resp. PU(n, 1)) when К = Μ (resp. K = C).
37 SU(n, 1) С Oc(Q) consists of those matrices which preserve Q and have determinant 1.

310 Chapter П. 10 Symmetric Spaces
where rn = d(x, xn). In the present setting, spheres about [xn] e KH" are the level
sets of the function
*/r пч (χ\χη)(χη\Χ)
Az(M) = r-r ·
(x\x)
If [xn] —> \y] e ЭКН", then the functions p„([x]) converge and the horospheres
centred at \y] have the form
Ηφ] = {[x] : t(x\x) = (x\y)(y\x)},
where t < 0. An alternative description of these horospheres is given in (10.28) and
(10.29).
Other Models for KH"
10.24 The Ball Model. We describe an alternative model for KH" corresponding to
the model of real hyperbolic space described in (1.6.2). The point set of this model,
which we denote KB", is obtained by taking the subset of K" that consists of elements
χ = (jci ,..., χ^) such that ^2" χ,χ; < 1. The distance between points x, у € KB" is
given by the formula
u^m ^ (i - (*ly))(i - №))
cosh d(x, y) = ,
(1 - (x\x))(l - (y\y))
where (x\y) = Σ" x,)', is the standard scalar product on K".
We leave the reader to check that the map which assigns to χ = {x\, ..., xn) e
KB" the point of KH" with homogeneous coordinates {x\, ..., xn, 1) is an isometry
and that this map extends to a homeomorphism from the closed unit ball in K" to
KH" U ЭКН".
10.25 The Parabolic Model. The main benefit of the parabolic model is that, like
the upper half space model for И", it permits a convenient and explicit expression
for the horocyclic coordinates associated to a point at infinity (10.29) and the
associated subgroup AN which stabilizes the given point (see 10.28). (This gives a
way of describing horospheres in KH" that is more amenable to calculation than the
description given in (10.23).)
Consider the form Q on K"+1 given by
η
ΰ{χ, у) = {АуУ = ~х\Уп+\ -хп+\у\ + Y^xtyi-
Q is equivalent to Q by a linear change of coordinates, namely jc, t-» x'j where
xfl = {x\ + xn+i)/V2, x'n+l = (x, - χπ+ι)/α/2, and x, = x'i if i φ 1, η + 1. The
underlying set of the parabolic model for KH" is the subset of KP" defined by
{[x] I Q!{x, x) < 0}. and the metric is defined by the formula in (10.5) with Q
replaced by Q.

Horocyclic Coordinates and Parabolic Subgroups for Hffl" 311
Recall that if χ e KP" has homogeneous coordinates (x\,... ,xn, 1) then its non-
homogeneous coordinates are, by definition, (jq,..., x„). These coordinates give the
classical identification of K" with a subset of KP".
For the purposes of calculation it is useful to describe the parabolic model in non-
homogeneous coordinates. The subset of KP" defined in homogeneous coordinates
by @{x, x) < 0 is precisely the set of points with non-homogeneous coordinates
(jci , ..., xn) satisfying the inequality
η
(*) У^ЗС;.*; < X\ + X\ = 2ΐί1χ\.
ϊ=2
Let oo denote the point of KP" with homogeneous coordinates (1,0, ..., 0). The
boundary at infinity is
ueK" | 2_. χίχι = x\ + x\
U{oo}
oo is a convenient point to use when giving explicit descriptions of the horocyclic
coordinates associated to a point at infinity. (Focussing on oo involves no loss of
generality because Isom(KH") acts transitively on 3KH".)
Horocyclic Coordinates and Parabolic Subgroups for KH"
By definition, oo e 3KH" is the endpoint of the geodesic ray given in homogeneous
coordinates by 11-» co(t) = (e', 0,..., 0, e~'). We shall need the following lemma
in order to determine the subgroup of Isom(KH") that fixes all of the horospheres
centred at oo.
10.26 Lemma. Let X be a CAT(0) space and let Gbe a group acting by isometries
on X. Suppose that h e G leaves invariant a geodesic line с : К —»■ X and that
h.c(t) = c(t + a) where a > 0. Let xq = c(0) and let N с G be the set of elements
g e G such that h~nghn.xo —*■ xo as η —> oo. Then
(1) N is a subgroup.
(2) N fixes c(oo) e dX and leaves invariant the Busemann function associated to
Proof. In order to prove (1), given g, g' e Gwe write gn = h nghnzaa.g'n = h "g'h"
and calculate:
d(g~lg'„.xo,xo) = d(g'n.x0, g„.x0)
< d{g'n.x0, xq) + d(xo, gn.xo);
the right side converges to 0 as η —» oo.

312 Chapter 11.10 Symmetric Spaces
For (2) we must prove that for any χ e X and g e N we have bc(x) = bc{g.x).
Remembering thatxo = c(0), we have:
bc{g-x) — bc{x) = lim d(g.x,hn.xo) — d(x,h".x0),
and
\d(g.x, hn.x0) - d(x, h".x0)\ = \d(x, g~lh".x0) - d(x, hn.x0)\
< d(g-lhn.x0, hn.x0) = d(h-"g-]hn.x0,xo),
which tends to zero as η tends to infinity. D
Let Οκ(β') be the subgroup of GL(n+1, K) that leaves invariant the form Q. The
natural action of 0^{Q) on KP" preserves the parabolic model of KH". Consider
the 1-parameter subgroup of 0^{Q) formed by the elements
/e' 0 0 \
A(t) = I 0 /„_, 0 .
\0 0 e-4
This subgroup leaves invariant the geodesic line given in homogeneous coordinates
by co(i) = (e', 0,..., e~'), andA(i).co(0) = co(t). (Recall that co(oo) = oo and note,
for future reference, that co(i) has non-homogeneous coordinates (e2', 0, ..., 0).)
We shall apply the preceding lemma with A(l) in the role of h and cq in the
role of c. Let N С 0^(Q') be the subgroup formed by the matrices ν such that
lim,_>.ooi4(—ί)ΐ"·ι4(0 is the unit matrix. Calculating A(-t)vA(t) and noting that the
off-diagonal entries must tend to 0 and the diagonal entries must tend to 1, one sees
that N consists of matrices of the form
/1 A/, 2 M,3\
ν = I 0 /„_, M23 .
\0 0 1 /
One further calculates (exercise!) that (v(jc)|vCy))' = (x\y)' for all x, у with (x\x)' < 0
and (У1У)' < 0 if and only if Mi 2 (which is a (1, и - l)-matrix) andM23 (which is
an (n — 1, 1)-matrix) satisfy
(♦) M23 = Ίίπ andM]2M23 = M]3 + M]3,
where Μ denotes the standard conjugation on К and Μ is the transpose of M.
Let Л denote the 1-parameter subgroup {A(t): t e Щ.
10.27 Lemma.
(1) AN := {A(t)v | t e Ж, ν e AN} is a subgroup ofOK(Q').
(2) N is normal in AN.
(3) 7/ΊΚ = M. then N is abelian, and г/К is С or the quaternions then N is nilpotent
but is not abelian if η > 1.

Horocyclic Coordinates and Parabolic Subgroups for KH.n 313
Proof. These are simple calculations. Π
The following proposition is a special case of a general phenomenon for
symmetric spaces (cf. 10.50 and 10.69).
10.28 Proposition.
(1) The subgroup AN acts simply transitively on KH".
(2) The orbits ofN are the horospheres centred at the point oo.
(3) The geodesic lines in the parabolic model that are asymptotic to cq are precisely
the geodesies t t-» v.co(t), where ν e N and cq is given in homogeneous
coordinates by co(t) = (e',0,... ,0, e~').
Proof. Recall that oo is the endpoint of cq in the forwards direction. We have already
seen (Lemma 10.26) that υ.οο = oo for all υ e N and that the orbits of N are
contained in the horospheres centred at oo. It is clear that A acts simply transitively
on Co(K) and hence on the set of horospheres centred at oo. Thus, in order to prove
(1) and (2) it suffices to show that given an arbitrary point x, with non-homogeneous
coordinates (x\,..., x„) say, there exists a unique t e К and a unique ν e N such
that χ is the image under vA(t) of the point given in non-homogeneous coordinates
byjc0 = (1,0, ...,0).
Writing ν in block form as above, vA(t).xo = χ becomes the system of equations:
x\ = e2t + M13
X2 = M23
where X2 = fe, · · ■, xn)· Thus Хг uniquely determines М2з and hence M]2 and
9ΐΜΐ3 = |л2Х2 (by (♦)). Moreover, X2X2 is, by definition, equal to the left side of
the equation (*) displayed in (10.25), and hence it is less than 2Ъ\х\. There therefore
exists a unique teM such that Wix\ = e2' + ЭШв, and M]3 is uniquely determined
by X] = e1' + M13. Thus the above system of equations has a unique solution
(t, v) e R x N. D
10.29 Horocyclic Coordinates in the Parabolic Model. Let χ be a point in the
parabolic model that lies on the horosphere through the point x, with homogeneous
coordinates (e',0,..., e~') and let ν be the unique element of N such that v.x, = x.
Let the submatrices Mis of ν be as above.
The horocyclic coordinates of χ are, by definition, the entries of the submatrix M23
together with the purely non-real part of Л в. Thus χ is assigned 2n - 1 (resp. An - 1)
real coordinates when К is С (resp. the quaternions). According to (10.28), these
coordinates together with t uniquely specify x.

314 Chapter П. 10 Symmetric Spaces
The Symmetric Space P(n, R)
We turn now to the study of higher-rank symmetric spaces. For the reasons
explained in the introduction, we shall concentrate our attention on P(n, R), the space
of positive-definite, symmetric (n, и)-matrices with real coefficients.
Our first task will be to metrize P(n, /?); we shall do so by describing a Riemannian
metric on it. We shall then describe the natural action of GL(n, K) on P(n, R) and
prove that this action is by (Riemannian) isometries. First, though, we pause to fix
notation and remind the reader of some basic definitions.
10.30 Notation. We shall write M(n, R) to denote the algebra of (n, «)-matrices with
real coefficients. GL(n, M.) will denote the group of invertible (и, «)-matrices. The
action of matrices on W will be on the left (with elements of K" viewed as column
vectors). The transpose of a matrix A will be denoted Ά.
Let S(n, W) be the vector subspace of M(n, R) consisting of symmetric matrices,
and let P(n, W) С S(n, W) be the open cone of positive-definite matrices. We remind
the reader that a matrix A is symmetric if and only if for all vectors v, w el'we
have (Av, w) = (v,Aw), where ( , ) is the usual scalar product on W; and A is
positive definite if, in addition, (Αν, υ) > 0 for every ν φ 0.
Given a symmetric matrix A one can always find an orfhonormal basis for W
consisting of eigenvectors for A. More precisely, there exists an orthogonal matrix
О e SO(ri) of determinant one such that 'ОАО is diagonal. The entries λι,..., λ„ of
this diagonal matrix are all positive if and only if Л is positive definite (they are the
eigenvalues of A).
P(n, R) as a Riemannian Manifold
10.31 The Riemannian Metric on P(n, K). As P(n, Ж) is an open set of S(n, K),
it is naturally a differentiable manifold of dimension n(n + l)/2. The tangent
space TpP(n, R) at a point ρ is naturally isomorphic (via translation) to S(n, K).
On TpP(n, K) we define a scalar product by:
(X\Y)p=Tt(p-]Xp-]Y),
whereX, Υ e TpP(n, R) = S(n, R)andTr(A)isthetraceofamatrixA. (IfX e S(n, Щ
has entriesxy, then (X\X), = \\X\\2 = £jc?·.)
This formula defines a Riemannian metric on P(n, K).
10.32 The Action of GL(n, K) on P(n, Ш). The group GL(n, W) acts on S(n, W) and
P(n, R) according to the rule
g.A := gA'g,
where g e GL(n, M.) and A e S(n, Щ. This action leaves P(n, Щ invariant.

P(n, R) as a Riemannian Manifold 315
10.33 Proposition. Consider the action ofGL(n, K) on P{n, Ж).
(1) The action is transitive.
(2) The action is by Riemannian isometries.
(3) The stabilizer of I e P(n, K) is 0{n).
(4) {±/} С GL(n, Ж) acts trivially on P(n, K) and GL(n, K)/{±/} acts effectively.
Note that in the light of this proposition we may identify P(n, Ш) with the
homogeneous space GL(n, Щ/0(п).
Proof. Given ρ e P(n, K), there exists О e SO(ri) such that D = ΌρΟ is a diagonal
matrix D with positive entries λ,-. Letpl/2 = (O D]/2 Ό), where D]/2 is the diagonal
matrix with entries χ/λ,-. Then ρ = pl^2.I. This proves (1).
Part (3) is obvious and (4) is an easy exercise. To prove (2) one calculates:
given g e GL(n, K) and X,Y e TpP(n, K), the derivative of g at ρ maps X onto
g.X = gX'geTg.pP(n,W), so
(8.χ\8.γ)8.ρ = ττ('8-]ρ-ι8-]8χ'8'§-]Ρ-]§-]§γι8)
= Tr(p-lXp-]Y) = (X\Y)p.
О
Recall that a symmetric space is a connected Riemannian manifold Μ where for
each point ρ e Μ there is a Riemannian isometry σρ of Μ such that σρ(ρ) = ρ and
the differential of σρ at ρ is multiplication by — 1.
10.34 Proposition. P(n, R) is a symmetric space.
Proof. Given ρ e P(n, R), the required symmetry σρ is q t-» pq~~]p. This map is the
composition of σ/ : q t-» g~' and q t-» /?.g. We have already seen that the latter is
an isometry. To see that σ/ is an isometry38, first note that its derivative at q sends
X e S(n, K) = Tq(P(n, K)) (the initial vector of the curve m q + tX) to X :=
q~lXq~] e S(n,R) = Tq-1P(n, K) (the initial vector of the curve t ь» (q + tX)~]).
Thus we have
(X|7)r. = Тг(#?У) = Тг^-'од-' ^"') = Tr^-'X^"1 У) = {X\Y)q,
and hence σ/ is an isometry. It is obvious that the differential of σ/ at / is multiplication
by — 1, and hence so is the differential of σρ at p, because σρ is the conjugate of at
by q ь->· p]/2.q. □
10.35 Remark. In a symmetric space the composition of two symmetries aq ο σρ is
called a transvection; it acts as a translation on any locally geodesic line containing
ρ and q. In the preceding proof we saw that the action of ρ e P{n, R), viewed as an
38 Under the natural identification P(n, M) = GL(rc, Ш)/0(п), the symmetry σ/ is induced by
the map д t-» '<?""'.

316 Chapter П. 10 Symmetric Spaces
element of GL(n, K), is the composition σρ ο σ/, so the action of P{n, Ж) С GL(n, K)
is by transvections.
The Exponential Map exp: M(n, R) -> GL(n, R)
Our next major goal is to prove that P(n, K) is a CAT(O) space. Since we have
described the metric in terms of a Riemannian structure, the natural way to prove
this is to use (1 A.2) from the appendix to Chapter 1. This requires that we understand
the divergence properties of geodesies issuing from a point in P(n, Ш). In order to
gain such an understanding we must examine the exponential map.
We have a concrete interpretation of T/P(n, K) as S(n, K), and the exponential
map can be interpreted in a correspondingly concrete manner, namely as the familiar
operation of exponentiation for matrices, exp : M(n, M.) —> GL(n, R), where
oo
ехрЛ :=J2A"/nL
We shall need the following properties of exp.
10.36 Lemma.
(1) ехр(Л) = '(exp A).
(2) If AB = BA, then А{ец>В) = (ехр5)Л and ехрЛ exp5 = ехр(Л + B). In
particular exp(-A) = (ехрЛ)-1.
(3) exp(gAg-]) = gexp(A)g-]forallg e GL(n, K).
(4) |(exp tA) = (exp tA)A for all t e R.
(5) The map exp is differentiable.
(6) exp maps S(n, K) bijectively onto P(n, K).
Proof. We shall only prove (6). It is obvious that if A is symmetric then expA is
symmetric. It is also obvious that every diagonal matrix in P(n, K) is exp of a diagonal
matrix, so since every ρ e P(n, M.) is O.A (action of 10.32) for some О е SO(ri) and
some diagonal Δ, the surjectivity of exp follows from (3).
A symmetric matrix A is uniquely determined by its eigenvalues and the
associated direct sum decomposition 01=1£'; of W into the eigenspaces of A. The
decomposition associated to exp Л has exactly the same summands, and if the action
of Л on Ei is multiplication by λ, then the action of exp Л on £, is multiplication by
εχρ(λ,). Thus one can recover the action of Л on W from that of ехрЛ, and hence
exp is injective on S(n, K). D
10.37 Remark. It follows from the preceding lemma that for every ρ e P(n, K) there
exists a unique symmetric matrix X such that expX = p. Let p]/2 = exp(X/2); this
is the unique q e P(n, K) such that q1 = p, and we shall see shortly that it is the
midpoint of the unique geodesic joining / to p.

The Exponential Map exp: M(n, Ш) -> GL(n, M) 317
For future reference (10.40), note that exp(X/2) acts on P(n, K) as the transvection
σρι/2 ο σ/ which sends ρ to /.
The following exercises contain some further useful facts concerning the
exponential map; some of these facts will be needed in the sequel.
10.38 Exercises
(1) For all А, В e M(n, K), we have
(ехрА)Я(ехр-А) = (exp(adA))(5),
where ad^ is the endomorphism of the vector space M(n, K) defined by
ΆάΑ(Β)= [Α, Β] :=ΑΒ-ΒΑ.
Hint: Express ad^ as La — Ra, where La (resp. Ra) is the endomorphism of M(n, K)
mapping С to AC (resp. CA). Note that LaRa = RaLa, hence
■ τΡώΊ
{La - RAf = Σ τ^Α^-Α-
p+q=k'
,p\q\
(2) Verify that the formula (A\B) = Tr(AS) defines a scalar product on M(n, K).
Show that if X e S(n, K), then ad* is a self-adjoint operator on M(n, K), i.e. for any
A, Be M(n, Ж), we have
ЫХА\В) = (Α\ΆάχΒ).
(3) Given X e S(n, K), consider the endomorphism τχ of Μ (η, К) defined by the
formula
τχ(Υ) = - exp(-X/2) exp(X + tY) exp(-X/2) .
dt i=o
Prove that τχ is self-adjoint.
(Hint: You may first wish to show that
σο ι
τχ(Υ) = Σ Τ\ Σ exp(-X/2)X"№ exp(-X/2)
k= 1 p+q=k-1
hence
(τχ(Υ)\Ζ) = Σ Σ Tr(exp(-X/2)X"№ exp(-X/2)'Z).
k=lp+q=k-l
Then use the fact that Tt(AB) = Tt(BA).)
(4) Show that expTr(A) = det(expA).

318 Chapter П. 10 Symmetric Spaces
P(n, R) is a CAT(O) Space
We are now in a position to prove:
10.39 Theorem. P(n, K) is а рюрег CAT(O) space.
As we remarked earlier, since we have described the metric on P(n, M.) by giving
the associated Riemannian structure, the natural way to prove this theorem is to apply
(1 A.2). In order to do so we must show that the restriction of exp to S(n, W) satisfies
the conditions on the map e in (1. A. 1). This is the object of the next lemma, for which
we require the following notation: given X e S(n, R), we define an endomorphism
adx/2 fr£ (2Л+1)!
10.40 Lemma.
(1) Consider the differential τχ atX e S(n, M.) of the composition of exp and the
transvection sending expX to I (see (10.37)). By definition, τχ is the
endomorphism ofS(n, Щ given by
Τχ(Υ) = j{ ( exp(-X/2) exp(X + tY) exp(-X/2))
We claim that
sinh adx/2
adx/2
(2) τχ does not diminish the norm of any tangent vector. More precisely, for all
Υ e S(n,R) we have \τχ(Υ)\ > \Y\ with equality if and only ifXY = YX.
(Recall that \Y\2 = Ύΐ(Υ2).)
Proof. First we prove (1). Given Υ e S(n, Ж) we consider the curve t t-» X(t) =
X + tY. By differentiating the identity
X(i)(expX(i)) = (expX(i))X(i)
at t = 0, writing E0 = jt expX(i)|(=o and noting that Υ = jtX(t)\t=0 and X = X(0),
we get:
Υ (expX) +XE0 = E0X + (expX) Y.
That is,
XE0 - E0X = (ехрХ)У - У(ехрХ).
By definition, τχ(Υ) = exp(—X/2)E0 exp(—X/2). Thus, multiplying both sides of
the above equation on both the left and the right by exp(—X/2), and noting that
exp(-X/2) commutes with X, we get:
!=0

Ρ(η, Ж) is a CAT(O) Space 319
Χτχ(Υ) - τχ(Υ)Χ = exp(X/2) У exp(-X/2) - exp(-X/2) Υ exp(X/2).
By definition, 2 sinh(adx/2) = exp(adx/2) — exp(ad_x/2)· Therefore, applying
exercise 10.38(1) to each term on the right-hand side of the above equation we get:
Χτχ(Υ) - τχ(Υ)Χ = 2 sinh(adx/2)(y).
Thus
sinh adx/2
(*) adx/2 οτχ = adx/2-
adx/2
Let F := τχ — (sinh adx/2)/adX/2. We shall show that F = 0 as an endomorphism of
M(n, K). Both ad^/2 and τχ are self-adjoint operators on M(n, K) (exercises 10.38
(3 and 4)), hence F is self-adjoint. Let N С M(n, K) be the kernel of adx/2; it is the
vector subspace {Y e M(n, W) : XY = YX}. By (*) we know that F maps M(n, Ж)
to N. It follows that the restriction of F to N1- (the orthogonal complement of N) is
0, because if χ e Ν1 then for all у е М(п, Щ we have (A(x)\y) = (x\A(y)) = 0. It
only remains to check that F vanishes on N. But this is obvious, because if XY = YX
then τχ(Υ) = Υ and (sinh adx/2)/adx/2(y) = Y.
In order to prove (2) we choose an orthonormal basis e\,..., e„ for W consisting
of eigenvectors for X; say (X/2)e, = λ,-e;. Let Ey e M(n, K) be the endomorphism
defined by E^eu = 5^e;, where Sy* = 1 if j = k and zero otherwise. (The elements
Eij form an orthonormal basis of M(n, №.).) We have Χ/2 = Σ"=1 λ,·£ ,7, so since
EijEa = §кЕц, it follows that ааХ/2(Еу) = (λ, — λ/)Ε^, and hence
sinh(l,-- λ;)
r*(£,y) = — 7-^-Ey
λ; — kj
If У е 5(и, Μ), then Υ = Σ"._, y,y£,y where y,y = >};, and we have
Y^ sinh(l; - λ,)
Therefore
■ ■ ι ^-i — ^-/
Mioi2 = тг(гх(у)2)=ς^-^τ^ > Σ4 = m2>
because |(sinhl)/l)| > 1. One gets equality in this expression if and only if y,j = 0
whenever λ,- φ λ,. And this happens if and only if the action of Υ on W leaves the
eigenspaces of X invariant, which is the case if and only if XY = YX. D
Proof of 10.39. In Lemma 10.36 we saw that exp : S(n, K) -> P(n, K) is a diffeo-
morphism, and the preceding lemma shows that at the point / e P(n, K), the map
exp satisfies the conditions required of e in Lemma 1A.1 (with ε > 0 arbitrary).
Since GL(n, K) acts transitively by isometries on P{n, K), it follows from (1 A.2) that

320 Chapter П. 10 Symmetric Spaces
P(n, W) is a CAT(O) space. And since exp does not decrease distances, P{n, Щ is
proper. Thus Theorem 10.39 is proved. D
As a consequence of (10.39) we have:
10.41 Corollary. Every compact subgroup of GL(n, M.) is conjugate to a subgroup
ofO(n).
Proof. If К С GL(n, К) is compact, then every if-orbit in P(n, M.) is compact and
hence bounded. It follows that К is contained in the stabilizer of the circumcentre ρ
of this orbit (2.7). The action of GL(n, K) is transitive, so there exists g e GL(n, M.)
such that g.I = p. The stabilizer of / is 0(ri) (10.33) and therefore g~]Kg С 0(п).
D
We also note two consequences of Lemma 10.40.
10.42 Corollary.
(1) The geodesic lines с : К —> P(n, К) with c(0) = ρ are the maps c(t) =
g (exp tX)'g, whereX e S(n, R) with Tr(X2) =\andp = g'g.
(2) If ρ = expX then d(I, pf = Tr(X2) = |X|2.
(3) More generally, the restriction of exp to any vector subspace of S(n, M.)
consisting of commuting matrices is an isometry.
10.43 Exercise. Let А С GL(n, K) be the group of diagonal matrices with positive
entries on the diagonal. The action of Л on P(n, M.) is by transvections and/ e Min(A)
(see 10.33 and 6.8(1)). Prove that Min(A) = A.I = A and that А С P(n, K) is
isometric to E" (cf. 6.8(5) and 7.1).
Flats, Regular Geodesies and Weyl Chambers
Flat subspaces play an important role in determining the geometry of symmetric
spaces. In this section we shall describe the flat subspaces in P(n, M.) (10.45). In
contrast to the rank one case, if a CAT(0) symmetric space Μ has rank > 2, for
example Μ = P(n, W) with η > 2, then Isom(M) does not act transitively on the set
of geodesies in Μ (unless Μ = E"), however it does act transitively on pairs (F, p),
whereFis a flat subspace of maximal dimension and/? e F. In the case Μ = P(n, K),
one way of seeing that Isom(M) does not act transitively on the set of geodesies in
Μ is to observe that some geodesies lie in a unique maximal flat while others do
not (10.45). The set of points q e F ^ {p} such that F is the unique maximal flat
containing [p, q] is open and dense in F; it is the complement of a polyhedral cone
of codimension one and its connected components are called Weyl chambers.
The description of the flat subspaces in P(n, M.) and their decomposition into
Weyl chambers will be important in later sections when we come to describe the
geometry of P(n, M.) at infinity.

Flats, Regular Geodesies and Weyl Chambers 321
10.44 Definition. A subspace F с P(n, Щ is called a flat of dimension к (more
briefly, a k-flat) if it is isometric to E*. (Thus 1 -flats are the geodesic lines in P(n, Щ.)
If F is not contained in any flat of bigger dimension, then F is called a maximal flat.
In (10.43) we described a particularly nice flat Л С P(n, К). This is the image
under exp of the subspace do С S(n, K) consisting of diagonal matrices. The
following proposition shows that the action of GL(n, Ж) conjugates every flat in P(n, K)
into a subspace of A.
10.45 Proposition.
(1) Every flat in P(n, M.) is contained in a maximal flat, and every maximal flat has
dimension n.
(2) GL(n, Ж) acts transitively on the set of pairs (F, p), where F с Р(п, Щ is a
maximal flat and ρ e F.
(3) The exponential map exp : S(n, K) —> P(n, Ж) induces a bijection from the set
of vector sub spaces {о С S(n, K) | XY = YXforallX, Υ e a} to the set of flats
F = exp α that pass through I.
(4) The geodesic line 11-» expiX through I is contained in a unique maximal flat
if and only if the eigenvalues ofX e S(n, K) are all distinct.
Proof. In any complete Riemannian manifold Μ of non-positive curvature, if F С М
contains ρ and is isometric to E*, then F is the image under the exponential map
of a ^-dimensional subspace in TpM. Thus part (3) of the present proposition is an
immediate consequence of 10.40(2).
Let oo С S(n, K) be the subspace of diagonal matrices. If a symmetric matrix
commutes with all diagonal matrices then it must be diagonal, so it follows from (3)
that Л = exp oo is a maximal flat in P(n, Ж).
Oo has dimension η and GL(n, K) acts transitively опР(и, К), so in order to prove
(1) and (2) it suffices (in the light of (3)) to show that if α is a vector subspace of
S(n, Ж) such that XY = YX for all X,Y e a, then there exists О е SO(n) such that
O.a С cio.
Given o, we choose a matrix X e о that has the maximum number of distinct
eigenvalues, к say. Let О e SO(n) be such that OX Ό is a diagonal matrix. We must
show that ΟΥ Ό is diagonal for every Υ e o. It suffices to show that every eigenvector
of X is an eigenvector of Y. (The columns of О are eigenvectors for X.)
Let W = @Ί=]Ει be the decomposition of K" into the eigenspaces of X
(corresponding to the distinct eigenvalues λι < · · · < λ*.). If Υ e о then Υ commutes
with X and hence leaves each £, invariant. If Υ e о is close to X, then for each i the
eigenvalues of Y\e-, must be close to λ,·. Since Υ has at most к eigenvalues and the λ,
are distinct, it follows that Y\El (which is diagonalizable) must be a multiple of the
identity. This completes the proof of (3).
In (4) it suffices to consider the case where X is diagonal. If two of the
diagonal entries of X are the same, then the permutation matrix which interchanges the
corresponding basis vectors of W commutes with X. On the other hand, if all of the
diagonal entries of X are distinct, then the preceding argument shows that the only

322 Chapter 11.10 Symmetric Spaces
symmetric matrices which commute with X are diagonal. Thus A is the only maximal
flat containing X if and only if the eigenvalues of X are all distinct. D
The final part of the preceding proposition shows that there are different types of
geodesies in P(n, R). The following definition provides a vocabulary for exploring
these differences.
10.46 Definition (Singular Geodesies and Weyl Chambers). Let Μ be a symmetric
space. A geodesic line or ray in Μ is called regular if it is contained in a unique
maximal flat, and otherwise it is called singular.
Let F С Μ be a maximal flat. A Weyl chamber in F with tip at ρ e F is a
connected component of the set of points q e F χ {ρ} such that the geodesic line
through ρ and q is regular.
If two geodesic rays с and d are asymptotic, then с is singular if and only if
d is singular (cf. 10.65). In particular, since singularity is obviously preserved by
isometries, the group of isometries of P(n, R) does not act transitively on дР(п, R)
if η > 2 (cf. 10.22 and 10.75).
It follows immediately from 10.45(4) that the Weyl chambers in any flat F С
P(n, R) are open in F. The following proposition gives more precise information.
10.47 Proposition. Let F с P(n, R) be a maximal flat, fix ρ e F.
(1) {q e F χ {ρ} | the geodesic line containing [p, q] is regular} is the complement
in F of\n{n — I) flats of dimension (n — 1) that pass through p.
(2) There are n\ Weyl chambers in F with tip at p.
(3) GL(n, R) acts transitively on the set of Weyl chambers in P(n, R).
Proof. In the light of 10.45(2) it suffices to consider the case where ρ = I and
F = A (the space of diagonal matrices with positive entries in the diagonal). Each
q e A can be written uniquely in the form diag(e(1,..., e'") e A; in other words q
is the image under exp of the diagonal matrix diag(?i,..., t„) e A; we regard the
t; as coordinates for oo (the subspace of S(n, R) comprised of diagonal matrices).
According to 10.45(4), the geodesic line through ρ and q φ ρ is regular if and only
if all of the i, are distinct, i.e. if and only if (t\,..., t„) does not lie in one of the
\n(n — 1) codimension-one subspaces of ao defined by an equation t, = tj, where
j > i. The (n — l)-dimensional flats in part (1) of the proposition are the images of
these subspaces under exp.
The Weyl chambers in A with tip at / are the images under exp of the n\ open
convex cones defined by ίσ(ΐ) > · · · > ίσ(„), where σ is a permutation of {1,..., n].
The subgroup of SO(n) consisting of monomial39 matrices acts transitively on these
Weyl chambers. Parts (2) and (3) now follow, because GL(n, R) acts transitively on
pairs (F, p) and takes Weyl chambers to Weyl chambers (because they are intrinsically
defined in terms of the metric on P(n, R)). Π
matrices with one non-zero entry in each row and column

The Iwasawa Decomposition of GL(n, Ш) 323
10.48 Exercise: P(n, K) as a space of ellipsoids in E". Given ρ e P(n, K) with inverse
(p'ij), let Sp be the ellipsoid in E" defined by the equation J^P'i^i^' = 1· Prove that
ρ м- £p is a GL(n, K)-equivariant map from P(n, K) to the set of (non-degenerate)
ellipsoids in E" centred at the origin.
This map sends / e P(n, M.) to the sphere Y^xJ = 1. Describe the images of
the maximal flats through /, the images of the Weyl chambers with tip at /, and the
images of the regular geodesies through /.
The Iwasawa Decomposition of GL(n, Ш)
Consider the following subgroups of GL(n, K):
• К = 0(n) is the group of orthogonal matrices.
• Л is the subgroup of diagonal matrices with positive diagonal entries.
• iV is the subgroup of upper-triangular matrices with diagonal entries 1.
Note that AN is the group of upper triangular matrices with positive diagonal
entries and that N is a normal subgroup in AN.
The subgroups K, A and N are intimately related to the geometry of the symmetric
space P(n, K): we saw in (10.41) that К and its conjugates are the maximal compact
subgroups of GL(n, K), and we saw in (10.43) that the orbit of / under the action of
Л is a maximal flat; N plays a key role in describing the horospherical structure of
P(n, Ж) (see (10.50) and compare with the groups A and N considered in (10.28)).
10.49 Proposition (Iwasawa Decomposition). The map Κ χ Α χ Ν -> GL(n, Ж)
sending (к, a, n) to кап is a diffeomorphism. More informally,
GL(n, R) = KAN.
Proof. This follows directly from the existence and uniqueness of the Gram-Schmidt
orthogonalisation process. If υι,..., v„ are the column vectors of a matrix g e
GL(n, Ж), there exist unique λ,^ e Μ for 1 < i < j < η such that the vectors
v[ = vi, v'2 = ληνι +V2, ..., v'n = λι„ι>ι + ■ · · + λ„_ι „υ„_ι + νη are mutually
orthogonal; if и' e N is the matrix whose (i, gentry is Хц for i < j, then gn' is the
matrix whose columns are v\, ..., v'n. Let a be the diagonal matrix whose entries in
the diagonal are the norms of the υ-. Then gn'a~l = к is an orthogonal matrix and
we have g = кап, where η = n'~ . This decomposition is unique and the entries of
k, a and η depend differentiably on the entries of g. D
10.50 Corollary. The map Α χ Ν -> P(n, K) sending (a, n) to an.I = anbia is a
diffeomorphism. In other words the subgroup AN ofGL(n, №.) acts simply transitively
on P(n, K).
Proof. We know (10.33) that GL(n, K) acts transitively on P(n, K), so given ρ e
P(n, Ж) the set Cp = {g e GL(n, №)\g.p = /} is non-empty. The stabilizer of /

324 Chapter 11.10 Symmetric Spaces
is К ~ 0(n), so Cp is a coset Kan and by the proposition a and η are uniquely
determined by p. □
We mention two other closely related and useful decompositions:
10.51 Exercises
(1) Prove that the map (k,p) ь» kp from 0(n) χ P(n, K) to GL(n, K) is a diffeo-
morphism (its inverse is the map g t-» {g{'gg)~xl1, Cgg)l/2)
(2) Let A be the set of diagonal matrices diag(Xi, ..., λ„) with λι > · ■ · >
λ„ > 0. Show that GL{n) = KA K. More precisely, show that every g e GL{n, Щ
can be written as g = как1 with k,ld e 0(n) and аеЛ and show that a is uniquely
defined by g.
(Hint: To show existence use (1). To show uniqueness, note that the eigenvalues
of a are the square roots of the eigenvalues of 'gg.)
The Irreducible Symmetric Space P(n, W)i
Following the outline of our treatment of hyperbolic spaces, we now turn our attention
to the study of convex submanifolds of P(n, K), where the range of such submanifolds
is much richer than in the hyperbolic case (see 10.58). The first such submanifold
that we consider is P(n, R)i, the subspace consisting of positive definite matrices
with determinant one.
A totally geodesic submanifold of P(n, K) is, by definition, a differentiable sub-
manifold Μ of P(n, K) such that any geodesic line in P(n, K) that intersects Μ in
two points is entirely contained in M. Such a submanifold is convex in the sense of
(1.1.3) and hence is a CAT(0) space. Moreover, if p lies in such a submanifold then
the symmetry σρ leaves Μ fixed, and hence Μ is a symmetric space.
10.52 Lemma.
(1) P(n, M)i с P(n, K) is a totally geodesic submanifold.
(2) SL(n, Щ с GL(n, K) leaves P(n, M)i invariant and acts transitively on it.
SL(n, K) acts effectively if η is odd and otherwise the kernel of the action is
{±/}.
(3) P(n, M)i = ехр5Хи, K)o, where S(n, K)o С S(n, K) is the vector subspace
consisting of matrices with trace zero.
Proof. P(n, M)i is obviously invariant under the action of SL(n, Ж), and the proof of
10.33(1) shows that the action of SL(n, K) is transitive; this proves (2). Part (3) is an
immediate consequence of 10.36(6) and the (easy) exercise 10.38(4).
It follows from (3) that any geodesic 11-» exp(iX) joining / to a point of P(n, R)i
is entirely contained in P(n, R)i, so by homogeneity (2) we see that P(n, M.)\ is totally
geodesic. D

The Irreducible Symmetric Space P(n, Μ), 325
Lemma 10.52 illustrates the fact that essentially the whole of our treatment of
P(n, R) works equally well if we replace P(n, R) by P(n, K),, GL(n, R) by SL(n, R),
and S(n, R) by S(n, R)q. In particular we have the natural identification
P(n, R)i = SL(n, R)/SO(n).
(Hence P(2, K)i = SL(2, R)/SO(2) is isometric to H2, up to a scaling factor.)
We also have the diffeomorphism
SO(n) x P(n, R)i -* SL(n, R)
given by (k, p) t-» kp, and the Iwasawa decomposition
SL(n, R) = KAN
where К = SO(n), the subgroup N is the group of upper-triangular matrices with
diagonal entries one and A is the group of diagonal matrices with positive diagonal
entries and determinant one. And following the proofs of the corresponding results
for the action of GL(n, R) on P(n, R), one sees that SL(n, R) acts transitively on the
set of Weyl chambers in P(n, K)i and on the set of pointed flats.
A further important point to note is that P(n, R)\ is one factor in a (metric) product
decomposition of P(n, R). To see this, first observe that the centre of GL(n, R) consists
of the scalar multiples of the identity and the quotient of this subgroup by {±/} acts
by hyperbolic translations on P(n, R). It follows from (6.15) and (6.16) that P(n, R)
has a non-trivial Euclidean de Rham factor, and it is not difficult to see that P(n, K)|
is the complementary factor. More precisely:
10.53 Proposition. The map (s, p) t-» es^'p gives an isometry
R χ P(n, M)i = P(n, R).
Moreover, P(n, R)\ cannot be expressed as a non-trivial (metric) product.
We leave the proof as an exercise for the reader. (One way to see that P(n, M)i
is irreducible is to consider how flats intersect in any product and then compare this
with the following description of Weyl chambers in P(n, R)\.)
10.54 Flats and Weyl Chambers in P(n, M)i. There is an obvious bijective
correspondence between maximal flats in P(n, K)i and maximal flats in P(n, M):amaximal
flat F\ in P(n, R)i corresponds (under the decomposition in (10.53)), to the maximal
flat F = R x F\ in P(n, R). Similarly, there is a correspondence between the Weyl
chambers in P(n, K)i and the Weyl chambers in P(n, R). Following the proof of
(10.47) we give an explicit description of the Weyl chambers in P(n, M)i.
Let о С S(n, R) be the vector space of diagonal matrices with trace zero. Then
A = exp о is a maximal flat in P(n, K)i and exp | α is an isometry. The Weyl chambers
in A with tip at / are the images under exp of the n\ open convex cones of the form

326 Chapter П. 10 Symmetric Spaces
{diag(ib · · ·, t„) | I> = 0 and ta(X) > > ta(n)], where σ e S„ (the group of
permutations of {1,..., и}).
The closure of the cone indexed by σ is the intersection of the (n — 1) open half-
spaces ίσ(;) — ίσ(ι+ΐ) < 0 with the hyperplane t\ + ■ ■ ■ + t„ =0. This is a simplicial
cone. Its faces of codimension k are obtained by replacing k of the inequalities
*σ(ΐ) > · · · > ta(Ji)} by equalities. Thus, for example, a codimension one face is
defined by the equations t\ + ■ ■ ■ + t„ = ίσ(,+ΐ) — ίσ(ι) = 0 and ίσ(/+ΐ) — ίσ(/) < 0 for
The Weyl chambers associated to the permutations σ and σ' have a codimension-
one face in common if and only if for some i e {1,..., и — 1} we have σ'(ί) =
σ(ί + 1), σ'(ί + 1) = σ(ΐ) and σ'φ = σ(/) for; i [i, i+l].
The obvious action of the symmetric group Sn on the coordinates i, induces a
simply transitive action of S„ on the above set of Weyl chambers. Sn is called the
Weyl group; the interested reader can check that it is the quotient of the subgroup of
SO(n) that leaves the flat Л invariant by the point-wise stabilizer of A.
In order to determine the shape of the Weyl chambers, we calculate the dihedral
angles between the codimension one faces of the Weyl chamber corresponding to
the trivial permutation. Let F, be the codimension one face defined by the equation
ti = ti+\ and let e\,..., e„ be the orthonormal basis of the space of diagonal matrices
corresponding to the coordinates i,. The unit normal to F-t is щ = -jtifii — e,+i) and
therefore (щ\щ) = —1/2 if 7" = i + 1 and (щ\щ) = 0 if j > i+l. Taking inverse
cosines, we see that the dihedral angle between F, and F,+i is π/З and the angle
between the faces F, and Fj is π/2 if \i —j\ > 2.
10.55 Example. In the notation of (10.54), if η = 3 then о can be identified with the
plane in M3 defined by the equation it + i2 + ?з =0. The Weyl group is S3, so there
are 6 Weyl chambers contained in A with tip at /; they are the images under exp of
the sectors in о which form the complement of three lines through 0 as indicated in
figure (10.1).
*2 > h = t3 ti = t2 > h
\ t2 > h > t3 /
*2 > *3 > *I \ / ti > *2 > ti
h = h> ^ Д h > t2 = t3
*3 > *2 > tl / \ ti>h>t2
/ t3 > ti > h \
h > h = *2 ti = *з > *2
Fig. 10.1 The Weyl chambers in a maximal flat of P(3, M), = 5L(3, M)/50(3)

Reductive Subgroups of GL(n, M) 327
Reductive Subgroups of GL(n, R)
In the introduction to this chapter we mentioned that every simply connected
symmetric space of non-positive curvature is isometric (after rescaling) to a totally geodesic
submanifold of P(n, R). We also alluded to the close connection between the study
of such symmetric spaces and the theory of semi-simple Lie groups. In this section
we shall present some totally geodesic submanifolds of P(n, R) and describe the
connection between such submanifolds and the subgroup structure of GL(n, R) (see
10.58). The subgroups with which we shall be concerned are the following.
10.56 Definition (Reductive and Algebraic Subgroups of GL(n, R)). A subgroup G
of GL(n, R) is reductive if it is closed and it is stable under transposition: if A e G
then Ά e G.
A subgroup G С GL(n, R) is algebraic if there is a finite system of polynomials
in the entries of M(n, R) such that G is the intersection of GL(n, R) with the set of
common zeros of this system.
Before describing the connection between reductive subgroups and totally
geodesic submanifolds of P(n, R), we mention some examples of reductive
algebraic subgroups of GL(n, R). Examples (2) and (3) explain why we have restricted
attention to real matrices, and (4) will allow us to embed KH" (rescaled) in P(n, R)
as a totally geodesic submanifold.
10.57 Examples
(1) SL(n, R), 0(n) and SO(ri) are all reductive algebraic subgroups of GL(n, R).
For instance 0(ri) is defined by the vanishing of 'MM — I, the entries of which are
quadratic polynomials in the entries of M.
(2) GL(n, C) as a reductive subgroup of GL(2n, R). Consider the map from
GL(n, C) to GL(2n, R) that sends A + iB, where A, В e M(n, K), to
(i "Л'
This map is obviously injective and its image is defined by the obvious equalities
between entries. Thus GL(n, C) can be realised as a reductive algebraic subgroup of
GL(2n, R). Note that transposition in GL(2n, M.) induces the conjugate transpose on
GL{n, C).
(3)GL(n, 7i) as a reductive subgroup of GL{An, R). Let7i denote the quaternions.
Following the construction of (2), we shall explain how to realize GL(n, Ti) as a
reductive algebraic subgroup of GL(4n, R).
First note that one can embed Ti in M(2, C) by sending q = Oq + a\ i + a^j-\- a^k
to the matrix
(d 70eM(2'C)>

328 Chapter 11.10 Symmetric Spaces
where с = ao + a\i and d = aj. + a^i (so q = с + dj). The map Τι -» M(2, C)
thus defined is an injective ring homomorphism and the conjugate of q is sent to the
conjugate transpose of the image of q. More generally, one can realize GL(n, Τι) as
a subgroup of GL(2n, C) by sending the matrix С + Dj e GL(n, Τι), with C,D e
M(n, C), to the matrix
(δ ~<?)^£<2n,C).
The conjugate transpose on GL(2n, C) induces the conjugate transpose on GL(n, Τι).
The composition of the above maps GL(n, Τι) ^-> GL(2n, C) ^ GL(4n, Ж)
embeds GL(n, Τι) as a reductive algebraic subgroup of GL(4n, K).
(4) Let К = 1, С or H. We saw in 10.20(4) that 0K(Q), the subgroup of
GL(n + 1, K) that preserves the form Q on K"·1, is closed under the operation of
conjugate transpose. Thus by means of the above embeddings we can realize Ok(Q)
as a reductive subgroup of GL(m, K), where m = (n + 1), (2n + 2) or {An + 4) in the
cases К = Μ, С and Τι respectively. Moreover, it also follows from 10.20(4) that
this reductive subgroup is algebraic.
More generally, instead of Q one can consider a form of type (p, q):
ρ р+ч
Qp,q(x, У) = Σ** ~ 11 *&'·
i=l i=p+\
The subgroup of GL(n + 1, K) that preserves Qpq is closed under the operation of
conjugate transpose (it can be described in a manner analogous to 10.20(4)), and its
image under the embeddings in (2) and (3) is a reductive algebraic group.
In the course of the following proof we shall need the fact that a closed subgroup
G of GL(n, M.) is a Lie group, i.e. it is a differentiable submanifold and multiplication
and passage to inverses in G are differentiable operations. (This is easy to verify in
the above examples.)
Observe that if G с GL(n, K), then a necessary condition for G.I с P(n, K) to
be totally geodesic is the following (cf. 10.59):
(*) if X € S(n, R) and expX e G then exp(iX) e G, Vj e K.
10.58 Theorem. Let G с GL(n, Ж) be a reductive subgroup satisfying (*). Let
K = GH 0(n) and let Μ = Gil P(n, Щ. Then:
(1) Μ is the G-orbit of I.
(2) Μ is a totally geodesic submanifold ofP(n, K); it is diffeomorphic to G/K.
(3) Μ is a CAT(0) symmetric space.
(4) К is a maximal compact subgroup ofG; up to conjugacy in G it is the unique
such subgroup.
(5) The map Κ χ Μ —> G sending (k, m) to km is a diffeomorphism.

Reductive Subgroups of GUn, M) 329
Conversely, if V is a totally geodesic submanifold of P(n, R) and I e V, then
G = {g e GL(n,M.) | g.V = V} is a reductive subgroup of GL(n,R) and V =
G П P(n, R).
Proof. If X e S(n, R) and/? = expX, then/? = g.I, where g := exp(X/2) e G. Thus
(1) is a consequence of (*).
G is a Lie group and the action of GL(n, R) is differentiable, so Μ = G.I is a
differentiable submanifold of P(n, R). And since the stabilizer of / in GL(n, R) is
0(n), Μ is diffeomorphic to G/K. The image of 11-» exp(iX) is the unique geodesic
line containing the segment [/, p] and (*) ensures that this line is entirely contained
in M, so by homogeneity Μ is totally geodesic.
Since Μ is totally geodesic, if ρ e Μ then the symmetry σρ : P(n, R) -> Р(и, R)
leaves Μ invariant. The existence of the symmetries σρ \м means that Μ is a symmetric
space.
К is the stabilizer of / in G. It is compact because G is closed in GL(n, R) and
0(n) is compact. Thus (4) follows from (2.7), as in (10.41).
We have seen (10.51) that the map 0(n) χ P(n, R) sending (k,p) to kp is a
diffeomorphism onto GZ,(«, R), so in order to prove (5) we need only check that
if kp e G then к and ρ belong to G. We have p2 = '(kp)(kp) e M, and writing
p2 = exp X we have ρ = exp(X/2). Thus (*) implies ρ e G, hence ieG and (5) is
proved.
If V с P(n, R) is a totally geodesic submanifold then for any ρ e У the symmetry
σρ leaves V invariant. Givenp, q e V, if m is the midpoint of [ρ, q] then am(p) = q.
Thus the subgroup G of GZ,(«, R) leaving V invariant acts transitively on V. Moreover,
since q t-» p.q is the composition of σ/ and σρ, we have V с G Π Р(и, R). And if
/? e G Π Р(и, R) then p.I = p2 e V, so /?, which is the midpoint of [/, p2], is in V.
Thus V = G П Р(и, R).
It remains to check that G is reductive; it is obviously closed so what we must
show is that G = 'G. Given g e G, write g = pk where fc e О(и) and /j e Р(и, R).
Then g./ = p.I = p2 e V, hence ρ eVCG, therefore к e G and 'g = fc-1/? e G.
D
There are a number of remarks that we should make regarding the above theorem.
The first important point to note is that it applies to all reductive algebraic subgroups
of GL(n, R):
10.59 Lemma. Let G с GL(n, R) be a subgroup and let X e S(n, R) be such that
expX eG.IfG is algebraic, then exp(iX) e Gfor all t e R.
Proof. It suffices to show that if a polynomial function on M(n, R) vanishes at exp(rX)
for some X e S(n, R) and all reZ, then it vanishes at exp(iX) for all t e R.
After conjugation by an element of 0(n), we may assume that X is a diagonal
matrix diag(X!,..., λ„). Thus it suffices to show that if a polynomial
P(xl,...,xtl) = YJCil...inx'l ...x^

330 Chapter II. 10 Symmetric Spaces
of degree m in η variables such that P(erX>,..., erX") = 0 for all reZ,
then P(e'Xl,..., e'x») = 0 for all t e №..
Let si,..., Sk be the distinct values taken by the monomials e'|X| ... e'"\ on the
set of (ii,..., i„) e N" such that ΐχ -\ + i„ < m. One can write
к
P{ea\...,e'K) = Y^ajs'j,
;=i
where a, is the sum of the coefficients c,·,...,-,, for which e'lX) ... e'"x" = Sj.
If we view (V. qSj =0 | r = 0, 1, ..., A: — l)asa system of к equations in the
к variables a.\, ... ,α^, then the determinant of the coefficient matrix is Π/>,(γ — J/)>
the Vandermonde determinant, which is non-zero. By hypothesis £\. ajsj = 0 for all
reZ, hence all the щ are zero. D
There are several other important remarks to be made concerning (10.58):
70.60 Remarks. In the notation of (10.58):
(1) In general G does not act effectively on Μ (for example if Μ — {/}).
(2) In general an isometry of Μ cannot be extended to an isometry of P(n, M). For
example if η > 2 and Μ is a maximal flat in P(n, Щ, then Isom(M) acts transitively
on the set of geodesies in Μ whereas G does not (no element of G sends a regular
geodesic to a singular one).
(3) If the action of К on the tangent space T/M is irreducible, then the subspace
metric on Μ С Р(п, К) is the unique G-invariant Riemannian metric on Μ up to a
scaling factor. To see this, first observe that any G-invariant Riemannian metric is
determined by the scalar product given on 7)M, because G acts transitively on M.
Fix an isometry from E* to T/M endowed with the metric induced by the Riemannian
metric on P(n, Ж). If there were a non-proportional G-invariant Riemannian metric
on Μ then its unit ball would be an ellipsoid in this Euclidean space different from
a sphere, and the subspace spanned by those vectors which had maximal length
in the fixed Euclidean metric would span a proper /ί-invariant subspace of T/M,
contradicting the fact that the action of К is irreducible.
(4) Embedding KH" and Other Symmetric Spaces. In the light of (4), by applying
(10.58) to the reductive algebraic subgroup G = <?к(0 described in 10.57(4) we
can realize KH" (rescaled) as a totally geodesic submanifold of P(m, Щ where
m = (n+ l)dimi{K.
This embedding illustrates a general phenomenon. We mentioned earlier that,
after rescaling the metric on its irreducible factors, one can isometrically embed any
symmetric space of non-compact type as a totally geodesic submanifold Μ с Р(п, Ш)
(see for instance [Eb96, 2.6.5]).
(5) The flats in Μ are the intersection with Μ of the flats in P(n, K). All of the
maximal flats in Μ have the same dimension (this is called the rank ofM) and the
group G acts transitively on the set of maximal flats.

Semi-Simple Isometries 331
Semi-Simple Isometries
In the remainder of this chapter we will concentrate on the horospherical structure of
P{n, Ж) and the geometry of дР(п, К). However, before turning to this topic we pause
to reconcile an ambiguity in our terminology: classically, a matrix g e GL(n, R) is
said to be semi-simple if and only if it is conjugate in GL(n, C) to a diagonal matrix;
on the other hand, we have been studying GL(n, K) via its action by isometries on
the CAT(0) space P(n, Щ, and for such isometries we have a different definition of
semi-simplicity, viz. (6.1). The following proposition shows that these definitions
agree. Indeed if g e GL(n, R) acts as a semi-simple isometry of any totally geodesic
submanifold of P(n, M.) then it is semi-simple in the classical sense.
10.61 Proposition. Let Μ be a totally geodesic submanifold of P(n, K). Suppose
that g e GL(n, K) leaves Μ invariant and let ~g e Isom(M) be the restriction of
g. The isometry ~g is semi-simple in the sense of (6.1) if and only if the matrix g is
semi-simple.
Proof. Assume that ~g is semi-simple. If ~g fixes a point ρ e M, then g is conjugate to
an element of 0(n), hence it is semi-simple. Otherwise ~g is hyperbolic, which means
that there is a geodesic с : К —» Μ and a > 0 such that g.(c(i)) = c(s + a), Vs e K.
Since Μ is totally geodesic, с is a geodesic line in P(n, Ж) and g is hyperbolic. After
conjugating g in GL(n, K) we may assume that c(0) = /. If we write X to denote the
element of S(n, K) such that exp(iX) = c(s), then g'g = exp(aX) and the equation
g.c(s) = c(s + a) becomes gexp(sX)'g = exp(iX)exp(aX). Differentiating with
respect to s at s = 0, we get gX'g = Xexp(aX) = Xg'g, hence gX = Xg. Thus
h := exp(aX/2) (which is semi-simple) commutes with g; let к = gh~x = h~xg.
Both g and h act as translation by a on c(K), therefore к fixes c(0) = /, in other words
к е 0(n). In particular к is semi-simple, and since it commutes with the semi-simple
matrix h, the product g = hk is also semi-simple.
To prove the converse, suppose that g e GL(n, M.) is a semi-simple matrix that
preserves a totally geodesic submanifold M. As Μ is a closed convex subspace, the
isometry of P(n, №.) defined by g will be semi-simple if and only if its restriction
to Μ is semi-simple (6.2(4)). Thus it is enough to show that g acts semi-simply on
P(n, K).
By hypothesis, g is conjugate in GL(n, C) to a diagonal matrix diag(Xi,..., λ,?);
we may assume that λ2, = λ2,·_ι if 0 < i < к and that λ,- is real if i > 2k. Standard
linear algebra yields elements k,h e GL(n, Ж) and a basis e\,..., en for W such
that g = kh = hk, where /г(е,) = |X,|e, for all i and к multiplies e, by λ,·/|λ;| if
г > 2k, and for i < к the action of к on the subspace with ordered basis {e2,-i, е-ц\
is rotation by arg(X,y |λ,-|). Conjugating g by a suitable element of GL(n, K), we may
assume that e\,..., en is the standard basis, and hence g = kh = hk where к е 0(n)
and h e P(n, Ж). Now, if h = /, then g e 0(n) acts as an elliptic isometry of P(n, K)
(fixing /). Otherwise, h = exp X where X e S(n, Ж) is non-zero; in this case h acts by
translation on the geodesic 11-» exp(iX) in P(n, Ж). This geodesic is invariant under
the action of к and hence g, therefore g acts on P(n, K) as a hyperbolic isometry. D

332 Chapter П. 10 Symmetric Spaces
While on the subject of reconciling terminology, we should note that in texts on
symmetric spaces, [Eb96] for example, the term "hyperbolic isometry" is reserved
for transvections. In such texts isometries that are hyperbolic in our sense but which
are not transvections are usually called loxodromic.
Parabolic Subgroups and Horospherical Decompositions
ofP(n,R)
In this section we shall describe the Busemann functions and horospheres in P(n, K)
and calculate the stabilizer G^ с GL(n, Ж) of each point ξ e дР(п, K). In its
broad outline this section follows our earlier discussion of the horospherical structure
of KH" (10.26 to 10.29). However, in the present (higher-rank) situation things
are less straightforward and more interesting because the parabolic subgroups and
horospherical structure associated to a point c(oo) e 8P(n, K) depend very much on
the nature of the geodesic с (whether it is singular versus regular for example).
10.62 Definition. Given ξ e дР(п, К), define
G4 :={geGL(n,R)|g.$ = $}.
If ξ ~ c(oo) where с is the geodesic line c(t) = exp(iX), define
Щ := (geG| | exp(-iX)g exp(iX) -> / as t -> oo}.
(We shall show that Щ is a normal subgroup of G%.)
G^ is called the, parabolic subgroup associated to ξ, and N$ is called the how-
spherical subgroup associated to ξ. It follows from (10.26) that the orbits of N$ are
contained in the horospheres centred at ξ. These orbits foliate P(n, K) and there is a
complementary orthogonal foliation (see 10.69) whose leaves are the totally geodesic
subspaces
F(c) := [J{c'(K) | d : Ш -> P(n, K) is a geodesic parallel to c],
where с ranges over the geodesies such that c(oo) = |.
We shall give explicit descriptions of G$, Щ and F(c). We shall see that the map
Νξ χ F(c) —> P(n, Щ sending (v, p) to v.ρ = νρ \) is a diffeomorphism, and we shall
use this map to describe the Busemann functions of P(n, K).
10.63 Examples. The structure of G^, Νξ and F{c) depends very much on the
nature of c(oo) = ξ. There are two extreme cases: if c{t) = e'^I (the direction
of the Euclidean de Rham factor (10.53)), then G^ = GL(n, K), N% is trivial and
F(c) = P(n, K); in contrast, if t (->· c(t) is a regular geodesic, then G% is conjugate to
the group of upper triangular matrices (10.64), while Νξ is the subgroup consisting of
matrices with ones on the diagonal, and F(c) is the unique maximal flat Л containing

Parabolic Subgroups and Horospherical Decompositions of P(n, Ш) 333
c(K). In this second example P(n, M.) is foliated by the maximal fiats whose boundary
at infinity contains ξ — c(oo) (cf. 10.68).
10.64 Proposition. Let ξ e dP(n, K) be a point at infinity, and let с : гь> exp(iX)
be the geodesic ray issuing from I with c(oo) = ξ, where X e S(n, K) and \\X\\ = 1.
Then
Gt = {g e GL(n,R) | lim exp(-iX)g exp(iX) exists}.
i->00
Conjugating by an element ofSO(n) we may assume that X is a diagonal matrix
diag(Xi,..., λ„) with λι > · · · > λ„. In this case, ifr\,...,rk are the multiplicities
of the eigenvalues ofX taken in decreasing order, then g e GL(n, K) belongs to G^
if and only if
(An An ■ ■■ Au\
0 A-n ... An
0 0 ... Аш'
where Aij is an (г,-, η)-τηαίηχ.
Proof. LetX = diag (λ1;..., λ„) where λ, > λ, if г < j. If the (г, gentry of a matrix
g is gy, then the (г,gentry of the matrix exp(—tX)gexp(tX) is e~ti7i'~^gij, which
converges as t —> 00 if and only if λ, > λ,-. In particular, the matrices such that
lim^oo exp(— tX) g exp(iX) exists are precisely those displayed above; it remains to
show that these constitute G^.
If lim^oo exp(—tX/2) g exp(iX/2) exists, then
d(exp(-tX/2)gexp(tX/2).I, i) = d(gexp(tX/2).I, exp(iX/2)./)
is bounded for t > 0, hence the geodesic ray 11-» g exp(iX/2)./ (t > 0) is asymptotic
to c(t) = exp(iX/2)./ = exp(iX); in other words g.| = ξ.
Conversely, if g e G^, then d(gexp(tX/2).I, exp(iX/2)./) is bounded for
t > 0, hence exp(—tX/2)gexp(tX/2).I remains in a bounded subset. It follows
that {exp(—tX/2)g exp(tX/2) | t > 0} С GL(n, K) is contained in a compact
subset, because the map GL(n, M) —> GL(n, R)/0(n) = P(n, K) is proper. Hence there
exists a sequence tn —> 00 such that the sequence of matrices exp(i„X)gexp(i„X)
converges in GL(n, K) as и -> oo. But, as in the first paragraph, this implies that
gij = 0 if λ, < λ^, and hence (by the reverse implication in the first paragraph)
lim^oo exp(iX) g exp(iX) exists. D
10.65 Remark. For all ξ e дР(п, Ж) the group G^ acts transitively on P(n, M).
Indeed if we conjugate G$ into the above form, then it contains the group AN of
upper triangular matrices with positive entries in the diagonal, and this acts simply
transitively on P(n, K) (see 10.49).
Similarly, if ξ is the endpoint of a ray in P(n, K)i, then G^ П SL(n, K) acts
transitively on P(n, K)i.

334 Chapter 11.10 Symmetric Spaces
10.66 Proposition.
(1) Щ is a normal subgroup of G^.
(2) LetX = diag(Xb ... ,Xk) be as in (10.64), letc(t) = exp(iX) and let ξ = c(oo).
Then g e Щ if and only if
In AX1 ... A\k\
0 In ... Alk
0 0 ... InJ
(3) Νξ leaves invariant the Busemann function b^ ifc'(oo) = ξ.
Proof. The inclusion Νξ с G^ is immediate from (10.64) or (10.26). It also
follows from (10.26) that Νξ leaves invariant the Busemann functions associated to
rays c' with c'(oo) = f. Part (2) is an easy calculation, so it only remains to check
that Щ is normal in G^. If g e G% and ν e Νξ then exp(—tX) ν exp(iX) -> / as
t —> oo and exp(—tX) g exp(iX) converges to some element of GL(n, K),
therefore exp(—tX)gvg~x exp(iX), which is the conjugate of exp(—tX)v exp(iX) by
exp(— tX) g exp(iX), converges to /. D
Part (3) of the above proposition implies that the orbits of Λ^ are contained in
horospheres about ξ. In general Νξ is not the largest normal subgroup of G^ satisfying
this property. For example, if c(t) = exp( -W) then Νξ is trivial but SL(n, K) preserves
the horospheres centred at c(oo).
Next we characterize F(c), which we defined to be the union of the geodesic lines
that are parallel to the given geodesic line с : Μ —> P(n, К).
10.67 Proposition. If c(t) = exp(iX) is a geodesic ray in P(n, K), then F{c) с
P(n, Щ consists of those positive-definite, symmetric matrices which commute with
expX.
IfX = diag (λι,..., Xf.) is as in (10.64), then F(c) is the set of matrices g =
(gij) e P(n, Щ such that g,y = 0 if λ, φ λ,- :
F(c)
This is a totally geodesic submanifold isometric to the product Π,= ι Дг<> ^)·
Proof Let ξ = c(oo) and let — ξ = c(—oo). Given ρ e P(n, M) we may choose
geGf such that ρ = g.I = g'g (see 10.65). Since t (->· g.c(t) is the unique geodesic
line through ρ such that g.c(oo) = ξ, by the flat strip theorem (2.13), ρ e F(c) if and
only if g.c(oo) = ξ and g.c(—oo) = —|; in other words, if and only if g e G^ DG_^.
We restrict our attention to the case c(t) = exp(iX) with X = diag(Xj,..., λ^.)
as in (10.64). Write g = (gy). By applying (10.64) to t ы> exp(iX) and to t ь>
,,Κ)
0
0
0
0
P(r2,R) .
0
0
• ° ^
0
0
• Р(п,Ш)/

Parabolic Subgroups and Horospherical Decompositions of P(n, Ш) 335
exp(i(—X)) we see that g e G^ П G_^ if and only if gy = 0 when λ, φ λ/. This
implies that/? = (py) is a symmetric matrix with/Jy = 0 when λ, φ λ,·.
The ρ e Р(и, К) that commute with exp X are precisely those with p^ = 0
when λ, φ λ,-. We have shown that F(c) is contained in this set of matrices.
Conversely, any such matrix can be written as ρ = exp Y, where У is a symmetric
matrix that commutes with X. In this case d(exp Υ exp(tX/2).I, exp(tX/2).I) =
u?(exp(—tX/2)expYe.xp(iX/2).I, I) = d(expY.I, I) is independent of i, hence
the geodesic t t-» exp У exp(tX/2).I, which passes through /j, is parallel to ί ь>
exp(tX/2).I = c{t). D
10.68 Corollary, //c is a regular geodesic in P(n, K) then F(c) is the unique maximal
flat containing c(K).
Proof. By conjugating we may assume that c(t) = exp(iX) where all of the
eigenvalues of expX are positive and distinct. By (10.45), the set of matrices in P(n, M)
which commute with expX is the maximal fiat Л = exp do. □
We know by (2.14) that F(c) С P(n, Ш) is convex and isometric to F(c)o χ Ш,
where F(c)o is the set of points of F(c) whose projection on c(K) is c(0) = /. The
horospheres centred at ξ = c(oo) intersect F(c) orthogonally in the parallel translates
ofF(c)0.
10.69 Proposition (Busemann Functions in P(n, K)). Let c(t) = exp(iX) be a
geodesic line in P(n, Ж), where χ e S(n, M) and \\X\\ = 1. Let ξ = c(oo). Then:
(1) The map Νξ χ F(c) —> P(n, K) sending (v,p)tov.p = vp'visa diffeomorphism.
(2) The Busemann function bc associated to the geodesic с is given by the formula
bc(v.p) = -Tr(AT), where ρ = exp Υ e F(c) and ν e Νξ.
Proof After conjugation we may assume that X is a diagonal matrix
/μι/r, 0 ... 0 \
I 0 μ2Ιη ... 0
Ιο о ... о
V 0 0 ... μ^η/
with μι > ■ ■ · > μ/c and r\ + ■ ■ ■ + rk = п.
To prove that the map N% χ F(c) -> P(n, R) is bijective, we shall argue by
induction on к, the number of distinct eigenvalues of X. In the case к = 1 this map is
trivially a bijection: N$ is trivial and F{c) = P(n, M). For the inductive step we write
(и, n) matrices g e M(n, K) in block form g = ( 8u 8n), where g22 e M(rt, M).
\g21 g22/
We want to show that, given ρ e Р(и, Μ), the equation υ/ V = ρ has a unique solution
(v,f) e Νξ χ F(c). In the light of (10.66) and (10.67), this equation takes the form:
(v\\ vxl\ifn 0\ /Vn °\ = (P" Pl2\
\0 In)\0 fnjy'vii In) \P2i Pii)'

336 Chapter П. 10 Symmetric Spaces
where/?2i ='ρη· Thus it is equivalent to the three equations:
/22 = P22
ΙΊ2/22 =P]2
Vl]fll'V]] + V]2f2l'V]2 =PU-
From the first two of these equations we get/22 = P22 and V]2 = /512/22'' anc* me
third equation becomes
Κιι/ιιΊΊΙ =Pll ~P\2P22lP2l-
If we can show that the symmetric matrix/?ц —рпРггР^ IS positive-definite then it
will follow from our inductive hypothesis that this last equation has a unique solution
Οι 1,/11) of the required form.
To say that ρ is positive definite means precisely that bcpx > 0 for all
nonzero (n, l)-matrices (column vectors) x. Given an (и — r^, 1)-matrix x\ φ 0, we
let X2 = —P22P21X1 and calculate bcpx with be = (bc\,'x2). This yields Ьс\рцх\ —
,χ\Ρ\7ΡΐΐΡ2\χ\ > 0' thus/?n — P12P22P21 is positive-definite.
Fix ρ e F(c). By 10.66(3) the Busemann function bc is constant on the orbits of
Щ, hence bc(v.p) = bc(p) for all ν e % By (10.67), ρ = exp Υ where [X, Y] = 0,
thus using 10.42(2) we get:
d(p, c{t)f = u?(exp Y, exp tX)2
= u?(exp(-iX/2)exp Fexp(-iX/2), I)2
= d(exp(Y - tX), I)2
= Tt((Y - tX)(Y - tX))
(ο) = Ίχ{ΥΥ) - 2tTx(XY) + t2.
By definition bc(p) = lim^^ d(p, c{t)) — t. And from the triangle inequality for
A(p, c(0, c(t)) we get Ιίτη,^^άφ, c(t)) + t)/2t = 1. Therefore
bc(p) = lim -(40c, c(t))2 - t2),
and hence (o) implies that bc{p) = — Тг(ХУ). П
70.70 Exercise. Deduce that if cx{oo) = сг(оо) = ξ then F(c\) and F{C2) are
isometric.

The Tits Boundary of P(n, M), is a Spherical Building 337
The Tits Boundary of P(n, K)i is a Spherical Building
The purpose of this section is to describe the Tits boundary of P(n, K), as defined in
(9.18). Our previous results lead one to expect that djP{n, Щ should enjoy
considerable structure: in Chapter 9 we saw that the geometry of flats in a complete CAT(O)
space influences the Tits boundary of the space, and in the present chapter we have
seen some indication of the extent to which the geometry of P(n, K) is dictated by
the nature of its flat subspaces.
10.71 Theorem. 77ге Tits boundary of P(n, K)i is (naturally isometric to) a thick
spherical building.
In an appendix to this chapter the reader will find a brief introduction to spherical
buildings, a remarkable class of CAT(l) spherical simplicial complexes discovered
by Jacques Tits. The defining properties of a spherical building involve the behaviour
of its maximal simplices (chambers) and certain distinguished subcomplexes
(apartments). In our setting, the role of chambers and apartments will be played by the
boundaries at infinity of Weyl chambers and maximal fiats (see 10.72).
An Outline of the Proof of Theorem 10.71. First we shall define certain subsets of
дР(п, К)] to be apartments and chambers (10.72). We shall then fix our attention on
a basic chamber A+(oo). In the Tits metric A+(oo) is isometric to a spherical simplex.
We metrize dP(n, K)i as a spherical simplicial complex by transporting the metric
(and face structure) from A+(oo) to its translates under the action of SL(n, M) (see
10.75). There are then two key points to check: the intrinsic metric associated to this
spherical simplicial structure on dP(n, K)i coincides with the Tits metric (10.78); and
the apartments and chambers of dP(n, K)i, as defined in (10.72), satisfy the axioms
for a (thick) spherical building (10A. 1). Both of these points are easy consequences
of Proposition (10.77).
10.72 Definition (Apartments and Weyl Chambers at Infinity). A subset of dP(n, K)i
is called an apartment if it is the boundary at infinity of a maximal flat in P(n, K)i.
A subset of dP(n, K)i is called a Weyl chamber at infinity if it is the boundary at
infinity of the closure of a Weyl chamber in P(n, K)i.
There is not an exact correspondence between Weyl chambers at infinity and Weyl
chambers in P(n, K)i. Indeed a Weyl chamber in P(n, K)i is contained in a unique
maximal fiat whereas a Weyl chamber at infinity is contained in infinitely many
apartments (10.74). On the other hand, the correspondence between apartments and
maximal fiats is exact: this is an immediate consequence of (10.53) and the following
lemma.
10.73 Lemma. Let A and A' be maximal flats in P{n, K). IfdA = dA' then A = A'.
Proof. Let с be a regular geodesic line in A and let d be a geodesic line in A' such
that c'(oo) = c(oo) = ξ. Note that c'(—oo) = c(—oo), because this is the unique

338 Chapter 11.10 Symmetric Spaces
point of dA = dA' at Tits distance π from ξ. Thus the geodesic lines с and d are
parallel, and hence F{d) = F{c). But F(c) = A, by (10.68), so А' С A. D
10.74 Simplices in djP(n, K)i. We wish to describe a spherical simplicial structure
on дтР(п, K)i in which the maximal simplices are the Weyl chambers at infinity and
the apartments are isometrically embedded.
The apartments of dTP(n, K)i are isometrically embedded copies of S"~2.
Following our discussion in (10.45) and (10.54), we know that every Weyl chamber in
P(n, K)i is a translate by some g e SL(n, M) of
η
A+ = (exp(iX) | s > 0, X = diag(i] ,...,t„), ]P t,■. = 0 and t\ > ■■· > t„}.
i=0
Therefore every Weyl chamber at infinity is a translate by some g e SL(n, Щ of the
closure A+(oo) of A+(oo) := (c(oo) | c[0, oo) с А+ a geodesic ray}, which in the
Tits metric is isometric to a spherical simplex of dimension (n — 2) (contained in the
copy of S"-2 that is the boundary of the maximal flat containing A+). We described
the shape of this simplex in (10.54).
The codimension-one faces of A+(oo) are obtained by replacing one of the
inequalities U > ti+\ by an equality, and faces of codimension к are obtained by
replacing к of these inequalities by equalities.
In order to show that the translates g.A+(oo) of A+(oo) by the elements g e
SL{n, Ж) are the (n — 2) simplices of a simplicial subdivision of dTP{n, K)bjwe must
check that every ξ e djP(n, K)i lies in a translate of A+(oo) and that g.A+(oo) Π
A+(oo) is a (possibly empty) face of A+(oo).
10.75 Proposition.
(1) The spherical (n — 2)-simplex A+(oo) is a strict fundamental domain for the
natural action ofSL(n, K) on dP(n, K)] (i.e., A+(oo) contains exactly one point
from each SL(n, W)-orbit).
(2) For every g e SL(n, K), the intersection g.A+(oo) Π Α+(οο) is a (possibly
empty) face ofA+(oo).
(3) SL(n, K) acts transitively on pairs (дА, С), where dA is an apartment and
С с dA is a Weyl chamber at infinity.
(4) For every ξ e dP(n, M)h the stabilizer of ξ in SL(n, Ж) coincides with the
stabilizer of the unique open simplex containing ξ.
(5) The orbits ofSL(n, K) in dP(n, K)t are the same as the orbits ofSO(n).
Proof. First we prove (5). Given ξ e dP(n, K)b let с : [0, oo) -> P(n, K), be the
geodesic ray with c(0) = / and c(oo) = ξ. Given g e SL(n, K), let d = g о с and let
I' = c'(oo) = g.|. There is an element h e SL(n, K) fixing ξ' such that h.c'(0) = I
(see 10.65). Therefore hg lies in the stabilizer of /, which is SO(ri) (10.33), and
hg-ξ = ξ'-

The Tits Boundary of P(n, Μ), is a Spherical Building 339
Every ξ e dP(n, M.) can be represented uniquely as c(oo) where c(t) = exp(iX)
is a geodesic through /. Now, since X is symmetric, there exists a unique diagonal
matrix X' with non-increasing entries along the diagonal such that X' = gX 'g for
some g e SO(n). It follows that c'(oo), is the unique point of A+(oo) in the SO{n)-
orbit of ξ, where c'(t) = exp(iX'). In the light of (5), this proves (1). Part (4) then
follows easily from (10.64). To prove (2), note that if χ lies in the given intersection,
by (1) we have g.x = χ and then by (4) we have that this intersection is a face.
It remains to prove (3). Since SL(n, Ш) acts transitively on the set of Weyl
chambers in P(n, K)i, it acts transitively on the set of Weyl chambers at infinity. Thus it
is sufficient to prove (3) for apartments containing Л+(оо). But in the light of (4),
mis follows immediately from the fact that for every ξ e dP(n, K)i the stabilizer of
ξ acts transitively on the set of geodesies с with c(oo) = ξ (10.65) and if ξ is in the
interior of A+(oo) then each such с is contained in a unique maximal fiat, therefore
G^ acts transitively on these fiats. Π
At mis stage we have proved:
10.76 Corollary. The Weyl chambers at infinity are the maximal simplices in a
spherical simplicial complex whose underlying set is дР(п, Ж)\. The natural action
ofSL(n, M.) on dP(n, K)i is by simplicial isometries.
It remains to check that the intrinsic metric associated to this spherical simplicial
structure coincides with the Tits metric. This is the first consequence that we shall
draw from the following proposition (whose proof will be completed in 10.80). This
proposition also implies that the simplicial structure that we have described satisfies
axiom (3) in the definition of a building (10A.1).
10.77 Proposition. Any two Weyl chambers at infinity are contained in a common
apartment.
Proof. Let Л С Р(п, Ш)\ be the maximal fiat consisting of diagonal matrices. In the
light of 10.75(3), it suffices to show that given any Weyl chamber at infinity С there
is an element of SL(n, M.) that leaves A+(oo) invariant and maps С into ЗЛ.
From 10.64 and 10.75(4) we know that the stabilizer of A+(oo) is the group of
upper triangular matrices in SL(n, K), and from 10.54 we know that any other Weyl
chamber in ЗЛ can obtained from A+(oo) by the action of a permutation matrix. Thus
the proposition follows from the fact that any element of SL(n, K) can be written in
the form gisg2, where g] and g% are upper triangular matrices and s is a monomial
matrix (this crucial fact will be proved in 10.80). For if С = g.A+(oo) and g = g\Sgi,
then gj"1 .C = i.A+(oo), whence both С and A+(oo) are contained in g\.dA. Π
10.78 Corollary. The Tits metric on дР(п, Ш)х coincides with the intrinsic metric of
the spherical simplicial complex obtained by dividing дР(п, К) ] into Weyl chambers
at infinity.

340 Chapter П. 10 Symmetric Spaces
Proof. Let dT denote the Tits metric on dP(n, W)} and let d be the intrinsic metric
for the spherical simplicial structure on дР(п, Ш)х. The latter is obtained by taking
the infimum of the lengths of piecewise geodesic paths in the spherical simplicial
complex. The length of any such path is the same when measured in both the Tits
metric and d, because the Tits metric coincides with the simplicial metric on the
closure of each Weyl chamber. Thus άτ(ξ, f) < rf(|, f') for all ξ, ξ' e дР(п, К),.
On the other hand, in the light of (10.77) we know that ξ, ξ' lie in a common
apartment. In the Tits metric this apartment is isometric to §"~2, and any Tits geodesic
joining ξ to f obviously has length άτ(ξ, ξ') when measured in the metric d. Thus
άτ{ξ, ξ') > ά(ξ, ξ'). О
The Final Step in the Proof of Theorem 10.71
The axioms of a building are given in (10A.1). In the light of the preceding results
only axiom (3) requires further argument.
First we consider the case where two apartments Ε = дА and Ε' = дА' contain
a chamber С in their intersection. In this case (10.75) implies there is a simplicial
isometry Φε,ε from E' onto Ε that restricts to the identity on C. We claim that Φε,ε
restricts to the identity on the whole of EilE'. To see this, note that if χ e ΕΠΕ' then
χ is a distance less than π from some pointy in the open set C; the restriction of Φε,ε
to an initial segment of the unique geodesic [y, x] is the identity, and since [y, x] is
the only geodesic of length d(x, y) in Ε that has this initial segment, Φε,ε(χ) = χ.
Now suppose that B\ and Bj. are (possibly empty) simplices in the intersection
of two apartments Ex and Е2. For i = 1, 2 we choose chambers С, с Е, such
that Bt с С,. By (10.77) there is an apartment E' containing both C\ and Cj.. The
composition of the maps Фё^е, anc* Φε,Ει sends Ej. isometrically onto Ex and restricts
to the identity on B\ U B2. О
Οχ Ρ (η, Ш) in the Language of Flags and Frames
The following interpretation of apartments and Weyl chambers at infinity provides
a useful language when one wishes to bring the tools of linear algebra to bear on the
study of djP(n, K). For example, the content of (10.80) was crucial in our proof of
(10.78).
A flag in K" is a sequence of subspaces V] с Vi · ■ · С Vn- \ such that V, has
dimension i. An unordered frame in K" is a set of η linearly independent subspaces
of dimension one, and an ordered frame is a sequence of η linearly independent
subspaces of dimension one. (Thus every unordered frame gives rise to n\ ordered
frames.) Associated to each ordered frame one has the flag V] с · · · С V„_b where
Vk is the subspace spanned by the first к elements of the ordered frame.
We write flo to denote the flag associated to the standard (ordered) basis ex,..., e„
for W, and we write fr0 to denote the associated unordered frame (the set of one-
dimensional subspaces (e,)). We shall also use the notation:

дтР(п, М) in the Language of Flags and Frames 341
Λ = the set of maximal fiats in P(n, R)\ (= the set of apartments in dP(n, R)i)
С = set of Weyl chambers at infinity
Tr = set of unordered frames in W
Tl = set of flags in K".
The obvious action of SL(n, K) on each of Tr and Tl is transitive, and we saw earlier
that the action of SL(n, K) on Л and С is transitive.
10.79 Proposition. There are natural SL(n, R)-equivariant bijections A—*- Tr and
С —> Tl, and an induced bijection from {(А, С) \ А е А, С с дА} to the set of
ordered frames in K".
Proo/ The group of upper triangular matrices in SL(n, M) is the stabilizer of both
A+(oo), the basic Weyl chamber at infinity, and the flag fio. Thus, since the action
of SL(n, M.) is transitive on both С and Tl, then g.A+(oo) м> g.fl0 is an equivariant
bijection from С to Tl.
Let Ao be the maximal flat in P(n, Ш)\ consisting of diagonal matrices. The
stabilizer in SL(n, K) of Л о (equivalently дА о) is the subgroup consisting of monomial
matrices, i.e. matrices having only one non-zero entry in each row and column. This
subgroup is also the stabilizer of the unordered frame fr0. Thus g.A м> g.fro is an
equivariant bijection from A to Tr.
SL(n, K) acts transitively on the set of ordered frames and also on the set of pairs
{(A, C) | A € А, С С дА}. The stabilizer of (Ao, A+(oo)) is the group of diagonal
matrices of determinant one, and this is also the stabilizer of the ordered frame
associated to the standard basis of K". Hence the required equivariant bijection. D
10.80 Lemma. Given two flags V] с · · · С Vn-X and V{ с · · · С V'n_v one can
find a basis vx,... ,v„ for Ш" and a permutation aof{\,...,n] such that V, is the
span of{vi | j < i}for each i < η and V, is the span of{va^ | j < i].
Sketch of proof. In order to prove this lemma one analyzes the standard proof of the
Jordan-Holder Theorem (see [Bro88, p.84]). We shall give the proof for η = 3 and
indicate how the different cases that arise in the proof are related to the geometry of
Weyl chambers as described in 10.55 (and figure 10.1). It follows from the description
given there that each apartment дА С djP(n, Μ)λ is an isometrically embedded circle
and the Weyl chambers at infinity contained in дА are six arcs of length π/З. The
action of the symmetric group 1S3 on the set of Weyl chambers in an apartment is
described in (10.55).
Let V, С V2 and V\ С V'2 be the two distinct flags in K" and let С and C" be the
corresponding Weyl chambers at infinity. We seek a basis {vx, vi, v$} of Ж3 and a
permutation σ such that V] (resp. V[) is spanned by υλ (resp. υσ(ΐ)) and V2 (resp. V^)
is spanned by {ub 112} (resp. {υσ(ΐ), υσ(2)})· As we noted above, the present lemma
implies that there is an apartment A containing both С and C", and the permutation σ
describes the action of S3 on A (see 10.55). With this geometric picture in mind, one

342 Chapter П. 10 Symmetric Spaces
expects there to be essentially three cases to consider: С and C" might be adjacent
chambers of A, they might be opposite, or they might be neither.
First case (adjacent chambers): V\ = V[ or V2 = V2.
If V, = v[, we choose the basis {v\, v2, 1>з} so that {vu v2} spans V2 and {ub 113}
spans V2. The permutation σ is the transposition (2, 3).
If V2 = V2, we choose the basis {ι>ι, ι>2, из} so that v\ (resp. v2) spans Vi (resp. V[)
and 113 φ V2. In this case σ is the transposition (1,2).
Second case (opposite chambers): Vi φ V[,V2 φ V2 and V2 П V2 φ V, or V\. This
is the generic case.
We choose the basis {v\, v2, Щ} so that Vi is spanned by V\, V2 Π V'2 by v2 and
V[ by 113. The permutation σ is (3, 2, 1). In this case there is only one apartment
containing the given Weyl chambers С and C.
Third case: V2 η V2 = V, φ V[ (the case У2 η У2 = V( / V, is similar).
We choose {ub v2, v^} so that V^ is spanned by i>b V2 by {иь иг} and V[ by 113.
The permutation σ is (3, 1, 2). In this case the Weyl chambers are neither adjacent
nor opposite in any common apartment. D
Appendix: Spherical and Euclidean Buildings
Spherical buildings were first introduced by Jacques Tits [Tits74] "as an attempt to
give a systematic procedure for the geometric interpretation of the semi-simple Lie
groups, in particular the exceptional groups". Euclidean buildings emerged from the
study of /j-adic Lie groups [IwMa65] and their theory was developed by Bruhat and
Tits [BruT72], who established that Euclidean buildings were CAT(O) in the course of
proving a version of the Cartan fixed point theorem. (They used the СAT(0) inequality
in the guise of 1.9(lc)). Tits later defined abstract buildings as combinatorial objects
(chamber systems) whose basic structure can be described in terms of Coxeter groups.
There is an extensive literature approaching the subject of buildings from various
perspectives and we shall not attempt to replicate it. Instead, we shall give a
rudimentary introduction to spherical and Euclidean buildings with the main objective
of proving that the axioms of a building imply upper curvature bounds (Theorem
10A.4). We refer the reader to the books [Bro88], [Ron89] and [Tits74] for a much
more comprehensive introduction to buildings. The survey articles of Ronan [Ron92]
also contain a lot of information. For a less complete but very readable introduction
we suggest [Bro91]. (A useful introduction to Coxeter groups, by Pierre de la Harpe
[Har91], can be found in the same volume as [Bro91].) We also recommend to the
reader the more metric approach to buildings presented by Mike Davis in [Da98].
The following definition of spherical and Euclidean buildings is not the most usual
one, but it fits naturally with the geometric viewpoint of this book. Our definition of
a thick building is equivalent to the usual one (see [Bro88, IV, VI] or [Ron89])

Appendix: Spherical and Euclidean Buildings 343
10A.1 Definition. A Euclidean (resp. spherical) building of dimension и is a piece-
wise Euclidean (resp. spherical) simplicial complex X such that:
(1) X is the union of a collection Л of subcomplexes E, called apartments, such
that the intrinsic metric dE on Ε makes (E, dE) isometric to the Euclidean space
E" (resp. the и-sphere §") and induces the given Euclidean (resp. spherical)
metric on each simplex. The и-simplices of Ε are called its chambers.
(2) Any two simplices В and β of X are contained in at least one apartment.
(3) Given two apartments Ε and E' containing both the simplices В and B', there
is a simplicial isometry from (E, dE) onto (E', dE) which leaves both В and β
pointwise fixed.
The building X is called thick if the following extra condition is satisfied:
(4) Thickness Condition: Any (n — 1 )-simplex is a face of at least three и-simplices.
10A.2 Remarks
(1) For the purposes of this definition, the O-dimensional sphere S° is defined to
consist of two points a distance π apart.
(2) In condition (3), the simplices В or B' can be empty.
(3) It is usual to say that a building of dimension η has "rank (n + 1)". This
superficially odd convention means that the rank of a symmetric space such as P(n, K)i is
the same as the rank of its Tits boundary. (See also 10A.7.)
10A.3 Examples
(1) Any metric space X such that the distance between any two distinct points
is it is a spherical building of dimension 0, where the apartments are the pairs of
distinct points of X.
(2) Let X be as in (1). The spherical suspension (as defined in Chapter 1.5) of
X is a spherical building of dimension 1 whose apartments are circles, namely the
suspensions of the apartments of X.
More generally, if X and X' are spherical buildings of dimension η and n', then
their spherical join X * X' is a spherical building of dimension (n + n' + 1) whose
simplices (resp. apartments) are the spherical join of the simplices (resp. apartments)
ofXandX'.
(3) If the edges a metric simplicial tree all have length 1 and every vertex of the
tree is adjacent to at least two (resp. three) edges, then the tree is a Euclidean building
(resp. a thick Euclidean building) of dimension one.
(4) In the final sections of the preceding chapter we showed that, when equipped
with the Tits metric, the boundary at infinity of the symmetric space P(n, Ж)\ is a
spherical building of rank (n — 1). In general the Tits boundary of any symmetric
space of non-compact type is a spherical building whose rank is the rank of the
symmetric space (see [Eb97]).
(5) Let к be a field and consider the simplicial graph X whose vertices are the
1-dimensional and 2-dimensional subspaces of къ, and which has an edge joining

344 Chapter П. 10 Symmetric Spaces
two vertices if and only if one of the corresponding subspaces is contained in the
other. In other words, X is the incidence graph of points and lines in the projective
plane over k.
If one metrizes each edge to have length π/З, then X is a spherical building
of dimension one in which the apartments are hexagons (circuits of combinatorial
length 6). If к is the field with 2 elements, then X is a graph with 14 vertices and
21 edges, and every vertex is of valence three, corresponding to the fact that every
line through the origin lies in exactly three planes and each of these planes contains
exactly three lines through the origin. (Exercise: draw this graph.)
More generally, one can consider the poset of proper non-zero vector subspaces
in k" ordered by inclusion. A suitable geometric realization of this set is again a
spherical building of rank (n — 1), whose apartments are isomorphic to the apartments
in дтР(п, М)ь the chambers are in natural correspondence with flags in к" and the
apartments correspond to the unordered frames of k" (cf. 10.79 and 10.80). There is
an obvious action of SL(n, k) on this building.
10A.4 Theorem.
(i) The intrinsic metric on a Euclidean (resp. spherical) building X is the unique
metric inducing the given metric ufe on each apartment E.
(ii) With this metric, X is a CAT(O) space (resp. a CAT(l) space).
(iii) IfX is thick, then all of the n-simplices in X are isometric to a single geodesic
η-simplex in E" (resp. S"). (In this case it follows from (1.7.13) that the intrinsic
metric is complete.)
Proof. The retraction pe.c- Let С be a chamber contained in an apartment E. If Ε
is another apartment containing C, then by 10A. 1(3) there is a simplicial isometry
Φε.ε of Ε onto Ε fixing С pointwise. This isometry is unique because the image of
each geodesic is determined by the image of an initial segment contained in С This
uniqueness forces the restriction of Φε,ε' to Ε Π Ε to be the identity.
We wish to define a simplicial retraction ρ = Pe.c, which is a simplicial map
from X onto E. By 10A.1(2), each point χ e X is contained in an apartment Ε that
contains С Define p(x) := Φε,ε'(χ)- This definition is independent of the choice of
Ε because if E" is another apartment containing С and x, then Φε.ε" fixes С and χ
and Φε,ε ° Φε·,Ε" = Φε,ε" because of the uniqueness established above. Note that ρ
is a simplicial map and that its restriction to each chamber is an isometry. Moreover
its restriction to each apartment containing С is an isometry.
The restriction of the intrinsic metric to Ε is dg. Let d denote the intrinsic metric
on X (see 1.7.4). Because ρ = pe,c maps simplices to simplices isometrically, it
preserves the length of piecewise geodesic paths (and m-strings). Thus, given x, у е
E, since p\e = idg, when calculating d(x, y) one need only quantify the infimum in
the definition of d over paths (strings) contained in E. And this infimum is clearly
equal to dE(x, y). This proves part (i) of the theorem. It also shows that X is a geodesic
space (the geodesic joining χ to у in Ε is a geodesic in X). In fact these are the only
geodesies.

Appendix: Spherical and Euclidean Buildings 345
10A.5 Lemma. Let Ε be an apartment in X, let x,y e E, and in the spherical case
assume d(x, y) < π. Then every geodesic in X connecting χ to у is contained in E.
Proof. Suppose that ζ € X is such that d(x, y) = d(x, z) + d(z, y). Let z1 be the point
on the geodesic segment [jc, y] joining χ to у in Ε with d(x, z) = d(x, ζ1). We must
show that ζ = ζ'. Let С be a chamber in Ε containing z' and let ρ = pE.c- Since ρ
does not increase the length of paths, d(x, p(z)) < d(x, z) and d(y, p(z)) < d(y, z).
And
d(x, y) < d(x, p(z)) + d(p(z), y) < d(x, z) + d(z, y) = d(x, y).
There must be equality everywhere, therefore ρ (ζ) = ζ!. On the other hand, there is an
apartment Ε containing both ζ and C, by 10A.1(2), and by definition p(z) = Φε.Ε'(ζ)
and z' = Φε.ε'(ζ')- Since φΕ.Ε' is injective, we have ζ = г'. П
It follows immediately from this lemma that there is a unique geodesic joining
each pair of points x, у e X (assuming d(x, y) < π in the spherical case). It also
follows that geodesies vary continuously with their endpoints, because if / is close
to у then there is a chamber С containing both у and /, and given χ there exists an
apartment containing χ and С The desired curvature bounds can thus be deduced
from Alexandrav's patchwork (proof of (4.9)) or from (5.4). However, instead of
appealing to these earlier results, we prefer to give a direct and instructive proof of
the fact that X is CAT(O) (Euclidean case) or CAT(l) (spherical case).
The СAT(0) and СAT( 1) Inequalities. Given x, y, z e X (with d(x, y)+d(y, z) +
d(z, χ) < 2π in the spherical case), let ρ be a point on the unique geodesic segment
[x, y]. Let Ε be an apartment containing [jc, y], let С С Ε be a chamber containingp,
and let ρ = pE,c- We have d(x, p(z)) < d(x, z\ d(y, p(z)) < d(y, z) and d(p, p(z)) =
d(p, г). Let г be a point in Ε such that d(x, z) = d(x, z) and d(y, z) = d(y, z). The
triangle A(x, y, z) С Ε is a Euclidean or spherical comparison triangle for A(x, y, z).
Thus it suffices to prove that d(p, z) < d(p, z)-
Let ζ еЕЪе, such thatdOc, z) = d(x, p(z)) andd(y, z') = d(y, z). As d(y, p(z)) <
d(y, z'),ше vertex angle at χ in A(x, y, p(z)) is less than that in A(x, y, z'), and hence
(by the law of cosines) d(p, p(z)) < dip, z)· Similarly, comparing A(x, y, f) with
A(x, y, z), we see that dip, z!) < dip, z), hence dip, z) = dip, p(z)) < dip, ζ). This
proves (ii).
The case of a thick building. Let С and С be distinct chambers in X. They lie in
a common apartment and therefore can be connected by a gallery, i.e. a sequence of
chambers С = Co, C\..., C„ = С such that each consecutive pair C,, C,-+) have an
(n — l)-dimensional face in common. Thus it suffices to consider the case where С
and С have an (n — l)-dimensional face В in common.
By thickness, there exists a chamber C", distinct from С and C, of which В
is a face. Let Ε and E' be apartments containing С U С and С U C" respectively.
The restriction to С of pe,c ° Pe\c sends С isometrically onto С and fixes B. This
completes the proof of Theorem 10A.4. D

346 Chapter II. 10 Symmetric Spaces
10A.6 Remark. It follows from (10A.4) that Euclidean buildings are contractible.
According to the Solomon-Tits theorem [Sol69], every spherical building of dimension
η has the homotopy type of a wedge (one point union) of и-dimensional spheres.
10A.7 Coxeter Complexes. We continue with the notation from the last paragraph
of the preceding proof. Let TL be the half-space of Ε that contains C" and has В in its
boundary. It is not difficult to show that the restriction of Pe.c ° Pe.c to Ε is simply
the reflection of Ε in the boundary of Η (see [Bro88, IV.7]). It follows that if X is
a thick building and Ε is an apartment, then the subgroup of Isom(F) generated by
reflections in codimension-one faces acts transitively on the chambers in E, Further
thought shows that any chamber is a strict fundamental domain for this action. This
group of reflections, which we denote W, is actually a Coxeter group (as defined in
12.31) with generating system S, where S is the set of reflections in the codimension-
one faces of a fixed chamber С (see [Bro88, IV.7]). Ε is (isomorphic to) the Coxeter
complex for (W, S) and \S\ is the rank of the building. If W is finite then the building
is spherical. If W is infinite then the building is Euclidean.
10A.8 Example. Consider the building X = дтР(п, Ж) ι, which has rank (n - 1).
In this case each apartment Ε is tesselated by (n — 2)-simplices (the closures of the
Weyl chambers at infinity), W is the symmetric group on η letters, and the preferred
generating set S is the set of reflections in the (n — 1) maximal faces of a fixed
chamber.
In the case η = 3, the apartments are regular hexagons (circles divided into
six equal parts — see proof of 10.80), S has two elements and W has presentation
(sus2 \s*=sl = (s\s2Y = 1).
10A.9RemarL·
(1) In the definition of a Euclidean building (10A.1), instead of insisting that X
be a simplicial complex, we might have assumed instead that the chambers of X were
convex Euclidean polyhedra. For instance we might take the cells to be cubes. This
leads to a more general notion of a Euclidean building, with the advantage that the
product of two such Euclidean buildings is again a Euclidean building (cf. [Da98]).
An examination of the proofs given above reveals that they all remain valid if
one adopts this more general definition of a Euclidean building.
(2) One can show that if X is a Euclidean building as defined in 10A.1, the
boundary dTX of X with the Tits metric is a spherical building (see [Bro88]), as it is
the case when X is a symmetric space of non-compact type. In his Habilitationsschrift
[Le97], Bernhard Leeb established characterized irreducible Euclidean building and
symmetric spaces in terms of their Tits boundary. He showed that if a proper С AT(0)
proper space X has the geodesic extension property (5.7), and the Tits boundary of
X is a thick, connected, spherical building, then X is either a symmetric space of
non-compact type or a Euclidean building. (X will be a symmetric space if and only
if every geodesic segment extends to a unique line.)

Chapter 11.11 Gluing Constructions
In this chapter we revisit the gluing constructions described in Chapter 1.5 and ask
under what circumstances one can deduce that the spaces obtained by gluing are of
curvature < к. Our purpose in doing so is to equip the reader with techniques for
building interesting new spaces of curvature < к out of the basic examples supplied
in Chapters 5, 10 and 12.
Gluing САТ(к) Spaces Along Convex Subspaces
If one glues complete CAT(/c) spaces along complete, convex, isometric subspaces,
is the result a CAT(/c) space? We shall show that the answer to this question is yes.
In the case of proper metric spaces this result is due to Reshetnyak [Resh60] and the
proof is rather short. In the general case one has to overcome the additional problem
that the existence of geodesies in the space obtained by gluing is far from obvious
(cf. 1.5.25(3)).
We maintain the notation established in Chapter 1.5; in particular, given two
metric spaces X\ and X2 and isometries j) : A —> Aj onto closed subspaces Aj с Xj,
we write X\ UA X2 for the quotient of the disj oint union of X\ and X2 by the equivalence
relation generated by [ϊχ (a) ~ 12(a) Va e A]. (Suppressing mention of the maps j) in
this notation should not cause any confusion.)
11.1 Basic Gluing Theorem. Let к е К be arbitrary. Let X\ and X2 be CAT(/c)
spaces (not necessarily complete) and let A be a complete metric space. Suppose
that for j = 1,2, we are given isometries г} : A —> Aj, where Aj с Xj is assumed to be
convex if к < 0 and is assumed to be DK -convex if к > 0. Then Xx uA X2 is a CAT(/c)
space.
Proof.
Stepl. We first prove the theorem assuming that X = Х\иАХг is a geodesic metric
space (which we know to be the case if X\ and X2 are proper 1.5.24(3)). We shall
verify criterion 1.7(4) for all geodesic triangles in X (with perimeter < 2DK if к > 0).
The only non-trivial case to consider (up to reversing the roles of Χι and X2) is that
of a geodesic triangle A([x\, y], \y, x2], [x\, x2]) with jq, у e Χι and x2 eX2\4
For this, one fixes points z, z' e Ai on the sides [jci, x2] and \y,x2] respectively.

348 Chapter II. 11 Gluing Constructions
Any such points have the property that [xu z] Я [x\, x2] and [y, z1] Я [у, x2] are
contained in X\, and that [z,x2] Я [х\,х2] and [г'.хг] Я [хьхг] are contained
in X2. In order to complete the proof one simply applies the gluing lemma for
triangles (4.10), first to A([xuz], [z, z'l [x\,z']) and Δ([ζ, z'L [ζ1, x2], [ζ, x2]) and
then to Δ([χ,, χ2], [χ2, ζ'], [χ\, ζΊ) and Δ([χ, ,y],\y, z'], [x\, г'])·
Step 2. We shall now prove that X is a geodesic metric space. As we do not assume
that Л1 is proper, according to exercise 1.5.25(3) we must use both the fact thatXt and
X2 are CAT(/c) and the fact that the Aj are convex in order to show that X = Χχ uA X2
is a geodesic metric space. It suffices to show that for all x\ e X\ and x2 e X2 (with
d(xi, x2) < DK if к > 0) there exists ζ € A \ such that d(xx, z) + d(z, x2) = d(xx, χ2).
We fix a sequence of points zn € A \ such that d(x\ ,zn) + d(z„, x2) — dOq, x2) < l/n.
Because the numbers d(xi, z„) are bounded, we may pass to a subsequence and
assume that d(xx, zn) -> t\ as и -> oo. Thus ύ?(χ2, ζ») -> ^2 := ^ь ^г) — ^ι as
и —> oo. We claim that (z„) is a Cauchy sequence; we will then be done because Л1
is complete and г = Нщ, zn has the desired property.
Before turning to the general case, we give a proof valid for к < 0. Given ε > 0,
letN > 0 be such that max{\d(xi, zn) — t\\, \d(x2,z„) — l2\] < ε for all и, т > N.
Let ρ be the midpoint of the geodesic segment [zn,Zm] Я А \. By the convexity of the
metric on X2 we have:
d{x2,p) < тах{й?(г„, х2), d(zm, x2)} <h + s,
hence d(x\ ,ρ)>ίι~ε. For a comparison triangle A(x\, zn, zm) in E2, an elementary
calculation with the Euclidean scalar product yields:
d(xupf = -d(xi,zn)2 + -d(x\,zmf ~ -d(zn,zm)2·
We also have d(x\, p) < d(xx, p), by the CAT(O) inequality in Χχ. Hence
d(zn, zmf < 2 d(xi ,ΖηΫ + 2 d(xi, zm? -4d(xupf
<2d(xuzn)2 + 2d(xuzm)2-4(ll -ε)2 < 16£,ε.
Step 3. We maintain the notation established in Step 2. The following proof that
(z„) is a Cauchy sequence is valid for any к e K. By reversing the roles of l\ and l2
if necessary, we may assume that £1 < d{x\, хг)/2 < DK/2.
We fix a number I with DK > I > ύ?^, хг). As in (1.2.25), it follows from
(1.2.26) that for every ε > 0 there exists 5 = δ(κ, Ζ, ε) such that, for all x,y,w e M2,
if d(x, y) < I and d(x, w) + d(w, y) < d(x, y) + 8, then d(w, [x, y]) < ε. Shrinking
5, we may assume d(x\, x2) + δ < I. We fix an integer N so that for all и > N
we have d(xuZn) < DK/2 and \d(xu zn) — U\ < ε, and d(xuz„) + d(x2,zn) <
d(x\, x2) + δ. If n, m > N, then d(zn, zm) < ά(χλ, z„) + d(xx, zm) < DK,so zn and
zm are connected by a unique geodesic segment (contained in the convex set Л 0 and
there are unique geodesic triangles At in Xx and Δ2 in X2 with vertices (jcb z„, Zm)
and (x2, zn, Zm) respectively. The perimeter of each of these triangles is bounded by
d(xi, zn) + d(x2, zn) + d(x\, Zm) + d(x2, zm) < 21 < 2DK. We consider comparison

Gluing САТ(к) Spaces Along Convex Subspaces 349
trianglesAt = A(i1,znizm)andA2 = Δ(χ2, zn,zm) in M^, joined along the common
edge [z„,zm] and arranged so that xx and X2 are on opposite sides of the line through z„
and zm- (We assume that the triangles At and Δ2 are non-degenerate; the degenerate
case is trivial.)
Consider the unique geodesic segment \χλ, x2] in M%. There are two cases:
(a) [x\, x2] meets [z„, zm] in one point y, or
(b) [xu x2] does not meet [zn,Zm].
In case (a), if у e [z„, zm] is the point a distance d(z„, y) from zn, then by the
CAT(/c) inequality,
ί?(*ι, *г) < ά{χλ, у) + d(y, x2) < ά{χλ, у) + d(y, x2) = ά(χλ, x2).
Hence
d(xuzn) + d(zn,x2) = d(xuzn) + d(zn,x2) < d(xux2) + S< d(xbx2) + S,
so by the definition of 5 there exists pn e{xu x2] such that d(zn,pn) < ε. Similarly,
there exists pm e [xb x2] such that d(zm,pm) < ε.
The difference between d(xi,z„) and d(xi,zm) is at most 2ε, so the distance
betweenpn andpm is at most 4ε. Hence d(z„, zm) = dQ,n, zm) < 6ε.
In case (b), the sum of the angles of Δ ι and Δ2 at one of the vertices zn OTzm is not
less than π; suppose this vertex is zn- According to 1.2.16(1), there exists a triangle
in M\ with vertices (*,, x2, zm) such that d(x}, zm) = d(Xl, zm), d(x2, zm) = d(x2, zm)
and d(xi,x2) = d(xuzn) + d(zn,x2). Let z„ e [ЗсьЗсг] be such that d(xuzn) =
d(xu zn). From (1.2.16(2)) we have d(z„, zm) > d(zn, zm) = d(zn, zm). And by the
definition of 5, thepoint|m lies in the ε-neighbourhood of [x\, x2]- So since \d(xx, z„)—
d{x\,zm)\ < 2ε, wehaved^Zm) < 3ε. D
11.2 RemarL·
(1) Under the hypotheses of (11.1), X\ and X2 are isometrically embedded in
X\ l_U X2 as DK-convex subspaces, and X is complete if and only if Χλ and X2 are
both complete.
(2) {Successive Gluing). In the notation of (1.5.26), if each of the metric spaces
X\,...,X„ is CAT(/c), and if each of the subspaces As along which they are glued
is complete and DK -convex, then by repeated application of (11.1) we see that the
space Yn obtained by successive gluing is CAT(/c).
We generalize the Basic Gluing Theorem (11.1) to the case of arbitrary families
of CAT(/c) spaces.
11.3 Theorem (Gluing Families of CAT(/c) Spaces). Let (Χχ, dx)x<=A be a family of
CAT(/c) spaces with closed subspaces Αχ с Χχ. Let A be a metric space and suppose
that for each λ e Λ we have an isometry ίχ : A —> Αχ. Let X = \_\ΑΧχ be the
space obtained by gluing the Χχ along A using the maps ix (see 1.5.23). If A is a

350 ChapterII.ll Gluing Constructions
complete CAT(/c) space (in which case each Αχ is DK -convex and complete), then X
is a CAT(/c) space.
Proof. We claim that for all lb λ2 € Λ, the natural inclusion of Χχ, l_U Χχ2 into X
is an isometry. Indeed if x, у e Χλι U Χχ2 are a distance less than DK apart in X then
there is a unique geodesic connecting them and this lies in Χλι l_U Χχ2. To see this,
note that because Αχ С Χχ is DK -convex, any geodesic in Χλ of length less than DK
with endpoints in Αχ lies entirely in Αχ, and thus any chain in X that has endpoints
in Χλι l_U Χχ2 and has length less than DK (in the sense of (1.5.19)) can be shortened
by deleting all of its entries that lie in Xx \ A for λ φ {λ ι, λ2}.
It follows from this description of geodesies that X is DK -geodesic and any
geodesic triangle Δ С X with perimeter less than 2DK and vertices x, e Χχ}, i =
1, 2, 3 is contained in (Χλ| l_UΧχ2) LU ·Χλ3 · Applying (11.1) twice, we see that (Χλ| l_U
Χλ,) UA Χχ3 is a CAT(/c) space and hence Δ satisfies the CAT(/c) inequality. D
Gluing Using Local Isometries
The preceding constructions were global: we glued CAT(/c) spaces using isometries
between complete DK -convex subspaces to obtain new С АТ(/с) spaces. We now turn to
the analogous local situation: we glue spaces of curvature < к using local isometries
between locally convex and complete subspaces with the expectation that the result
will be a metric space of curvature < к. We also seek more general constructions
that will allow us to glue along subspaces which are not locally convex.
At an intuitive level, much of what we expect from local gluing is clear—one can
visualize the result of gluing two hyperbolic surfaces along two closed local geodesies
of equal length for example. There are, however, a number of technical difficulties
associated to showing that the quotient metric has the anticipated properties. These
are eased by the following special case of (1.5.27).
11.4 Lemma. Let X be a metric space, A a closed subspace ofX and σ : A —» A
a local isometry such that σ2 is the identity of A and σ(α) φ a for each a & A. Let
X be the quotient ofX by the equivalence relation ~ generated by a ~ σ(α). Then,
the quotient pseudometric on X is actually a metric, and for every χ eX there exists
ε(χ) > 0 such that: if χ φ A then B(x, ε(χ)) is isometric to B(x, ε(χ)); if χ e A then
the restriction ofa to B(x, 2ε(χ)) Γ) A is an isometry and B(x, ε(χ)) is isometric to the
ball of radius ε(χ) about the image of χ in the space obtained by gluing B(x, 2ε(χ))
to Β(σ(χ), 2ε(χ)) using this isometry.
11.5 Lemma. Let X, A and σ be as above. IfX has curvature < к and A is locally
convex and complete, then X also has curvature < к.
Proof. Given χ e A, by shrinking ε = ε(χ) if necessary (notation of (11.4)) we may
assume that B(x, 2ε) and Β(σ(χ), 2ε) are CAT(/c) and that their intersections with A

Gluing Using Local Isometries 351
are convex and complete. Writing I = Α Γ\Β(χ, 2ε), by (11.4) we know that B(x, ε) is
isometric to the ball of radius ε about the image of χ in B(x, 2ε) U/ Β(σ(χ), 2ε), where
the inclusion / —> Β(σ(χ), 2ε) is the restriction of σ. Thus, by the Basic Gluing
Theorem (11.1), B(x, ε(χ)) is СЩк). О
11.6 Proposition (Simple Local Gluing).
(1) Let X be a metric space of curvature < к. LetA\ andAi be two closed, disjoint
subspaces ofX that are locally convex and complete. If i: A\ —> A2 is a bijective
local isometry, then the quotient ofX by the equivalence relation generated by
[αϊ ~ i'(ai), Vai e A\ ] has curvature < к.
(2) Let X\ and Xi be metric spaces of curvature < к and let Α ι с Χι and Aj. С Хг
be closed subspaces that are locally convex and complete. Ifj:A\ —> Aj. is a
bijective local isometry, then the quotient of the disjoint union X = X\ Ц Xj. by
the equivalence relation generated by \_ax ~ j(a{), Vaj e A\ ] has curvature
< к.
Proof. One proves (1) by applying the preceding lemma with A = A\ U A 2 and
а|л, = i and а\л2 = Г1. And (2) is a special case of (1). D
We highlight three simple but important examples of (11.6), each of which will
play a role in Chapter Ш.Г.
11.7 Examples
(1) (Mapping Tori). Let X be a metric space. The mapping torus of φ e Isom(X)
is the quotient of Χ χ [0, 1] by the equivalence relation generated by [(x, 0) ~
(φ(χ), 1), Vjc e X], it is denoted Μφ. If X is non-positively curved, then so is Мф.
(2) (Doubling along a Subspace). Given a metric space X and a subspace Y, we
write D(X; Y) to denote the metric space obtained by taking the disjoint union of two
copies of X and forming the quotient by the equivalence relation that identifies the
two copies of Y. The two natural copies of X in D(X; Y) are isometrically embedded
and the given identification between them induces a map D(X; Y) —> X which is a
left-inverse to both inclusion maps. If X has curvature < к and У is a closed subspace
that is locally convex and complete, then D(X; Y) has curvature < к.
(3) (Extension over a Subspace). This construction is a special case of (11.13).
Given a metric space X and a subspace Y, we write X*Y to denote the quotient of
Χ \\(Υ χ [0, 1]) by the equivalence relation generated by [y ~ (y, 0) ~ (y, 1), Vy e
У]. The natural inclusion Χ <^-> X*r is an isometry and there is a retractionX*r —» X
induced by the map (y, t) м> у from У χ [0, 1] to У. If X has curvature < к and Υ is
a closed subspace that is locally convex and complete, then X*Y has curvature < к.
11.8 Exercise. For each of the above examples, show that if X is complete then the
space obtained by the given construction is complete.
11.9Remark. Onecaniteratetheconstructionof (11.6(2)).Let(X,, d,-),i = 1, 2..., η
be a sequence of metric spaces; assume that a bijective local isometry/2 from a closed

352 Chapterll.ll Gluing Constructions
subspace Аг of X2 onto a closed subspace/г(Лг) of X\ is given; let Y2 be the quotient
of the disjoint union of X\ and Χχ by the equivalence relation generated by a ~fi(a)
for all a e Аг\ proceeding inductively, suppose that for every i > 2 a bijective local
isometry^ from a closed subspace At of X, onto a closed subspace of F,-_i is given
and define У; to be the quotient of the disjoint union of X,- and Yt- \ by the equivalence
relation generated by a ~fi(a) for all a e Л;.
If each X, has curvature < к and A, is locally convex and complete, then Y„
has curvature < /c; according to the description of neighbourhoods in the above
proposition, a small ball about each point of Yn can be obtained by successive gluing
of balls inXb .. .,X„.
In (11.4) we required the map σ to be an involution, i.e. we considered the
quotient of a metric space X by the equivalence relation associated to a free action
of Z2 by local isometries on a closed subspace Л С X. We restricted our attention to
Z2 actions in order to move as quickly and cleanly as possible to the proof of (11.6).
However (1.5.27) applies equally well to the free action of any group Г by local
isometries on a closed subspace Л С X, provided that this action has the property
that for each a e A there exists ε > 0 such that B(a, ε) Π Β(γ.α, ε) is empty if γ ψ 1
(where B(a, ε) is a ball in X, not just A).
In this more general setting, the ball of radius ε about the image of α in the
quotient is obtained by gluing the family of balls Β(γ.α, ε) using the isometries
B(a, £)ПЛн> Β(γ.α, ε) Π A obtained by restricting the action of Γ.
Using this description of ε-balls in place of that given in (11.4), and replacing
the appeal to (11.1) in the proof of (11.5) with an appeal to (11.3), we obtain:
11.10 Proposition. Let X be a metric space of curvature < к, let А с X be a closed
subspace that is locally convex and complete, and suppose that the group Г acts
freely by local isometries on A so that for each a e A there exists ε > 0 such that
B(a, ε) Π Β(γ.α, ε) is empty for ally φ 1. Then the quotient ofX by the equivalence
relation generated by [a ~ y.a, Vy e Γ Va e A] has curvature < к.
Similar arguments apply to equivalence relations that do not arise from group
actions. We mention one other example.
11.11 Proposition (Gluing with Covering Maps). Let X) and Xj. be metric spaces
of curvature < к, and for i = 1,2 let At С X; be a closed subspace that is locally
convex and complete. Let ρ : A\ —> Λ2 be a local isometry-that is a covering map,
and suppose that for each у е Аг there exists ε > 0 such that B(a, ε) is CAT(/c) and
B(a, ε) Π Β(α', ε) is empty for all distinct points a, a' e A1 withp(a) = p(a') = y.
Then the quotient of X\ Ц Х2 by the equivalence relation generated by [a ~
p(a), Va e Л1] has curvature < к.
Proof. Let X be the quotient of Xi Ц Х2 by the stated equivalence relation. First
suppose that Л2 = Хг. If ρ is a Galois covering, then this proposition reduces to the

Gluing Using Local Isometries 353
previous one. The argument in the non-Galois case is essentially the same: one uses ρ
and its local sections to glue the balls {B(a, ε) | a e p~l(y)} along their intersections
with Α ι (having shrunk ε to ensure that B(a, 2ε) Π Α ι is convex and complete for
each a). The space obtained by this gluing is CAT(/c), by (11.3), and B(y, ε/2) с Х
is isometric to a ball in this space.
In the general case one first forms Χι = (ΧΊ JJ Лг)/~ and then one constructs X
by gluing Χι to Xi along the obvious copies of Aj.. The image of Аг in X\ is closed
and locally convex and complete, so (11.6) applies. D
11.12 Examples
(1) Let X, A be as in (11.10). If a group Г acts properly by isometries on X
and leaves A invariant, then provided its action on A is free, then the quotient X/~
described in (11.10) will be a metric space of curvature < к.
(2) Suppose that X has curvature < к and let Л be the image of an injective local
geodesic с : §' —> X. Given any positive integer n, one can apply (11.10) to the
action of the cyclic group Z„ = (τ) on Л, where τ acts as c(9) м> с(9 + 2π/η).
(3) Let X and A be as in (2) and let У be a space of curvature < к that contains an
isometrically embedded line L. Suppose that there exists ε > 0 such that B(y, ε) is
CAT(/c)forevery;y e L. lip : Z, -» A is a local isometry, then by (11.11) the quotient
of X Ц Υ by the relation generated by \y ~ p(y), Vy e L] has curvature < к.
There are many situations in which one wishes to glue along subspaces that are
not locally convex, for example totally geodesic submanifolds with self-intersections.
When gluing along such subspaces one needs to use gluing tubes.
In the following proposition the restriction к > 0 is necessary because of the
need to introduce a product metric on the tube.
11.13 Proposition (Gluing With a Tube). Let X and A be metric spaces of curvature
< к, where к > 0. If A is compact and φ, ψ : A —> X are local isometries, then the
quotient ofX Ц(Л χ [0, 1]) by the equivalence relation generated by [ (a, 0) ~ φ(α)
and (a, 1) ~ ψ(α), VaeA] has curvature < к.
Proof. If a point in the quotient is not in the image of A x {0, 1} then it obviously has
a neighbourhood isometric to a neighbourhood of its preimage in X or A x [0, 1].
Consider the image у of a point у e A x {0, 1}. By compactness, the preimage
in XjJ(A χ [0, 1]) of у consists of a finite number of points, and it follows from
(1.5.27) that if 5 is sufficiently small then B(y, 5) is obtained from the union of the
5-neighbourhoods of these points by successive gluing. Hence (11.2(2)) applies. D
By taking X to be the disjoint union of Xo and X\ we obtain:
11.14 Corollary. Let Xo>-^i and A be metric spaces of curvature < к, where к >
0. If A is compact and φ;; : A —> Xi is a local isometry for i = 0,1, then the

354 Chapter П. 11 Gluing Constructions
quotient ofXo Ц(Л χ [0, 1 ]) Ц Χι by the equivalence relation generated by [ (a, 0) ~
φο(α), (α, 1) ~ φ\(α), Va e A ] has curvature < к.
11.15 Examples
(1) LetX0 ΆηάΧχ be metric spaces of curvature < к with к > 0, and for i = 0, 1
let c,- : §' —> X, be closed local geodesies of length £ (each c; may have many self-
intersections). Let S(i) be a circle of length £ and choose arc length parameterizations
c, : S(i) -> X,:oi thee,. Then, the quotient of Х0Ц(ЗДх[0, 1]) [J Xi by the relation
generated by [(Θ, 0) ~ cq(6), (Θ, 1) ~ ci(6), V0 e 5(£)] is a space of curvature
< к. The isometry type of this space depends not only on the curves c;(Sl) but also
on the parameterizations c, chosen.
(2) (Torus Knots). A torus knot is a smoothly embedded circle which lies on the
boundary of an unknotted torus in M?. The fundamental group of the complement of
such a knot has a presentation of the form Гп,т = (χ, у \ χ" = У"), where n and m
are positive integers (see [BuZi85], for example). Let Xq be a circle of length η and
let X] be a circle of length m. Let cq be a local geodesic that wraps m times around
Xo and let c\ be a circle that wraps η times around X\. The construction in (1) yields
non-positively curved 2-complexes whose fundamental groups are Г„ т.
11.16 Exercises
(1) Let Τ = S1 χ S1, a flat torus. Let c\ and сг be two isometrically embedded
circles in Τ that meet at a single point. Let X be the quotient obtained by gluing two
disjoint copies of Τ by the identity map on c\ U c%. Prove that X is not homeomorphic
to any non-positively curved space.
(2) Let F be a closed hyperbolic surface. Let C\ and Ci be isometrically embedded
circles in F that meet in a finite number of points. Prove that the metric space Υ
obtained by gluing two disjoint copies of F by the identity map on с ι U сг (no tube)
is not non-positively curved. When is Υ homeomorphic to a space with curvature
< 0 ? And curvature < -1 ?
(Hint: If Fhas genus g then one can obtain a hyperbolic structure on it by realizing
it as the quotient of a regular 4g-gon Ρ of vertex angle 7i/2g in H2 with side pairings
(1.5.29(1)); the vertices of the polygon are identified to a single point, υ say. If one
takes a regular 4g-gon in И2 whose edges are shorter than those of P, then the sum of
the vertex angles increases, so if one makes the same edge pairings as for Ρ then the
metric becomes singular at v, where a concentration of negative curvature manifests
itself in the fact that the cone angle (i.e. the length of the link of υ) is greater than 2π.
Indeed the cone angle tends to (4g — 2)π as the area of the covering polygon tends
to zero. When the cone angle is greater than Απ, there exist sets of four geodesic
segments issuing from ν in such a way that the Alexandrov angle between any two
of them is π; in such a situation, the union of the geodesic segments is locally convex
near v.)
In Sections ΙΙΙ.Γ.6 and 7 we shall return to the study of constructions involving
gluing in the context of determining which groups act properly and cocompactly

Equivariant Gluing 355
by isometries on CAT(O) spaces — Proposition 11.13 will be particularly useful in
that context. We refer the reader to those sections for explicit examples of spaces
obtained by gluing as well as general results of a more group-theoretic nature. We
include one such result here because it was needed in our discussion of alternating
link complements at the end of Chapter 5, and because it is a precursor to (11.19).
The definition of an amalgamated free product of groups is recalled in (ΙΙΙ.Γ.6).
11.17 Proposition. If each of the groups G\ and G2 is the fundamental group of a
compact metric space of non-positive curvature, then so too is any amalgamated free
product of the form G\ *% G2.
Proof. Let X\ and X2 be compact non-positively curved spaces with it\Xj = G,. The
generators of the subgroups of G\ and G2 that are to be amalgamated determine
free homotopy classes of loops in X\ and X2 respectively, and each of these classes
contains a closed local geodesic (1.3.16), which we denote C;. By rescaling the metric
on Χι we may assume that c\ and c2 have the same length and apply the construction
of (11.15(1)) to X = X\ U X2, i.e. we glue X\ and X2 with a tube whose ends are
connected to c\ and c2 by arc-length parameterizations. By the Seifert-van Kampen
theorem, the fundamental group of the resulting space is (isomorphic to) the given
amalgam G\ *iG2. G
Whenever one is presented with a result concerning the group Ζ it is natural to
ask whether an analogous statement holds for all finitely generated free groups or
free abelian groups. In (ΙΙΙ.Γ.6) we shall see that (11.17) admits no such extension.
Equivariant Gluing
One can regard (11.17) as a construction for gluing together the actions of G, on X,;
we were free to describe this construction in terms of the quotients X, because these
actions are free. One would like to have similar results concerning proper actions
that are not necessarily free. Actions which are not free are not fully described by the
associated quotient spaces alone, so in order to combine such actions one is obliged
to work directly with appropriately indexed copies of the spaces on which the groups
are acting — that is what we shall do in this section.
Amalgamated free products will be used extensively in Section ΙΙΙ.Γ.6. For the
purposes of the present section we shall only need their geometric interpretation in
the language of the Bass-Serre theory [Ser77]. The Bass-Serre tree Τ associated to
an amalgamated free product Γ = Го *н Γι (we regard Γ0, rt and Η as subgroups
of Γ in the usual way) is the quotient of the disjoint union Γ χ [0, 1] (copies of [0, 1 ]
indexed by Γ) by the equivalence relation generated by
(УУ0, 0) ~ (y, 0), (γη, 1) ~ (y, 1), (yh, t) ~ (y, t)
forally e Γ, yo e Γ0, yi e Γι, h e H, t e [0, 1]. The action of Γ by left translation
on the index set of the disjoint union permutes the edges and is compatible with the

356 ChapterII.ll Gluing Constructions
equivalence relation, therefore it induces an action Γ on Γ by isometries. The quotient
of the tree Τ by this action is an interval [0, 1]; the subgroup Η С Г is the stabilizer
of an edge in Τ and the stabilizers of the vertices of this edge are the subgroups
Го С Г and Г, с Г.
11.18 Theorem (Equivariant Gluing). Let Го, Γι and Η be groups acting properly
by isometries on complete CAT(O) spaces Xq, X\ and Υ respectively. Suppose that for
j = 0, 1 there exists a monomorphism fy : Η <^-> Tj and a (fy-equivariant isometric
embedding f :Y^-Xj. Then
(1) the amalgamated free product Г = Го *н Г] associated to the maps (fy acts
properly by isometries on a complete CAT(O) space X;
(2) if the given actions ο/Γο, Γι and Η are cocompact, then the action of Γ on X
is cocompact.
Proof. We wish to define а С AT(0) spaceX on which Γ will act properly by isometries.
We begin with disjoint unions of copies of Xq,X\ and [0, 1] x Y, each indexed by Г:
(Γ χ X0) [](Γ χ [0, 1] χ Υ) \J(Y χ Χ,).
Let X be the quotient of the displayed disjoint union by the equivalence relation
generated by:
(УУо>*о) ~(У, Уо-*о), (YY\,x\) ~(y, Yi-xi), (Yh,t,y)~(Y,t,h.y),
(YjoM) ~ (У, 0, у), (у,Ш) ~ (У, 1, у)
for all у е Г, y0 e Г0, yt e Гь he Η, х0 еХ0, χλ е Х\, t e [0, 1], уеУ.
For j = 0,1, let Xj be the quotient of Γ χ Xj by the restriction of the above
relation, and let Yj be the quotient of Γ χ [0, 1] χ Υ. Note that X0 is isometric to a
disjoint union of copies of Xq indexed by Г/Го: each {γ} χ Xq contains one element
of each equivalence class in Γ χ Xq\ if γ and γ' lie in different cosets then (y, xq)
and (y', jcq) are unrelated for all xq, x*0 e Xq. Similarly, X\ is isometric to a disjoint
union of copies of X\ indexed by Γ/Γι, and Υ is isometric to the disjoint union of
copies of [0, 1] χ Υ indexed by Y/H.
By definition X is obtained from Xq JJ Υ JJ X\ by the following gluing: for each
γ Η e Y/H, one end of the copy of [0, 1] χ У indexed by γ Η is glued to the copy of
Xo indexed by у Го using a conjugate of the isometry/o, and the other end is glued to
the copy of X\ indexed by уГ] using a conjugate of the isometry/i. It is clear from
our earlier results that X is non-positively curved; in order to see that it is a complete
CAT(0) space we shall argue that the above gluings can be performed in sequence
so that each gluing involves attaching CAT(0) spaces along isometrically embedded
subspaces — we shall use the Bass-Serre tree to guide us in choosing the order in
which we do these gluings.
The action of Г by left multiplication on the index sets (i.e. the factors Г) in the
above disj oint union permutes the components of the disj oint union and is compatible
with the equivalence relation. Thus we obtain an induced action of Г by isometries

Equivariant Gluing 357
on X. There is a natural Γ-equivariant projection ρ : X —» Τ, where Τ is the Bass-
Serre tree described prior to the statement of the theorem: the equivalence classes of
(γ, xq), (γ, χι) and (γ, t, у) are sent by ρ to the equivalence classes of (γ, 0), (у, Ι)
and (у, t) respectively, for all у e Γ, x0 e X0, x\ eXt,is [0, 1] and у е Y.
Forj = 0, 1, the components of Xj are the preimages in X of the vertices of Τ that
are the equivalence classes of (y,j), and the edges of Τ correspond to the components
of Y. It follows that the gluing of Υ to Xq Ц X\ that yields X can be performed in
the following order: begin at the component of Xq above a fixed vertex иеГ, glue
to this the components of Υ that correspond to the edges of Τ issuing from v, then
glue to the resulting space those components of X\ that correspond to the vertices a
distance one from ν e T; work out radially from v. Repeated application of (11.3)
shows that X is a CAT(O) space. (If we were working just with simply connected
topological spaces rather than CAT(O) spaces, then essentially the same argument
would show that X is simply connected.)
It is clear that the action of Γ on X is proper: the subgroups of Γ leaving the
components of Xj С X invariant are conjugates of I}, these act properly on the
components, and if у e Γ does not leave a given component of Xj invariant then it
moves it to another component, which means each point gets moved a distance at
least 2; a similar argument applies to points in Y.
Finally, the maps Xj ^ Xj ^ Yj\Xj and Ύ -> [0, 1] χ Υ -* [0, 1] χ (Η\Υ)
induce a continuous surjection from Г0\Хо Ц ([0. 1] x (H\Y)) Ц Γι \Χ] to Γ\Χ, so
if the actions of Г, and Η are cocompact, then the action of Γ on X is cocompact. G
The following result generalizes (11.17).
11.19 Corollary. If the groups Yq and Yx act properly and cocompactly by isometries
on CAT(0) spaces Xq and X\, and if С is a group that contains a cyclic subgroup
of finite index, then any amalgamated free product of the form Г = Го *с Y\ acts
properly and cocompactly by isometries on a CAT(0) space.
Proof. If С is finite, then by (2.8) the image of С in Г^ fixes a pointy e Xj forj = 0,1.
Thus we may apply the theorem with Υ equal to a single point and Υ —» Xj the map
with image pj.
If С contains an infinite cyclic subgroup of finite index, then it contains a normal
such subgroup (τ); let Xj be the image of τ in Γ,. The action of С on Xj leaves an axis
q : К —> Xj of Xj invariant (6.2). A virtually cyclic group cannot surject to both Ζ
and Z2 * Ία, so the image of С in Isom(^) either acts on сДК) as an infinite dihedral
group for j = 1, 2, or else it acts on both of these axes as an infinite cyclic group.
In each case, the action is determined up to equivariant isometry by the translation
length of any element у eCof infinite order. If we rescale the metric on Xq so that
| το | = | x\ |, and if we change the parameterization of cq by a suitable translation, then
c\ (t) м> C2(t) is a C-equivariant isometry. Thus we may apply the theorem, taking
Υ = К and^(i) = Cj(t), where each γ e С acts on Υ by y.t = Cj~l(y.Cj(t)). D

358 ChapterII.ll Gluing Constructions
11.20 Remark. The above argument extends to actions that are not cocompact if one
adds the assumption that the images of С in rt and Го contain hyperbolic elements.
An argument entirely similar to that of (11.18) yields the following result. (See
section Ш.Г.6 for basic facts concerning HNN extensions.)
11.21 Proposition. Let Yq and Η be groups acting properly by isometries on CAT(O)
spaces Xq and Υ respectively Suppose that for j = 0, 1 a monomorphism <fy : Η <^-> Γ
and a <fy-equivariant isometric embedding f : Υ —> Xq are given. Then the HNN
extension Γ = Γβ*# associated to this data acts properly by isometries on a CAT(O)
space X.
Proof. We recall that the HNN extension Γ = Г0*н is generated by Го and an element
s such that фо{Щ = ί$ι(/ϊ)ί_1 for all h e H, and that Γ0 is naturally identified to a
subgroup of Г (i.e. the base group, see Ш.Г.6.2). The associated Bass-Serre tree Τ
is the quotient of the disjoint union Γ χ [0, 1] (copies of [0, 1] indexed by Γ) by the
equivalence relation generated by
(У. t) ~ (ytfo(A), 0. (У. 0) ~ (γγο, 0) ~ (ys, 1)
for all γ e Γ, γο e Γ0, h e H, t e [0, 1]. The action of Г by left translation
on the index set of the disjoint union permutes the edges and is compatible with
the equivalence relation, therefore it induces an action of Г on Г by isometries. The
quotient of the tree Τ by this action is a circle; the subgroup фо(Н) С Г is the
stabilizer of an edge in Τ and Tq is the stabilizer of a vertex.
The space X will be the quotient of the disjoint union
(ГхХ0)Ц(Гх[0, l]x Y)
by the equivalence relation
(УУ0, x) ~ (У, Υύ-x), (У0о№, t, у) ~ (у, t, h.y),
for all γ e Γ, , yo e Γ0, he Η, te [0, 1].
The rest of the proof is entirely analogous to the proof of (11.18). G
As in (11.19) we deduce:
11.22 Corollary. Let Го be a group that acts properly by isometries on a CAT(0)
space, let С and С be subgroups ofT0 that contain cyclic subgroups of finite index
and let φ : С —>- С" by an isomorphism. If С has an element γ of infinite order then
suppose that γ and φ(γ) are hyperbolic isometries with the same translation length.
Then the HNN extension Γ = Γο*^ acts properly by isometries on a CAT(0) space,
and if the action о/Го is cocompact then so is the action ο/Γ.

Gluing Along Subspaces That Are Not Locally Convex 359
11.23 Remark. The hypothesis of properness was only used in (11.18) to deduce that
the action of Γ was proper. The theorem remains valid if one replaces properness by
semi-simplicity or faithfulness (as both an hypothesis and a conclusion).
Gluing Along Subspaces That Are Not Locally Convex
Our next set of examples illustrates the fact that there are circumstances in which
one can obtain a CAT(O) space by gluing spaces (which need not be CAT(O)) along
subspaces which are not locally convex.
We showed in (1.6.15) that, as Riemannian manifolds, spheres of radius r in W
are isometric to spheres of radius sinh r in E". Let D С Е" be a closed ball of radius
sinh r and let D' с И" be an open ball enclosed by a sphere S of radius r. Endow
Υ = И" ν ТУ with the induced path metric and let i : 3D —> S be a Riemannian
isometry. If X is the quotient of Υ JJ D by the equivalence relation generated by
[x ~ i(x), Vjc e 3D ], we say "X is obtained from W by replacing a hyperbolic ball
with a Euclidean ball".
In the course of the next proof we shall need the following observation.
11.24 Remark. Let (X, d) be a metric space and let (Z, d) be a subspace endowed
with the induced length metric. If с and d are geodesies in X that have a common
origin, and if the images of с and d are contained in Z, then Z(c, d) as measured in
(Z, d) is no less than Z(c, d) as measured in (X, d), because d > d. In particular, if a
geodesic triangle Δ С X is contained in Ζ and satisfies the CAT(O) angle condition
(1.7(4)) with respect to d, then it also satisfies condition (1.7(4)) with respect to d.
11.25 Proposition. IfX is obtained from W by replacing a hyperbolic ball with a
Euclidean ball in the manner described above, then X is a CAT(O) space.
Proof. We begin by describing the geodesies in X. We maintain the notation
established in the discussion preceding (11.24).
S is convex in Y, by 2.6(2), and the map i : 3D —> S preserves the length of
curves. Since the restriction to 3D of the metric on D is Lipschitz equivalent to the
induced path metric on 3D, it follows that X is a proper length space (homeomorphic
to K") and hence a geodesic space (see 1.5.20 and 1.5.22(4)).
For each pair of distinct points p, q e 3D, wehaveu?E(p, q) < dY(i(p), i(g)).This,
together with the fact that S is convex in Υ (and the definition of quotient metrics
in terms of chains, 1.5.19), yields the following description of geodesies in X: if
x, x1 e D С X, the Euclidean segment [x, x1] is the unique geodesic joining χ to x1 in
X; if jc e D с Xandy eFc X, each geodesic joining jc to у in X is the concatenation
of a Euclidean segment [jc, x/] С D and a hyperbolic segment [x1, у] С Υ that meets
S only at xf; and any geodesic in X joining a pair of points y, / e Υ is either a
hyperbolic segment in Υ or else the concatenation of three segments, two hyperbolic
ones [y, x], [x/, /] С Υ separated by a Euclidean one [jc, x/] С D.

360 Chapter П. 11 Gluing Constructions
We shall verify that all geodesic triangles Δ с X satisfy the CAT(O) condition as
phrased in (1.7(4)). If Δ is Euclidean, i.e. is contained in D С X, then Δ obviously
satisfies the CAT(O) condition, and likewise (in the light of 1.12) if Δ is hyperbolic,
i.e. Δ С Υ ч S. (If n > 3 then А с Υ does not imply that the convex hull of Δ is
contained in Y, but this does not effect the calculation of vertex angles.)
We define a dart to be a geodesic triangle in X that has one edge in D and two
edges in Υ (figure 11.1). Assume for a moment that we know that darts satisfy the
CAT(O) inequality. Figure 11.2 (where the cases are to be taken in order) indicates
how arbitrary geodesic triangles in X can be decomposed into smaller triangles that
are already known to satisfy the CAT(O) inequality. By applying the gluing lemma
for triangles (10.4) one sees that the given triangle satisfies the CAT(O) inequality.
Fig. 11.1 Darts
Fig. 11.2 Decomposing arbitrary triangles for (11.25)

Gluing Along Subspaces That Are Not Locally Convex 361
It remains to prove that every dart Δ satisfies the CAT(O) inequality. Let Η be
the intersection of Υ with the hyperbolic 2-plane in Ш" spanned by the vertices of
Δ and let X$ с X be the union of Η and the 2-disc in D spanned by the boundary
of H. With the induced length metric, Xq is isometric to the space obtained from
H2 by replacing a hyperbolic disc by a Euclidean disc. The inclusion Xq <^-> Χ does
not increase distances and its restriction to each side of Δ is an isometry, hence it
is sufficient to show that Δ verifies the CAT(O) inequality in Xq (with the induced
length metric). By making two applications of the gluing lemma for triangles (in Xq),
we can reduce to the case of darts in which the two sides that are hyperbolic segments
meet the circle tangentially (see figure 11.1) — suppose that Δ = A(p, q, r) has this
property.
The first inequality in Exercise 1.6.19(4) shows that the vertex angles at ρ and
q in any (isosceles) comparison triangle Δ(ρ, q, г) С Е2 are greater than those at
p,q e Δ с Хо- And the second inequality in 1.6.19(4) shows that the angle at r in
a comparison triangle A(p, q, г) с И2 is less than the vertex angle at r e Δ. By
(1.12), the angle at r in Δ(ρ, q, г) С И2 is less than the angle at r in Δ(ρ, q, г) С Е2,
so we are done. D
11.26 Exercises
(1) To obtain an alternative proof of (11.25), one can replace the above
considerations of darts by the following argument for the case η = 2.
Show that if η = 2, then X is a Gromov-Hausdorff limit of the CAT(O) spaces Xn
obtained from H2 by deleting a regular 2«-sided polygon inscribed in S and replacing
it with a regular Euclidean polygon with the same side lengths. (To see that X„ is
CAT(O), assemble it from 2и sectors by a sequence of gluings.)
(2) Let с : S1 —> E2 be a simple closed curve of length 2π that is not a circle.
Show that the space obtained by replacing the interior of the disc bounded by с with
a round Euclidean disc of radius 1 is not a CAT(O) space.
(Hint: Enclose с in a large square. If the described surgery on the plane gave a
metric of non-positive curvature, we could get a metric of non-positive curvature on
the torus by identifying opposite sides of the square. The Flat Torus Theorem would
then imply that the surgered plane is isometric to the Euclidean plane. To see that this
cannot be the case, recall that a simple closed curve of length 2π in the Euclidean
plane encloses an area at most π, with equality if and only if the curve is a round
circle.)
(3) The space obtained by removing a round disc from the Euclidean plane and
replacing it with a hyperbolic disc of the same circumference is not non-positively
curved.

362 Chapter II. 11 Gluing Constructions
Truncated Hyperbolic Space
In this section we shall describe the geometry of truncated hyperbolic spaces,
i.e. geodesic spaces obtained from real hyperbolic space by deleting a disjoint
collection of open horoballs and endowing the resulting subspace with the induced length
metric. The following result is a special case of the Alexander-Berg-Bishop
characterization of curvature for manifolds with boundary [ABB93b]. We present this
special case in detail because it provides a concrete example derived from a familiar
context, and also because it admits an elementary and instructive proof. The
construction does not involve gluing, but it is natural to include it at this point because
the arguments involved are very similar to those employed in the preceding section.
11.27 Theorem. Let X cfffea subspace obtained by deleting a family of disjoint
open horoballs. When endowed with the induced length metric, X is a complete
CAT(O) space.
This result does not extend to more general rank 1 symmetric spaces, as one
can see by looking at the stabilizers of horospheres (11.35). Indeed, because of the
nilpotent subgroups corresponding to such stabilizers, non-uniform lattices in rank
1 Lie groups other than SO(n, 1) cannot act properly by semi-simple isometries on
any CAT(O) space (7.16). In contrast, from (11.27) we get:
11.28 Corollary. Every lattice Г с SO(«, 1) acts properly and cocompactly by
isometries on a CAT(O) space X.
Proof. If Г is a cocompact lattice, take X = W. If Г is not cocompact, then one
removes a Γ-equivariant set of disjoint open horoballs about the parabolic fixed points
of Г; the action of Г on the complement X is cocompact and by the above theorem
X is CAT(O) (cf. page 266 of [Ep+92]). D
Throughout this paragraph we shall work with the upper half space model for
W. Thus W is regarded as the submanifold of W consisting of points {x\,... , xn)
with xn > 0, and this manifold is endowed with the Riemannian metric ^, where ds
is the Euclidean Riemannian metric on K"+1.
11.29 Horoballs in W. An open (resp. closed) horoball in H" is a translate of
Bq = {(x\,... ,xn) | xn > 1} (resp.x„ > 1) by an element of Isom(]H"). Ahorosphere
is any translate of the set Щ = {{xx,... ,x„) \ x„= 1). The closed horoballs in W
are the subspaces xn > const and the Euclidean balls tangent to dW. In the Poincare
ball model, the horoballs are Euclidean balls tangent to the boundary sphere.
11.30 Exercises
(1) Prove that every disjoint collection of closed horoballs in W is locally finite,
i.e., only finitely many of the horoballs meet any compact subset of W.

Truncated Hyperbolic Space 363
(Hint: In the Poincare ball model, if a horoball intersects the ball of Euclidean
radius r about the centre of the model, then the horoball has Euclidean radius at least
(1 - r)/2.)
(2) Let X be as in the statement of Theorem 11.27. Prove that X, equipped with
the induced path metric, is simply connected.
In the light of the Cartan-Hadamard Theorem (4.1) and the preceding exercise,
in order to prove Theorem 11.27, it suffices to show that the truncated space X is
non-positively curved. Since this is a local problem, there is no loss of generality
in restricting our attention to the case where X is obtained from Ш" by deleting the
single horoball Bq = {(χι,... , xn) \ xn > 1}. Let Xq = W ч Bq, endowed with the
induced path metric from Ш", and let Щ = {(x\,... , x„) | x„ = 1}.
11.31 Lemma. X0 is uniquely geodesic.
Proof. Fix x, у e Xq. We extend the natural action of 0(n — 1) on R"_1 to K" so
that the action is trivial on the last coordinate. This action preserves the Riemannian
metric j- and hence restricts to an action by isometries on W and Xq. Transporting χ
and у by a suitable element of 0(n — 1), we may assume that they lie in Ρ = {0} χ К2.
Because the metric on Ш" arises from the Riemannian metric —, it is clear that the
map W —> W Π Ρ given by (x\,... , xn) ы> (0,... ,0, xn-\, xn) strictly decreases
the length of any path joining χ to Υ unless the path is entirely contained in P. Thus
Ρ Π Xq is a (strictly) convex subset of Xq.
We have reduced to showing that there is a unique geodesic connecting χ to у in
Ρ Π Xq . But Ρ Π Xq is isometric to H2 minus an open horoball (with the induced path
metric), and this is a CAT(-l) space (1.16(5)). D
11.32 Lemma. The bounding horosphere Щ is a convex subspace ofXo, and with
the induced path metric it is isometric to E"_1.
Proof. The convexity of Щ is a special case of observation 2.6(2) — the point is
that the projection Xq —> Щ given by я : (χι, ... ,x„)i-> (χι, ■.. , 1) decreases the
length of any path not contained in Щ. The length of paths in Щ is measured using
the Euclidean Riemannian metric ds (because x„ = 1 on Щ), so Щ is isometric to
E""1. D
As in (11.31), one can reduce the proof of the next lemma to the case η = 2,
where the result is clear.
11.33 Lemma. If the hyperbolic geodesic connecting two points x,y e Xq is not
contained in Xq, then there exist points χ1, у' е Щ such that the unique geodesic
connecting xtoy in Xq is the concatenation of the hyperbolic geodesic joining χ to χ1,
the (Euclidean) geodesic joining x1toy1 in Щ, and the hyperbolic geodesic joining
У to y. Moreover the hyperbolic geodesies [xf, x] and [y1, y] meet Щ tangentially.

364 Chapter II. 11 Gluing Constructions
Proof of Theorem 11.27. We must show that for every geodesic triangle Δ =
A(p,q, r) in Xq the angle at each vertex is no larger than the angle at the corresponding
vertex of a comparison triangle in E2. We may assume that/?, q and r are distinct.
The proof divides into cases according to the position of the vertices relative to the
horosphere Hq.
Case 0. In the case where Δ is contained in Hq there is nothing to prove, because
Hq is isometric to E"_1. The case Δ cXq^Hq is also obvious (in the light of (1.12)).
Case 1. Suppose that just one vertex, q say, of the triangle Δ(ρ, q, r) lies on
Hq, and suppose that the sides of the triangle incident at this vertex are hyperbolic
geodesies. (This happens precisely when ρ and r lie on or below the unique hyperplane
in Ш" that is tangent to Щ at q.) In this case the convex hull in X of the points p, q, r
is equal to their convex hull in Ш", which is CAT(O).
Case 2. Now assume that Δ has one vertex in Xq \ Щ and two vertices on Hq.
Suppose also that each of the geodesic segments [p, q] and [p, r] is hyperbolic, i.e.
intersects H0 only at q and r respectively. Let Υ be the intersection of Xq with the
unique 2-plane (copy of H2) in W that contains p, q and r. If the boundary of this
2-plane contains the centre of the horosphere Xq, then we are in the setting of 1.16(6)
and otherwise Υ Π Hq is a circle — assume that we are in the latter case. Let D С Hq
be the Euclidean disc bounded by the circle Υ Π Hq, and let Ζ = Υ U D. Lemma
11.33 shows that Δ is contained in Z, and (11.25) shows that Z, equipped with the
induced path metric from X, is a CAT(O) space. Thus the vertex angles of Δ are no
greater than the corresponding comparison angles (cf. 11.24).
We list the remaining cases and leave the reader the exercise of subdividing the
given triangles (using Lemma 11.33) so that by using the gluing lemma one can
reduce to earlier cases (cf. figure 11.2).
Case 3. Assume that ρ e Xq \ H0 and q, r e Hq, and that at least one of the
geodesic segments [p, q] and [p, r] meets Hq in more than one point.
Case 4. Suppose that q e Hq and p, r e Χ \ ί?ο but we are not in Case 2.
Consider the cases [p, г] П Hq φ 0 and [p, r] Π Hq = 0 separately.
Case 5. p,q,r eX0\ H0 but Δ(ρ, q, г)Г)Ноф0. D
11.34 Corollary (Geodesies in Truncated Hyperbolic Spaces). Let X с И" be as in
Theorem 11.27. A path с : [α, b] —> X parameterized by arc length is a geodesic in
X if and only if it can be expressed as a concatenation of non-trivial paths c\,... , c„
parameterized by arc length, such that:
(1) each of the paths c, is either a hyperbolic geodesic or else its image is contained
in one of the horospheres bounding X and in that horosphere it is a Euclidean
geodesic;
(2) ifc-, is a hyperbolic geodesic then the image ofci+ \ is contained in a horosphere,
and vice versa;
(3) when viewed as a map [a, b] —> Ш", the path с is C1.

Truncated Hyperbolic Space 365
Proof. Since X is a CAT(O) space, it suffices to check that this description
characterizes local geodesies in X. And for this local problem it is enough to consider the
case X = Xq, where the desired result is a restatement of Lemma 11.33. D
11.35 Remark. If one removes an open horoball from a rank 1 symmetric space
other than Ш" then in the induced path metric on the complement the corresponding
horosphere will be convex (2.6(2)). But this horosphere admits a proper cocompact
action by a nilpotent group that is not virtually abelian (cf. 10.28) and therefore the
horosphere is not a CAT(O) space (7.16). Hence the symmetric space minus an open
horoball is not a CAT(O) space.
11.36 Remark (The Geometry of Cusps). If Г С SO(«, 1) is a torsion-free
nonuniform lattice, then Μ = Г\И" is a complete manifold of finite volume. The space
Xci" described in (11.18) projects to a subspace N с М that is a compact manifold
with boundary. The boundary components of N are in 1-1 correspondence with the
Γ-orbits of the horospheres in Ш" that bound X: each boundary component С of N
is the quotient of such a horosphere by the subgroup of Г that stabilizes it; С is a
closed manifold on which the induced path metric is flat (i.e. locally Euclidean), and
С <^> Wis a local isometry (where N is equipped with the induced path-metric from
M). The complement of N in Μ consists of a finite number of cusps, one for each
boundary component of N. If С is the quotient of a horosphere S by Го С Г, then
the cusp corresponding to С is the quotient by Го of the horoball bounded by S; this
cusp is homeomorphic to С χ [0, oo).
By replacing each of the horoballs in the definition of X by a smaller concentric
horoball in a Γ-equivariant manner, one obtains a Г-invariant subspace X' с И" that
contains X and hence a subspace N' = Г\Х' С М containing N. The components
of the complement of N in N' correspond to the components of dN; the component
corresponding to С С dN is homeomorphic to С χ (0, 1]. If С" С dN' corresponds
to C, then the path metric on C" is a scalar multiple of the path metric on С Thus,
replacing N by a suitable choice of N', one can decrease the volume of each boundary
component of N arbitrarily by extending down the corresponding cusp of M.
As one extends N down the cusps of Μ in the above manner, the shape40 of each
boundary component С remains constant. In order to vary the shape of the cusps
while retaining non-positive curvature, one must consider more general metrics on
N. The appropriate techniques are described in [Sch89]41: one regards each cusp as a
warped product С χ [0, oo) with the Riemannian metric ds2 = e~2t ds2, + dt2, where
dso is the given Riemannian metric on C. One can vary this metric on С χ [1, oo) so
that negative curvature is retained and for suitable Го > 1 the metric on С х [Го, оо)
becomes ds2 = e~2t ds2 + dt2, where dsi is a chosen flat metric on C. A local version
of (11.27) shows that the length metric on the manifold with boundary obtained by
deleting С χ (Γ, oo) from Μ is non-positively curved if Γ > Tq.
40 The shape of С is the isometry type of the flat manifold of volume one that is obtained from
С by multiplying the metric with a constant.
41 Schroeder informs us that a detailed account of the construction required in the present
context was written by Buyalo as an appendix to the Russian edition of [Wo67].

366 Chapter П. 11 Gluing Constructions
11.37 Exercises
(1) Let Г be as in (11.36) and suppose that the boundary components of N = Г\Х
are all tori. Show that if a group G acts properly and cocompactly by isometries on
CAT(O) space, then so too does any amalgamated free product of the form Г *А G,
where Л is abelian.
(Hint: If A is not cyclic, then since X is locally С AT(— 1) away from its bounding
horospheres, the Α-invariant flats F yielded by (7.1) must be contained in one of these
horospheres. Adjust the size and shape of A\F using the last paragraph of (11.36)
and apply (11.18).)
(2) The purpose of this exercise is to extend (11.28) to all geometrically finite
subgroups Г С SO(«, 1). Let Г С SO(«, 1) be an infinite subgroup, let Λ(Γ) be its
limit set (the set of points in ЗИ" that lie in the closure of an orbit of Г) and let
C(T) be the convex hull in W of the union of geodesic lines with both endpoints in
Л(Г). If Г is geometrically finite then С(Г) is non-empty and there is a Γ-equivariant
collection of disjoint open horoballs, with union U, such that the action of Г on
X = С(Г) П (W \ U) is proper and cocompact.
Prove that when endowed with the induced path metric, X is a CAT(O) space.
(We refer the reader to the papers of Bowditch for the general theory of
geometrically finite groups, and to Chapter 11 of [Ep+92] for results related to this
exercise.)

Chapter 11.12 Simple Complexes of Groups
In this chapter we describe a construction that allows one to build many interesting
examples of group actions on complexes (12.18). This construction originates from
the observation that if an action of a group G by isometries on a complex X has a strict
fundamental domain42 Y, then one can recover X and the action of G directly from
Υ and the pattern of its isotropy subgroups. (The isotropy subgroups are organised
into a simple complex of groups (12.11).)
With this observation in mind, one might hope to be able to start with data
resembling the quotient of a group action with strict fundamental domain (i.e. a
simple complex of groups) and then construct a group action giving rise to the
specified data. Complexes of groups for which this can be done are called strictly
developable. A simple complex of groups consists of a simplicial complex with
groups Ga associated to the individual simplices σ, and whenever one cell τ is
contained in another σ a monomorphism ψτσ : Ga —> Gx is given. If this data
arose from the action of a group G then the inclusion maps φσ : Ga —> G would
be a family of monomorphisms that were compatible in the sense that φτ ψτσ — φσ
whenever τ С σ. The main result of this section is that the existence of such a family
of monomorphisms is not only a necessary condition for strict developability, it is
also sufficient. More precisely, given any group G, the Basic Construction described
in (12.18) associates to each compatible family of monomorphisms φσ : Ga —> G
an action of G on a simplicial complex such that the (strict) fundamental domain and
isotropy groups of this action are precisely the simple complex of groups with which
we began.
The question of whether the local groups of a simple complex of groups inject
into some group in a compatible way is a property that appears to require
knowledge of the whole complex. Remarkably though, in keeping with the local-to-global
theme exemplified by the Cartan-Hadamard theorem, there is a local criterion for
developability: if one has a metric on the underlying complex, and this metric has
the property that when one constructs a model for the star of each vertex in a putative
development, all of these local models are non-positively curved, then the complex
of groups is indeed developable (12.28). This will proved in Part III, Chapter Q.
The construction outlined above is due essentially to Tits [Tits75 and Tits86b],
and has been used extensively by Davis and others in connection with constructions
that involve Coxeter groups and buildings (see in particular [Da83]). We shall give
i.e. a subcomplex that meets each orbit in exactly one point.

368 Chapter 11.12 Simple Complexes of Groups
a variety of concrete examples in an attempt to illustrate the utility of some of the
ideas involved in these and other applications. If the reader wishes to move quickly
through the main points of this chapter, then he should direct his attention first to
the basic definitions in 12.11-15, the main construction in 12.18 and the list of its
properties in 12.20.
As well as providing us with a basic tool for constructing examples of non-
positively curved spaces, this chapter also serves as an introduction to Chapter III.C,
where the same issues are addressed in a more serious fashion: a more sophisticated
notion of complexes of groups is introduced and used as a tool for (re)constructing
group actions that may not have a strict fundamental domain. (Requiring a group
action to have a strict fundamental domain is a rather restrictive condition.)
Stratified Spaces
Most of the examples that we shall consider will involve groups acting on simplicial
or polyhedral complexes, but it is both useful and convenient to work in the more
general setting of stratified spaces. A useful example to bear in mind is that of a
metric polyhedral complex К in which the characteristic map of each closed cell
is injective: one can stratify К by defining the closed cells of К to be the strata, or
alternatively one can take the dual stratification of К (see (12.2(1)).
12.1 Definition (Stratified Sets and Spaces). A stratified set (Χ, {Χσ}σ€-ρ) consists
of a set X and a collection of subsets Χσ called strata, indexed by a set V, such that:
(1) X is a union of strata,
(2) ϊΐΧσ =Χτ thena= τ,
(3) if an intersection Χσ Π Χτ of two strata is non-empty, then it is a union of strata,
(4) for each χ e X there is a unique σ (χ) eP such that the intersection of the strata
containing χ is Χσ<·χ\
The inclusion of strata gives a partial ordering on the set V, namely τ < σ if and
only if Χτ с Χσ. We shall often refer to (Χ, [Χσ}σ<=ν) as "a stratified space X with
strata indexed by the poset V" or (more casually) "a stratified space over V".
Stratified topological spaces: Suppose that each stratum X" is a topological space,
that Χσ ПГ is a closed subset of both Χσ and Xе for each σ, τ e V, and that the
topologies which Xе and Χτ induce on Χσ Π Χτ are the same. We can then define
a topology on X that is characterized by the property that a subset of X is closed if
and only if its intersection with each stratum Χσ is a closed subspace of Χσ. (This is
called the weak topology associated to (Χσ}σ(=·ρ, [Spa66, p.5].)
If X has the additional property that each stratum contains only a finite number
of strata (which ensures that any union of strata is a closed set), then X is called a
stratified topological space.
Stratified simplicial complexes: If a stratified set X is the geometric realization
of an abstract simplicial complex and each stratum Χσ is a simplicial subcomplex,

Stratified Spaces 369
then X is called a stratified simplicial complex. If we endow each of the strata with
its weak topology (1.7 A.5), then the topology on X arising from the stratification will
coincide with the (usual) weak topology on X. We shall implicitly assume that all
simplicial complexes considered in this section are endowed with this topology, but
it is also worth noting that all of the statements that we shall make remain valid if
one replaces the weak topology on X by the metric topology of (1.7 A.5).
Stratified MK-polyhedral complexes: If a stratified set X is an MK-polyhedral
complex with Shapes(X) finite, and if each stratum Xе is a subcomplex (i.e. a union
of closed faces of cells), then X is called a stratified Μ «-polyhedral complex. As in
the simplicial case, one has to decide whether to use the weak topology or metric
topology in this case, but all statements that we shall make will be valid for either
choice of topology. (Note that in the simplicial and polyhedral cases one does not
need to assume that each stratum contains only finitely many strata in order to ensure
that arbitrary unions of strata are closed.)
Throughout this section, the term stratified space will be used to mean one of the
three special types of stratified sets defined above. Morphisms of stratified spaces
are defined in (12.5).
st(jc): Let X be a stratified space and let χ e X. The complement of the union of
strata which do not contain χ is an open set; this set is denoted st(jc), and is called
the open star ofx. (In the case where X is simplicial or polyhedral, this terminology
agrees with that of (1.7.3) if one takes as strata the closed cells of X.) The closure of
st(x) will be denoted St(x).
Note that st(jc) = (y e Χ | Υσ, у e Χσ => χ e Χσ}.
12.2 Examples
(1) Let К be a simplicial complex oranM^-polyhedral complex К with Shapes(iT)
finite. Let V be the poset of the cells of К ordered by inclusion. As we noted above,
К has a natural stratification indexed by V, namely one takes as strata the cells of
K. Dual to this, one has a stratification indexed by Pop (the set V with the order
reversed): the stratum indexed by the cell σ of К is the union of those simplices in
the barycentric subdivision of К whose vertices are barycentres of cells containing
σ (figure 12.1).
(2) One can regard a 2-dimensional manifold X with a finite set of disjoint
closed curves as a stratified space whose strata are the curves and the closures of the
connected components of the complement of their union (figure 12.2 and 12.10).
(3) Let Μ be a compact manifold with boundary and let L be a triangulation of its
boundary as in 5.23. The stratification associated to this situation in 5.23 is indexed
by a poset Q. The stratum M0 is equal to M. The stratum Μσ (denoted L" in 5.23)
is the union of simplices in the barycentric subdivision LI of L; a simplex of Li is in
M" if all of its vertices are the barycentres of simplices in L that contain σ.

370 Chapter II. 12 Simple Complexes of Groups
Fig. 12.1 A stratum and its dual in the union of two triangles
Fig. 12.2 A stratified surface and the geometric realization of the associated poset
12.3 Affine Realization of Posets
Associated to any poset V one has a simplicial complex43 whose set of vertices
is V and whose fc-simplices are the strictly increasing sequences σο < · · · < ok
of elements of V. The affine realization of this simplicial complex will be called
the affine realization^ of the poset V, denoted \V\. It has a natural stratification
indexed by V: the stratum |Ρ|σ indexed by σ e V is the union of the fc-simplices
σο < · · · < Ok with Ok < a. The closed star St(a) is the union of the simplices with
σ as a vertex, and the open star st(a) is the union of the interiors of these simplices.
For each σ e V we consider the following sub-posets:
Ρ(σ) := (τ e V \ τ < σ or τ > σ},
Р°:=(геР|т < σ}, and
Lka{V):={peV\p>a].
Lka(V) (often abbreviated to Lka) is called the upper link of σ. The affine realization
of V is the stratum \V\", while the affine realization of Ρ(σ) is St^). The affine
realization of Lka is related to the others by means of the join construction, which is
defined as follows.
43 See the appendix to Chapter 1.7 for basic notions concerning simplicial complexes.
44 Sometimes we shall use "geometric realization" instead of "affine realization".

Stratified Spaces 371
Given posets V and V, one can extend the given partial orderings to the disjoint
union V Ц V by decreeing that σ < σ' for all σ e V, σ' e V; the resulting poset,
denoted V * V, is called the join of Ρ and P'. The affine realization of Ρ * V is the
join of the simplicial complexes |P| and |P'| (cf. 1.7 A.2). For all σ e P, we have
Ρ(σ) = ρσ * Ζ1σ, and hence St(a) is the simplicial join of \V\" and |Ζ1σ |.
A map between posets/ : Ρ —> Q is called a morphism if it is order preserving.
Such a morphism is called поп-degenerate if for each σ e Ρ the restriction of/ to Ρσ
is a bijection onto Q^(tJ) (this condition ensures that the affine realization [Я : |P| ->
| Q\, which was defined in (1.7A.3), sends each stratum \P° | homeomorphically onto
the stratum | Qfa)).
Note that, as simplicial complexes, |P| = |Pop|, but the natural stratifications on
these complexes are different. We write |Ρ|σ to denote the stratum in |Pop| indexed
by σ.
12.4 Examples of Poset Realizations
(1) Let К be an M^-polyhedral complex for which the inclusion maps of the cells
are injective (e.g. a simplicial or cubical complex), and let Ρ be the poset of the cells
of К ordered by inclusion. Then |P| is simplicially isomorphic to the barycentric
subdivision of K; the stratum \V\a is the cell σ с AT, while the stratum |Ρ|σ = \V°P\"
is the dual stratum described in (12.2(1)).
(2) Let и be a positive integer. Let S be a finite set with η elements, let S be the
poset of its subsets ordered by reverse-inclusion: fora, τ e S, if σ с r then τ < σ.
Let T be the subposet consisting of the proper45 subsets of S. The affine realization
of Τ is the barycentric subdivision of an (n — l)-simplex (see figure 12.3).
The affine realization |5| is a simplicial subdivision of the и-cube spanned by an
orthonormal basis (e^esinlR"; the subseta = {jb ..., Sk] corresponds to the vertex
J2i=\ es,- This subdivision is isomorphic to the simplicial cone over the barycentric
subdivision \T\ of the (n — l)-simplex; the cone vertex corresponds to subset S с S.
For σ = [s\,.. .,st} с S, the stratum |5|σ is the (n — fc)-dimensional sub-cube
consisting of those points χ = ^2ssSxses of the cube with xSl = · · · = xSk = 1.
The dual stratum |5|σ is the ^-dimensional sub-cube spanned by the basis vectors
est, ■ ■ ■ , esk ■
(3) Consider the poset of subsets of the set of vertices of a simplicial complex L,
ordered by reverse-inclusion. Let Q be the subposet consisting of the empty subset
and those subsets which span a simplex of L. The affine realization of Q is the
simplicial cone over the barycentric subdivision L' ofL. As in the previous example,
one can identify the cone over the barycentric subdivision of each (k — l)-simplex of
L with a certain simplicial subdivision of a fc-cube, and thus \Q\ = CL' is isomorphic
to the cubical complex F considered in 5.23. The stratum \Q\a is the subcomplex
denoted F° in example 5.23 (if σ = 0, then \Qf = \Q\ = F).
By convention, the empty subset is proper; 5 itself is not.

372 Chapter 11.12 Simple Complexes of Groups
Fig. 12.3 The affine realization of the poset of proper subsets of {1,2, 3}, and of the subposet
of all subsets.
Group Actions with a Strict Fundamental Domain
The main objects that we wish to study with the techniques developed in this chapter
are group actions on polyhedral complexes where the action respects the cell
structure. If a subcomplex Υ contains exactly one point of each orbit for such an action,
then Υ is called a strict fundamental domain. In this section we generalize the notion
of cellular action to strata preserving action, and define strict fundamental domains
for such actions.
12.5 Strata Preserving Maps and Actions. Given two stratified sets (Χ, (Χσ}σ(=·ρ)
and (У, {Yt}T£q), a map/ : X -> Υ is called strata preserving if it maps each
stratum Χσ of X bijectively onto some stratum of Y. The map/ : Ρ —» Q obtained
by defining/(σ) to be the index of the stratum f(X") is a morphism of posets, as
defined in (12.3).
If X and Υ are stratified spaces of the same type, then/ is called a strata preserving
morphism if its restriction to each stratum preserves the specified structure (i.e. in the
topological case the restriction must be a homeomorphism, in the simplicial case it
must be a simplicial isomorphism, and in the polyhedral case it must be a polyhedral
isometry). If there is a strata preserving morphism inverse to/, then/ is called an
isomorphism of stratified spaces.
The action of a group G on a stratified space (Χ, {Χσ}σ5-ρ) is said to be strata
preserving if for each g e G the map χ ы> g.x is a strata preserving morphism.
12.6 Exercise. Let G be a finite subgroup of 0(3). For a subgroup Η с G, let M.jj
(resp. S^) be the set of points χ in K3 (resp. §2) such that the isotropy subgroup of χ
is H. Show that the closures of the connected components of K^ (resp. S^), Η с G,
form a stratification of K3 (resp. S2), and that the action of G is strata preserving.

Group Actions with a Strict Fundamental Domain 373
12.7 Definition (Strict Fundamental Domain). Let G be a group acting by strata
preserving morphisms on a stratified set (Χ, {Χσ}σ5-ρ). A subset FCXis called a
strict fundamental domain for the action if it contains exactly one point from each
orbit and if for each σ e V, there is a unique ρ(σ) e V such that g.X° = Χρ(σ) с Υ
for some g e G. (Note that У is a union of strata and hence is closed.)
The set of indices Q с V for the strata contained in У is a strict fundamental
domain for the induced action of G on V in the following sense: Q is a subposet of
V which intersects each orbit in exactly one element, and [σ e Q, τ < σ] implies
τ e Q. Conversely, given a strata preserving action of G on X, if the induced action
on V has a strict fundamental domain Q in this sense, then the action of G on X has
a strict fundamental domain, namely Υ = (JCT<=g X° ■
12.8 Lemma. Consider a strata-preserving action of a group G on a stratified
space (Χ, {Χσ}σ5-ρ) with strict fundamental domain Υ = {J(jSqX't- Let χ m> р(х)
(resp. σ ы> ρ(σ)) be the map that associates to each χ e X (resp. σ e V) the unique
point of Υ {resp. Q) in its G-orbit. Then:
(1) χ м> p(x) is a strata preserving morphism X —■► Y.
(2) σ ы> ρ(σ) is a non-degenerate morphism ofposets V —> Q.
(3) For each σ e V, the following subgroups of G are equal: Ga = {g e G \
g.X" =Xa}and{g eG\ g.x = x, Vjc e Χσ}.
(4) IfX1 с Χσ then Ga с Gx.
(5) The isotropy subgroup (stabilizer) of each χ eXis{g eG| g .st(x)f)st(x) фЩ.
Proof. (1) and (2) are immediate from the definitions. If g e Ga and χ e Χσ, then
/j(jc) = p(g.x), since the image of ρ contains only one point from each orbit. But
χ н> p(x) is a bijection from X" to Xp(tJ) and both χ and g.jc are in Χσ, therefore
jc = g .jc. This proves (3), from which (4) follows immediately.
For (5), recall that st(x) = {ζ | Υσ, ζ e Χσ =Φ· j e Г}. We can assume
that χ e Y. Suppose that g.z = z' for some ζ, ί e st(x), and that ζ e Χσ. Then
jc e Χσ П g.X°. Let й e G be such that й.Хст с Υ. We have Лл = χ. As ftg"1 maps
g.X" to F, we also have hg~l.x = x, thus g.x = x. D
12.9 Examples
(1) Consider the tesselation of the Euclidean plane by equilateral triangles (see
figure 12.4). We regard E2 as a stratified space whose strata are the simplices of the
barycentric subdivision of this tesselation. Let G be the subgroup of Isom(E2) that
preserves the tesselation. The action of G is strata preserving and each 2-simplex is
a strict fundamental domain for the action.
Let Go be the orientation-preserving subgroup of index two in G and let G\ be the
subgroup of index six in G generated by the reflections in the lines of the tesselation.
Any triangle of the tesselation serves as a strict fundamental domain for the action
of Gi, but there does not exist a strict fundamental domain for the action of Go.
(2) Consider the construction due to Davis that we described in 5.23 and the
associated stratifications described in 12.(3) and 12.2(3). In this setting the action of

374 Chapter II. 12 Simple Complexes of Groups
Fig. 12.4 A strict fundamental domain for the action of G in 12.9(1).
G on К (resp. N) is strata preserving and has as a strict fundamental domain F (resp.
M).
(3) Let Σ(η) be the group of permutations of thesetfl, ..., «}.LetE(«)actonK"
by permuting the coordinates. The subset of K" consisting of points χ — (x\, ..., x„)
with x\ > ■ ■ ■ > xn is a strict fundamental domain for this action: it is a convex cone
C, the intersection of the (n — 1) half-spaces defined by the equations x, > x;+\. Let
Hi be the hyperplane defined by jc, = xi+i, where i = 1,..., η — 1. The boundary of
С is the union of the faces С Π Η;. Let s; e Σ(η) be the transposition exchanging i
and / + 1: it acts on R" as the reflection fixing the hyperplane #,. The angle between
two consecutive hyperplanes #,· and H-l+\ is π/З, and if \i — j\ > 1 then the angle
between #,· and Щ is π β. Let Sn~2 be the intersection of the unit sphere in W with
the hyperplane J2"=, xt = 0. The intersection of the cone С with S"~2 is a spherical
(n — 2)-simp]ex Δ, with dihedral angles π/З and π/2, which is a strict fundamental
domain for the action of Σ(η) restricted to S"~2. The translates of Δ by the action of
Σ(«) are the (n — 2)-simplices of a triangulation Τ of S"~2 and the action of Σ(η) is
strata preserving with respect to the stratification whose strata are the simplices of
this triangulation.
(4) Consider the Tits boundary dTP(n, R) { of the symmetric space P(n, 1). It has a
natural stratification by the Weyl chambers at infinity and their faces (see 10.75). The
natural action of SL(«, K) is strata preserving. Any Weyl chamber at infinity serves
as a strict fundamental domain for this action, for instance the set of points at infinity
of the subset consisting of diagonal matrices diag(e'',..., eu) with tx > ■ ■ ■ > tn and
Σ ti = 0. This fundamental domain is isometric to the spherical (n — 2)-simplex Δ
described above.
Notice that it follows from part (4) of the following lemma that if a group action
on a connected stratified space is proper and has a strict fundamental domain, then

Simple Complexes of Groups: Definitions and Examples 375
the group is generated by elements of finite order. (In particular, a torsion-free group
never admits such an action.)
12.10 Lemma. Consider a strata-preserving action of a group G on a stratified
space (Χ, {Χσ }as-p) with strict fundamental domain Υ = (Js X". Let Go с G be the
subgroup generated by the isotropy groups Ga, σ e Q, and let Xo = Go.Y.
(1) Ifg.Y(lYφ0,thengeGσforsomeσeQ.
(2) Ifg.Xo ПХоф0, then g e G0, hence g.X0 = X0.
(3) Xo is open and closed in X.
(4) X is connected if and only if Υ is connected and G = Go-
Proof. Part (1) follows immediately from the fact that each orbit intersects Υ in only
one point.
(2) If jc e g.XoDXo, then there are points у, у' e У and elements go, go e Go such
that* = g0.y = gg'o-У'- As in (1), wethenhavey = / and hencego~lggO e Ga(y).
(3) Xo is a union of strata, hence it is closed. Given χ e X" not in Xo, choose
g e G so that g.X" с Υ. If Χσ η X0 were not empty, then g.X0 П X0 would not be
empty, so by (2) we would have g .Xq = Xo> which is nonsense because χ e g .Xo. It
follows that X \ Xo is a union of strata, hence it is closed.
(4) Suppose that Υ is connected. Each g e Go is a product of elements g\, ...,gk
from the isotropy subgroups of the points of Y. Let y„ = g\.. .gn. Each pair of
consecutive sets in the sequence yk-Y, ■ ■ ■, γχ.Υ, Υο-Υ = Υ have a non-empty
intersection, hence their union is connected. Therefore g.Y is in the same connected
component of X as Υ is, and so Xo = Go. У is connected. If Go = G then Χ is also
connected.
Conversely, if X is connected then X = Xo, by (3), and hence G = Go, by (2).
Moreover, since У is a continuous image of X (12.8(1)), it too is connected. D
Simple Complexes of Groups: Definitions and Examples
In the introduction we indicated how simple complexes of groups were to be used
to reconstruct group actions on stratified spaces from a knowledge of the isotropy
subgroups on a strict fundamental domain. A more sophisticated notion of complex
of groups, adapted to describe group actions that do not have strict fundamental
domains, will be described in Part III, Chapter C.
12.11 Definition (Simple Complex of Groups). A simple complex of groups^
G(Q) = (GCT, ψτσ) over a poset Q consists of the following data:
46 Equivalent definition: A poset can be regarded as a small category (III.C) whose set of
objects is Q and which has an arrow σ —> τ whenever τ < σ. A simple complex of
groups is a functor from Q to the category of groups and monomorphisms. In this definition
we assume that the homomorphisms ψτσ are injective to ensure that simple complexes of
groups are always locally developable (12.24). In all respects other than local developability,
the theory would go through without any essential changes if we did not assume ψτσ to be
injective.

376 Chapter П. 12 Simple Complexes of Groups
(1) for each σ e Q, a group GCT, called the local group at σ;
(2) for each τ < σ, an injective homomorphism ψτσ : Ga ^> GT such that if
τ < σ < p, then
^r/> = ΨτσΨσρ-
Two simple complexes of groups {Ga, i/rr(7) and (G^., ψ'τσ) over a given poset Q
are said to be simply isomorphic if there is a family of isomorphisms φσ : Ga —> G^
such that ψ'τσφσ = φτψτσ for all τ < σ.
Let G be a group. A simple morphism φ = (φσ) from G(Q) to G, written φ :
G(Q) —» G, is a map that associates to each σ e Q a homomorphism φσ : Ga —» G
such that if τ < σ, then φσ = φτψτσ. We say that φ is injective on the local groups
if φσ is injective for each σ e Q.
12.12 Definition. Associated to any simple complex of groups G(Q) over Q, one
has the group
G(Q):=limGff,
which is the direct limit47 of the system of groups and monomorphisms (GCT, ψτσ).
The group G(Q) is characterized up to canonical isomorphism by the universal
property given below. It is obtained by taking the free product of the groups Ga and
making the identifications т^гст(^) = h, Vh e Ga, Vt < σ. Thus, given presentations
Ga = {A, \ ΤΙσ), one can present G(Q) as (Ц. Λσ | ΊΙσ, ψτσ(ά) = а, У а е
Λα, Υτ < σ) (where ψτσ(α) is given as a word in the generators Λτ).
The natural homomorphismSiCT : Ga —> G(Q) give a canonical simple morphism
ί : G(Q) ^ G(Q), where t = (tCT). In general ισ is not injective (see for example
12.17(5)).
12.13 The Universal Property. Given any group G and any simple morphism φ :
G(Q) —> G, there is a unique homomorphism φ : G(Q) —> G iwcft ίΑαί φ οι = φ
(i.e. φ ο ισ = φσ for every σ e Q). _____
(Conversely, every homomorphism φ : G(Q) —> G induces a simple morphism
φ : G(Q) -> G, gi'vew foy φσ := φ ο ισ.)
12.14 Remark. There exists a simple morphism φ from G(Q) to some group G that
is injective on the local groups if and only if ι is injective on the local groups.
12.15 Strict Developability. Consider a strata preserving action of a group G on a
stratified space (Χ, (Χσ}σ(=·ρ) with a strict fundamental domain Y. Let Q = {σ e
Ρ | Χσ £ Υ}. To this action and choice of Y, one associates the simple complex of
groups G(Q) = (GCT, ψτσ) over Q, where GCT is the isotropy subgroup of the stratum
ГСГ and ψτσ : Ga -» Gr is the natural inclusion associated to Хг СГ. The
inclusions φσ : Ga —> G define a simple morphism ^ : G(Q) —» G that is injective
on the local groups.
Sometimes called the amalgamated sum or amalgam.

Simple Complexes of Groups: Definitions and Examples 377
A simple complex of groups G(Q) is called strictly developable if it arises from
an action with a strict fundamental domain in this way. A necessary condition for the
strict developability is that there should exist a simple morphism G(Q) —> G that
is injective on the local groups. In (12.18) we shall show that this condition is also
sufficient.
12.16 Remark
The notion of a simple complex of groups is a particular case of the more general
notion of complex of groups explained in III.C. In the more general framework, one
considers morphisms G(Q) —> G that are not simple and there is a correspondingly
weaker notion of developability (cf. III.C, 2.15-16). If the affine realization of Q is
not simply connected, then it can happen that a simple complex of groups over Q is
developable but not strictly developable (this is the case in 12.17(5ii)).
12.17 Examples
(1) Triangles of Groups. Let Τ be the poset of proper subsets of the set S3 =
(1,2, 3}, ordered by reverse-inclusion. A simple complex of groups G(T) over Τ
is called a triangle of groups. (A ^-simplex of groups is defined similarly.) Thus a
triangle of groups is a commutative diagram of group monomorphisms as in figure
12.5. Such diagrams have been studied by Gersten and Stallings [St91] and others.
Tits considered more generally simplices of groups, i.e. complexes of groups over the
poset of faces of a simplex. In general triangles of groups are not (strictly) developable
(e.g. example(6)).
G{1,2}
Fig. 12.5 A triangle of groups
(2) A 1-simplex of groups is a diagram G\ <— Η —» Gj. of monomorphisms
of groups. The direct limit of this diagram is called the amalgamated free product
of G\ and G2 over Η and is often denoted G\ *H G2 (see III.Γ.6.1). The natural
homomorphisms of G\ and G% in G\ *н G% are injective and G\ *н G% acts on a
simplicial tree [Ser77], called the associated Bass-Serre tree, with strict fundamental
domain a 1-simplex (see 11.18). The isotropy subgroups of the vertices and the edge

378 Chapter 11.12 Simple Complexes of Groups
of this simplex are respectively G\, G2 and Я (see 12.20(4)). More generally, any
simple complex of groups whose underlying complex is a tree is strictly developable.
(3) Consider the action of G on E2 described in 12.9(1). Let S\, i2, S3 be the
reflections fixing the sides of a fundamental triangle, where the labelling {1}, {2}, {3}
is chosen as in figure (12.5); the vertex angles are π/6, π/3, π/2. The triangle of
groups G(T) associated to this action has the following local groups: G(,j is the cyclic
group of order two generated by the reflections j,·, for i= 1, 2, 3; the local groups at
the vertices G{i,2], G(2,3], G(i,3] are dihedral48 with orders 12, 6 and 4 respectively;
and the group G0 in the middle of the triangle is trivial (see figure 12.6).
(svs2) = D6 < (Sl) > D2=(alfa3>
Fig. 12.6 The triangle of groups associated to the action in fig. 12.4
(4) The complex of groups associated to the action of SL(«, К)опЭР(и, K)i with
the Tits metric (stratified by the Weyl chambers at infinity and their faces.cf. 10.75)
is an (n — 2)-simplex of groups. For instance for η = 4, if we choose as fundamental
domain the boundary at infinity of the Weyl chamber consisting of diagonal matrices
of the form diag(e",..., e'4) with t\ > ■ ■ ■ > Ц, we get a triangle of subgroups of
SL(4, K) as indicated in figure 12.7.
(5) A Non-Developable Complex of Groups. Let Q be the poset consisting
of 5 elements {p, ab σ2, τι, Тг}, with the ordering τ, < oj < ρ for i,j = 1, 2.
Consider the following complex of groups over Q: let Gp be the trivial group, let
Gr, = GCT| = GCT, = Z2, and let Gr, be a group Η containing two distinct elements
of order two, t\ and i2 say; fori = 1, 2, the homomorphism ψτισ; : GCT; —> Gr, sends
the generator of GCT, to r,. This complex of groups is not strictly developable, because
the kernel of the canonical homomorphism ir, : Gr, —>- G(<2) contains t\t^ ; indeed
G(Q) is the quotient of Я by the normal subgroup generated by ίχϊχ '. In particular,
if we take Я to be a simple group such as Л5, then G(Q) will be the trivial group.
(See figure 12.8.)
48 Our convention will be to write Dn to denote the dihedral group of order 2л.

Simple Complexes of Groups: Definitions and Examples 379
/.
*
0
V°
*
*
0
0
*
*
*
0
*N
*
*
*
/.
0
0
V°
*
*
0
0
*
*
*
*
*
*
*
*
Fig. 12.7 The triangle of groups associated to the action of SL(4, R) on dP(4, R),
Let Q be the subposet obtained from Q by removing p. The simple complex
of groups G(Q') obtained by restricting G(Q) to Q' is developable in the sense of
(III.C.2.11), but it is not strictly developable. Indeed, although G(Q') does not arise
from an action with strict fundamental domain, it is the complex of groups associated
to the simplicial action of a group #*z, on a tree such that the quotient is the affine
realization of Q' and the isotropy subgroups correspond to the local groups in Q'.
(There is some ambiguity concerning the choice of maps ψΧίσΓ)
Z2 = (s)
/ t \
H3 (S,t) <— {1} —* Z2
\ 4 /
Z2 = (t)
/
Η 3 (ί, t)
\
Z2 = (s)
Z2 = (t)
\
z,
/
(I)
Fig. 12.8 (I) A non-developable complex of groups with G(Q) =
which is not strictly developable
(Π)
Z2. (II) A graph of groups
(6) и-Gons of Groups. Let η > 3 be an integer. An n-gon of groups is a
commutative diagram of groups and monomorphisms of the type shown in Figure 12.9.
More precisely, it is a complex of groups over Q„, where Q„ is the poset of faces of
a regular и-gon Ρ in E2 ordered by inclusion.
In order to give examples, we introduce the notation Qn = {ρ, σ,, τ, | i e Ζ
mod и}, where the σ, and τ, correspond to the sides and vertices of Ρ and ρ is Ρ

380 Chapter П. 12 Simple Complexes of Groups
itself; the indices are chosen so that τ, < σ, < ρ and τ, < σ,+ι < p. (See figure
12.9.)
An interesting class of examples is obtained by taking Gp = {1}, Ga, = (*,·) = Ζ
and GTj = (xi, Xi+i | χ,-χ,+ι*,"1 = xj+l), and taking homomorphisms ψτ.σι and ψτισ,+,
that send the generators jc, e GCT, andjc,+i e GCT;+I to the elements of GTj that have the
same name49. Consider G(Qn) = (χι,..., xn \ χ,χί+\χ~{ = xf+l, i e Ζ mod ή).
G{Qz) is the trivial group (mis is not obvious!). If и > 4 then the local groups Gr,
inject into G(Qn), and hence G(Q„) is developable. The case η = 4 was studied in
detail by Higman, who showed that G(Q4) has no proper subgroups of finite index (see
[Ser77, 1.4]). Because G{Qa) is infinite, one can construct finitely generated infinite
simple groups by taking the quotient of G{Qa) by a maximal normal subgroup.
(*2>
II
(х2,хз) = Gr, <— GCT, —> Gr, = (xi,x2)
t \ t / t
(хз) = Ga3 <— GP={1} —► Gff, = (x,)
(x3,^4> = Gl3 <— GCT4 —> Gr4 = (xa,x\)
II
(*4>
Fig. 12.9 A 4-gon (square) of groups
(7) Extending an Action with Strict Fundamental Domain. Let G(Q) be the
simple complex of groups associated to a strata preserving action of a group G on
a stratified space X with strict fundamental domain Y. Let φ : G(Q) —> G be the
associated simple morphism. Assume that X and its strata are arcwise connected and
that Υ is simply connected. Let ρ : X —>- X be a universal covering of X. Let G be
the set pairs (ft, g), where g e G and ft : X —> X is a homeomorphism such that
p{h.x) — g.p(x) for all χ e X. The operation (gi, fti).(g2, ^2) = C?ig2, ^1^2) defines
a group structure on G. This group acts on X by (g, ft).ic = ft(jc).
Standard covering space theory (see e.g. [Mass91]) tells us that if (g, ft), (g, ft') 6
G then ft-'ft' acts as a covering translation, and the map π : G —> G that sends (ft, g)
to g e G is a surjective homomorphism whose kernel is naturally isomorphic to the
fundamental group of X:
1 -► 7Ti(X) -^ G Λ G -^ 1.
49The group (y,x | yxy~' = x2) has several interesting manifestations, for example as the
subgroup of SL(2, Z[j]) generated by у — I . .1 and л = I „ . 1, which acts as
affine transformations on R by x(t) = ί + 1 and y(t) — 2t.

The Basic Construction 381
If we use π to define an action of G on X, then ρ becomes a G-equivariant map. As
Υ is simply connected, there is a continuous map s : Υ —>- X such that/? о s is the
identity map on Y.
Let (Χ, {Χσ}) be the given stratification on X. We stratify X by choosing as strata
the connected components of the inverse images of the strata of X. Note that the
action of G is obviously strata-preserving and s(Y) is a strict fundamental domain
for this action. Moreover, each stratum of X is contained in some translate of s(Y),
which is assumed to be simply connected, and hence ρ maps each stratum of X
homeomorphically onto a stratum of X.
For each σ e Q, consider the isotropy subgroup Ga of s(Ya). The restriction of л
to όσ is an isomorphism onto Ga. These isomorphisms on local groups give a simple
isomorphism (in the sense of (12.11)) from G(Q) to the simple complex of groups
associated to the action of G on X with strict fundamental domain s(Y). And the
monomorphisms φσ : Ga —» G obtained by composing the isomorphisms Ga S Ga
with the natural inclusions όσ ^ G define a simple morphism φ : G(X) —» G that
is injective on the local groups, and φ — πφ.
The Basic Construction
We shall now describe the Basic Construction which we talked about in the
introduction. This construction allows one to reconstruct a group action with strict
fundamental domain from the associated simple complex of groups. It also shows
that a simple complex of groups G( Q) is strictly developable if and only if the natural
simple morphism ι : G(Q) —» G(Q) is injective on the local groups.
Following the Basic Construction we shall give a number of explicit examples to
try to give a sense of how one uses it, then we shall gather some of its basic properties.
Further examples will be given in subsequent sections.
12.18 Theorem (The Basic Construction). Let (Υ, {Υ"}σ^ο) be a stratified space
indexed by the poset Q. (The three types of stratified spaces were described in
(12.1).) Let G(Q) = (Ga, ψΤσ) be a simple complex of groups over Q. Let G be a
group and let φ : G(Q) —> G be a simple morphism that is injective on the local
groups. Then:
(1) Canonically associated to φ there is a poset D(Q, φ), called the development
of Q with respect to φ, and a stratified space D(Y, φ) over D(Q, φ) called the
development of Υ with respect to φ.
(2) Υ is contained in D(Y, φ) and there is a strata preserving action ofG on D(Y, φ)
with strict fundamental domain Y.
(3) The simple complex of groups associated to this action is canonically
isomorphic to G(Q), and the simple morphism G(Q) —■► G associated to the action is
φ. Therefore G(Q) is developable.
(4) IfY=\Q\thenD(?,<p)=\D{.Q,<p)\.

382 Chapter 11.12 Simple Complexes of Groups
Proof. As the simple morphism φ : G(Q) —»■ G is injective on the local groups, we
can identify each Ga with its image φσ(ΰσ) in G; if τ < σ, then the subgroup Ga is
contained in GT.
Define V — D(Q,<p) :— \\Q G/Ga, the set of pairs (gGa,a) with σ e Q
and gGa e G/Ga, a coset of Ga с G. We define a partial ordering on V by:
(gGr, τ) < (g'Ga, σ) if and only if τ < σ and g~lg' e Gr. We have a natural
action of G on D(Q, φ) defined, for g' e G, by g'.(gGa, σ) — (g'gGa, σ). The map
σ м> (GCT, σ) identifies Q with a subposet of D(<2, φ) which is a strict fundamental
domain for this action.
Let D(Y, φ) be the set which is the quotient of G χ У by the equivalence relation:
[(g,y) ~ (#'>/) ч*=^ У = У andg"'g' e GCTty)], where Χσ&,) is the smallest
stratum containing y. We write [g, y] to denote the equivalence class of (g, y). The
set G χ Υ is a disjoint union JJG Υ and as such inherits the (topological, simplicial or
polyhedral) character of Y. We endow D(Y, φ) with the quotient structure. D(Y, φ) is
then a stratified space over V where the stratum indexed by (gG„, σ) is Uyey [£> /J-
The group G acts by strata preserving automorphisms according to the rule g'.[g,y] —
[g'g,y]. And if we identify Υ with the image of {1} χ Υ in D(Y, φ) (where 1 is the unit
element of G), then У is a strict fundamental domain for the action, and (modulo the
natural identifications of the Ga with subgroups of G, and of Υ with the fundamental
domain) the associated complex of groups is G(Q).
To prove (4), we note that the fc-simplices in \D(Q, φ)\ that project to the ^-simplex
of \Q\ with vertices σ0 < · · · < ak have vertices (gGao, σ0) < · · · < (gGat, ak),
where g e G is well-defined modulo Gat. In particular these simplices are in bijection
with the elements of G/Gak. D
12.19 Examples of the Basic Construction
(1) Let Q — {σ, τ}, with τ < σ. Let У be the 2-dimensional disc with two strata,
the disc itself Υσ and its boundary YT. Let G(Q) be the complex of groups over Q
with Ga — {1} and Gr = Z2. There is only one simple morphism φ : G(X) —> Z2
that is injective on the local groups. The stratified space D(Y, φ) is the 2-sphere with
three strata, namely the two closed hemispheres and the equator. The generator of
Z2 fixes the equator and exchanges the hemispheres. G(Q) S Z2-
(2) Let Q — {σ, το, τι}, with τ0 < σ and τι < σ. Let Υ be the segment [0, 1]
with three strata Υσ - [0, 1], Υτ° - {0}, and Υτ> = {1}. Consider the complex of
groups G(Q) with Ga — {1}, Gro = Z3, and Gr, = Z2. Let φ : G(t) -> Gbea
simple morphism that is injective on the local groups. Then D(Y, φ) is the barycentric
subdivision of a graph with vertices of valence 2 and 3; the vertices of valence 3 are
indexed by pairs (yGTo, το) (notation as in the proof of (12.18)); the vertices of
valence 2 are indexed by pairs (yGTl, τι); the edges are indexed by pairs (yGa, cr).
For instance, suppose that G S Ζβ is generated by γ and define φ by sending
a fixed generator of Z3 (resp. Z2) to γ2 (resp. y3). Then D(Y, φ) is the barycentric
subdivision of the graph obtained by taking the simplicial join of the discrete sets
{^o,o> Το,ι} and {τι.ο> ri,b ^1,2}· The generator γ acts on D(Y, φ) by exchanging the
vertices το,ο and τ0,ι and cyclically permuting {τι0, Τι,ι, τ^}·

The Basic Construction 383
As another example, let G — (s,r \ s2 = 1, r3 = 1) = D3 and define φ :
G(Q) —» G by sending a generator of Z3 to r and the generator of Z2 to s. Then
D(Y, φ) is the same complex as above, but the action is different: r fixes the vertices
Το,ο and το, 1 and cyclically permutes the others, while s exchanges Το,ο and To, 1, fixes
one of the Ti,,, and exchanges the other two.
The group G(Q) is the free product Z2 * Z3. The space D(Y,) is the barycentric
subdivision of a regular tree (cf. 12.20 (4)) with two kinds of vertices, those of valence
2 adjacent to those of valence 3.
(3)Let Q = {ρ, σι, σι, σ3}, with σ, < ρ fori = 1, 2, 3. Let У be a pair of pants,
i.e. the complement in a 2-sphere of the interior of three disks whose closures are
disjoint. We stratify Υ over Q, with Υ» = Υ and each Y"' a boundary circle. Let G(Q)
be the simple complex of groups with Gp — {1} and GC[ = Z2. In this case G(Q)
is a free product Z2 * Z2 * Z2, and D(Y,) is an infinite surface stratified by pairs of
pants and their boundary curves.
Let G — Z2 χ Z2, with generators a and b. Let φ : G{Q) —> Z2 χ Ζ2 be the simple
morphism that sends the generators of Gai, GG2, and GCT3 to a, b, and ab respectively.
The geometric realization of D(Q, φ) is the barycentric subdivision of the 1-skeleton
of a regular tetrahedron. The non-trivial elements of G act as rotations around axes
joining the midpoints of opposite edges. D(Y, φ) is homeomorphic to the boundary
of a tubular neighbourhood of the 1-skeleton of the regular tetrahedron in K3. Thus
D(Y, φ) is a closed surface of genus 3, stratified as the union of four pairs of pants
(see figure 12.10).
Let φ' : G(Q) —> G be the simple morphism that sends the generators of GCT| and
Gai to a, and sends the generator of GCT, to b. Then Υφ, is also a surface of genus 3,
but the decomposition in pairs of pants is different — it is shown in figure 12.2. The
affine realizations of D(Q, φ) and D(Q, φ') are not homeomorphic.
•J,V
Fig. 12.10 The surface D(Y, φ) in 12.19(3) and the affine realization of D(Q, φ)

384 Chapter II. 12 Simple Complexes of Groups
(4) The examples of Mike Davis described in 5.22 and 5.23 are developments
D(F, φ) and D(M, φ) of the spaces F and Μ with the stratifications described in
12.4(3) and 12.2(3) respectively. To see this, take Q to be the poset associated to
these stratifications, let G(Q) be the complex of groups that associates to σ =
(jb ■ ■ ■, sk) e Q the group Ga = Π*=ι GSi defined in 5.23, and let φ : G(Q) -> G
be the morphism defined by the inclusions φσ of the groups Ga into the group
G = YlszsGs.
(5) Constructing Tesselations of H2. Let Ρ be a k-gon in the hyperbolic plane
with vertices τ\,..., τκ (in order). Assume that the angle at the vertex τ,· is of the
form ττ/pi, where/?, is an integer > 2 (such polygons exist if к > 4 and in general
if J2i 1/Pi < к — 2). Let σ, be the side of Ρ joining the vertex τ,_ι to the vertex τ,
(the indices are taken modulo k), and let i, be the reflection of H2 fixing σ,. Let G be
the subgroup of Isom(H2) generated by ji ,..., j*. We claim that G acts on H2 with
strict fundamental domain Ρ (and thus the images of Ρ by the elements of G form a
tesselation of H2 by polygons congruent to P). This is a particular case of a classical
theorem of Poincare.
To prove our assertion, we consider a complex of groups G(Qk) over the poset
Qk = {p, cr,, τ,} of faces of Ρ where ρ corresponds to P. Define Gp = 1, GCT;. to
be the subgroup of G generated by s,, and GT. the subgroup generated by i,_i and
s;. Let φ : G(Q) —> G be the morphism defined by the inclusions into G. The
development D(P, φ) is a connected hyperbolic polyhedron. There is a natural G-
equivariant projection/ : D(P, φ) —> И2 mapping [g, y] to g.y and we claim that
this projection is a local isometry. This is trivial except at the vertices τ, = [1, τ,] of
Ρ С D(P, φ), where it follows from our hypothesis that the angle of Ρ at τ,- is of the
form π I pi. Indeed this hypothesis implies that Gt, is the dihedral group of order 2k
which acts on И2 with a strict fundamental domain which is the sector bounded by
the rays issuing from τ, and containing the sides σ,_ι and σ,. As D(P, φ) is complete,
the map/ is a covering of И2, hence an isometry (cf. 1.3.25), which means that we
can identify D(P, φ) with И2. ___
The same construction using the morphism ι : G(Qk) —> G{Q0^would lead to
the same conclusion, and therefore G is isomorphic to the group G(Qk) obtained by
amalgamating the groups Gt; and GGl as in (12.12) (see also 12.22).
The same argument works when the hyperbolic plane is replaced by S2 or E2.
Indeed we could replace Ρ by any convex polyhedral и-cell Ρ in MnK such that the
dihedral angle between each pair of (n — l)-dimensional faces is of the form π/m
where m is an integer > 2. The reflections fixing the (n — l)-faces of Ρ generate a
subgroup G of Isom(M£) and the images of Ρ by the elements of G form a tesselation
of M''. The argument is similar to the preceding one and uses induction on the
dimension η (for details, see de la Harpe [Har91, p.221-238]).
12.20 Proposition (Properties of the Basic Construction). We maintain the notation
established in (12.18). Thus we have a simple complex of groups G(Q), a stratified
space (Υ, {Υσ}σ<=ο), and we consider simple morphisms φ : G(Q) —> G that are
injective on the local groups. (G is an arbitrary group.)

The Basic Construction 385
(1) (Uniqueness) Let X be a stratified space over Q-IfG acts by strata preserving
morphisms on X with strict fundamental domain Y, and the associated simple
complex of groups is simply isomorphic to G(Q), then the identity map on Υ
extends uniquely to a strata preserving G-equivariant isomorphism D(Y, φ) —>
X.
(2) Let Go с G be the group generated by the subgroups (pa(Ga). If Υ is connected,
then the set of connected components ofD(Y, φ) is in bijection with the set of
cosets G/Gq. In particular D(Y, φ) is connected if and only if Go = G.
(3) (Functoriality) Let π : G —» G' be a surjective homomorphism of groups.
Let φ' : G(Q) —» G' be the simple morphism defined by φ'σ := πφσ. If φ' is
injective on the local groups, then the identity map on Υ extends uniquely to
a strata preserving G-equivariant map ρ : D(Y, φ) —> D(Y, φ'). The map ρ is
a covering projection, and the kernel of π acts freely and transitively on the
fibres of p.
(4) Suppose that Υ is connected and simply connected and that each stratum
of Υ is arcwise connected. Then the development D(Y, t) of Υ with respect
to the canonical morphism ι : G(Q) —» G(Q) is simply connected. If the
canonical homomorphism φ : G(Q) —> G associated to φ is surjective, then
Tt\(D(Y, φ)) = ker φ. (In the topological case one has to assume that D(Y, φ)
is locally simply connected.)
Proof. We use the notation established in the proof of (12.18).
(1) The map [g,y] —> g.y is a stratum preserving G-equivariant isomorphism
from D(Y, φ) to X.
(2) follows from (12.10).
(3) Let ρ : D(Y, φ) -> D(Y, φ') be the map sending χ = [g, y] to x1 = [71(g), y];
it is G-equivariant and maps the stratum indexed by (g^CT(GCT), σ) onto the stratum
indexed by (n(g)<p'a(Ga), σ). The inverse image of x1 is the set of points of the form
к .χ, where к e ker π. Let Ya(y) be the smallest stratum of Υ containing у and let Gy=
(Pa(y){GG(y)). If st(y) is the open star of у in Y, the open star st(x) of χ in D(Y, φ) is
gGy.st(y) (see 12.8(5)). Similarly star(x!) = n(g)G'y.st(y), where G'y = <ω(σσω).
Thus ρ maps st(x) homeomorphically onto st(x!), and p~' (st(xf)) is the disjoint union
of the open sets k.st(x), where к e ker it. Therefore ρ is a Galois covering with Galois
group ker it.
(4) We use the construction of 12.17(7) applied to the action of G on X = D(Y, φ).
Thus we get a group G which is an extension of G by the fundamental group of
D(Y, φ), a stratification of the universal covering D(Y, φ) of D(Y, φ), and a strata
preserving action of G on D(Y, φ) with a strict fundamental domain one can identify to
Υ so that the associated complex of groups is canonically isomorphic to G(Q). Let φ :
G(Q) -» G be the associated morphism and let φ : G(Q) -> G be the homomorphism
associated to φ. By (1), there is a G-equivariant isomorphism D(Y, φ) -» D(Y, φ),
and by (3), a G(Q)-equivariant covering map D(Y,i) —> D(Y, φ); as D(Y, φ) is
simply connected, this is an isomorphism. Hence D(Y, ή is simply connected and

386 Chapter 11.12 Simple Complexes of Groups
G{Q) -> G is an isomorphism. The last claim follows from (3) applied to the
homomorphism φ. D
12.21 Corollary. Let G(Q) be a simple complex of groups over a poset Q. Let
Υ = \Q\. If the canonical simple morphism ι : G(Q) —> G(Q) is injective on the
local groups and if Υ is simply connected, then the space D(Y, ι) given by the Basic
Construction is also simply connected. (And by 12.18(5) it is the realization of the
poset D(Q, φ).)
The proposition can sometimes be used to find presentations of groups.
12.22 Corollary. Let G be a group acting on a simply connected stratified space
X with a strict fundamental domain-Y. Then G is the direct limit (amalgam) of the
isotropy subgroups of the various strata ofY.
Proof. LetG(Q)bethecomplexofgroupsassociatedtothisaction,let#> : G(Q) —» G
be the corresponding morphism and let φ : G(Q) —> G be the associated
homomorphism. According to (1), the space X is G-equivariantly isomorphic to D(Y, φ), by
(2) φ is surjective, and by (3) the φ-equivariant map D(Y, i) —> D(Y, φ) is a covering.
As D(Y, φ) is simply connected, this last map is an isomorphism, and so too is φ. D
12.23 Examples
(1) We revisit the group action and triangle of groups described in 12.17(3),
maintaining the same notation. Thus we have a group G generated by reflections
si,S2,s$ acting on E2. As E2 is simply connected, we can appeal to 12.22 (or to
12.19(5)) to see that the group G is isomorphic to G(Q), which has presentation
(sus2, S3 I s2 = s\ = s] = (sis2)6 = (s2s3Y = (ίιί3)2 = 1>·
Let φ : G(t) —> D^ be the simple morphism mapping s\, s2 to the corresponding
generators of the dihedral group D^ and s$ to s2(S[S2)2. If Υ is the triangle chosen
as strict fundamental domain for the action, then D(Y, φ) is the 2-torus obtained by
identifying opposite sides of the parallelogram formed by two adjacent triangles of
the tesselation. The group De acts on this 2-torus with a strict fundamental domain
isometric to Y. In general, given any surjective homomorphism φ : G(Q) = G —> Η
that is injective on the local groups, if ker φ is non trivial, then D(Y, φ) is isometric to
a 2-torus obtained by multiplying the distance function on the one described above
by a positive integer.
(2) We can apply corollary 12.22 to check that the group of permutations Σ(η)
(in the notations of example 12.9(3)) has the following presentation for η > 4:
it is generated by the elements s\,..., i„_i, subject to the relations sf = 1 for
i < и — 1, (ЗД+ι)3 = 1 for г < и — 1 and (ί,ί))2 = 1 fovj Φ i ± 1.
Let Τ be the triangulation of S"~2 described in 12.9(3), and let X be the 2-
dimensional subcomplex of the barycentric subdivision formed by the union of the
simplices whose vertices are barycentres of those simplices of Τ that have dimension

Local Development and Curvature 387
η — 2, η — 3 or η — 4. This complex X is the union of the cells of dimension < 2 in
the dual subdivision of T, hence it is simply connected. The action of Σ(η) on S"~2
preserves X and has a strict fundamental domain Χ Π Δ. The isotropy subgroups of
faces of dimension η — 3 (resp. η — 4) of Δ are the cyclic subgroups generated by
the Si (resp. the dihedral groups generated by pairs (i,, Sj), i φ j). By applying 12.22
to the action of Σ(«) on X we get the desired presentation.
Local Development and Curvature
The results of the previous section show that one can construct interesting group
actions from simple complexes of groups if there exist simple morphisms that are
injective on the local groups. However, we have not yet developed an effective
criterion that is sufficient to show that the local groups inject; that is the purpose of
this section. We shall explain how one can use the local structure of a simple complex
of groups to understand what the closed star of a point in the development would look
like if it were to exist - this is called local development. It is a remarkable fact (akin
to the Cartan-Hadamard Theorem) that if at every point in the complex the putative
star is non-positively curved (and the local groups act by isometries), and if \Q\ is
simply connected, then the complex is strictly developable. This will be proved in
Part III.
12.24 Construction of the Local Development. We shall describe an explicit model
for the closed star St(cr) of the vertex σ = [1, σ] in the affine realization of the
posets D(Q, φ) constructed in the Basic Construction (12.18). This description is
independent of φ. The local group Ga acts naturally on St(cr) with a strict fundamental
domain naturally isomorphic to St(a).
We define the complex St(a) to be the affine realization of the subposet of D( Q, φ)
whose elements are pairs that are either of the form (gisap(Gp), p), where ρ > σ
and g e Ga, or else are of the form (Ga, τ), where τ < σ (in such pairs, the first
member is considered as a coset in Ga). The partial ordering is defined by:
for σ < p,p' : (girap(Gp), p) < (g'irap>(Gp>), p') if ρ < ρ' and g~lg' e irap{Gp),
fort, τ'<σ: (Ga, τ) < (Ga, τ') if τ < τ',
for τ < σ < ρ : (Ga, τ) < (g^ap(Gp), p).
Using the notations of 12.3, this poset can also be described as follows.
Let G(Lka) be the restriction of the simple complex of groups G(Q) to the
subposet Lka(Q) = {p e Q : ρ > σ}. One has a canonical simple morphism
ψσ : G(Lka) —> Ga given by the injective homomorphisms (ψσ)ρ := ψσρ : Gp —>
Ga. The upper link of σ = (Ga, σ) in the poset D(Q, φ) can be defined as
Lk5 :=DiLka,fa),

388 Chapter II. 12 Simple Complexes of Groups
and St(a) is the affine realization of the poset
Q° *D{Lka,ira).
The map that associates to each ρ > σ the pair ρ = (isap(Gp), p), and to each
τ < σ the pair τ = (Γσ, τ), gives an identification of St(a) with a subspace of St(cr).
There is also a natural projection ρσ : St(a) —» St(a), induced by the poset map
sending each pair to its second coordinate.
The group Ga acts in an obvious way on St{a). Moreover it is clear that the
preceding construction does not involve the homomorphism φ. Thus we have the
following local developability property in complete generality.
12.25 Proposition. Let G(Q) = (GCT, ψτσ) be a complex of groups over a poset Q.
Canonically associated to each σ e Q there is a poset on whose affine realization
St(a>) the group Ga acts with strict fundamental domain St(a). In this action, the
isotropy subgroup of the stratum indexed by each τ < σ is Ga, and the isotropy
subgroup of the stratum indexed by ρ > σ is isap(Gp).
St(cr) together with the action of Ga is called the local development of G{Q) at
the vertex σ.
Assume that the affine realization \Q\ of Q is endowed with the structure of
an MK-simplicial complex with finite shapes. Then for each σ e Q, the simplicial
complex St(a) is also an MK-simplicial complex, and the projectionρσ is an isometry
on each simplex.
12.26 Definition (Simple Complex of Groups of Curvature < к). Let к е №.. A
complex of groups G(Q) over an MK -simplicial complex | Q\ is said to be of curvature
< к if the induced metric on the open star st(a>) in the local development is of
curvature < к for each σ e Q.
We shall be most interested in the case к = 0 where, of course, we say "non-
positively curved" instead of curvature < 0.
12.27 Remarks
(1) If | Q | is one dimensional, and the length of each of its edges is one, say, then
any complex of groups over Q is of curvature < к for every к е №..
(2) Suppose that Q is the poset of cells of an MK -polyhedral complex К with
Shapes(AT) finite and suppose that there are no identifications on the boundaries of
cells in K. Then the affine realization \Q\ of Q is isomorphic to the barycentric
subdivision K' of K, and as such inherits the structure of an MK -simplicial complex.
And for each vertex τ of K, the geometric link Lk(r, K) inherits a spherical metric:
it is a (spherical) geometric realization of the subposet Lkt С Q.
To check that a simple complex of groups G(Q) over Q is of curvature < κ,
it is sufficient to check that for each vertex τ e K, the affine realization of Lki =
D(LkT, Vr)> witn its natural structure as a spherical complex, is CAT(l) (see 1.7.56
and 5.2).

Local Development and Curvature 389
The following theorem, first proved by Gersten-Stallings [St91] in the particular
case of triangles of groups (see also Gromov [Gro87, p. 127-8], Haefliger [Hae90 and
91], Corson [Cors92] and Spieler [Spi92]), will be proved later in a more general
framework (lll.G)-
12.28 Theorem. Let Qbe aposet whose affine realization \Q\ is simply connected,
and suppose that \Q\ is endowed with an MK-simplicial structure with finite shapes,
where к < 0. Let G{Q) be a simple complex of groups over Q which is of curvature
< к. Then G(Q) is strictly developable.
If Q is finite and each local group Ga is finite, this implies that G(Q) acts
cocompactly on the САТ(к) space \D(Q, i)\ and that any finite subgroup ofG(Q) is
conjugate to a subgroup of one of the local groups Ga.
12.29 The Local Development at the Corner of an n-Gon of Groups. We wish to
describe the local development at the vertices of polygons of groups. If one focuses
on the corner of a polygon of groups, one has the vertex group, the two incoming
edge groups, and the local group of the 2-cell. With this picture in mind, we consider
simple complexes of groups over the poset Qo = {ρ,σ\, σ2, τ} with ρ < σ\, ρ < σ2
and σι < τ, аг < τ. This is equivalent to considering a group G (playing the role
of Gt), with two subgroups G\ and Go (corresponding to σχ and σ2) and a subgroup
Η с d П G2 (namely Gp).
/ N.
/ .^G
σ, Η > G,
Fig. 12.11 A simple complex of groups at a corner of an n-gon
According to 12.24, the link Ьк{т) of the vertex corresponding to τ in the local
development is the barycentric subdivision of a bipartite graph Q whose set of vertices
is the disjoint union of G/G\ and G/G2, and whose set of edges is G/H; the edge
corresponding to gH joins the vertices gG\ and gG2. The group G acts on Q (by
left multiplication) with a strict fundamental domain which is an edge. This graph is
connected if and only if G is generated by G\ U G2. The girth girth{Q) of this graph
is by definition the minimal number of edges contained in a simple closed curve in
Q (in the case of a bipartite graph, the girth is even or infinity). For instance the girth
is 2 if and only if Η φ G\ Π G2, and it is infinity if and only if the subgroup of
G generated by G\ U G2 is the amalgamated product G\ *H G2. If G\ = (s\) and
G2 = (s2) have order 2 and G is the dihedral group D„ of order 2n generated by
{ii, s2}, then the girth is In. If G = G\ χ G2, the girth of Q is 4.
The affine realization of the poset <20 is the union of two triangles glued along
the side joining ρ to τ. If we metrize |Q0| so that it is the union of two triangles

390 Chapter 11.12 Simple Complexes of Groups
in M2 such that the total angle at the vertex τ is a, then the development at τ is of
curvature < к if and only if 2я > a girth(Q). (Stallings [St91] calls 2π/girth{Q)
the (group theoretic) angle of the и-gon of groups at the vertex τ.)
For instance, if Ρ is a regular и-gon in E2, the specific и-gon of groups considered
in 12.17(6) is non-positively curved if и > 4, because the girth of the link of the local
development at each vertex τ, is 4 (a circuit of length four is formed by the edges
labelled 1, jc,, jc,jc;+i, jc,jc,+ijc,~' = xf+l) and the angle at a vertex is > π/2. If we
retain the same underlying simple complex of groups but now regard Ρ as a regular
и-gon in the hyperbolic plane with vertex angles greater than π/2 (this is possible if
и > 5), then this complex of groups is of curvature < — 1.
More generally, we might consider a complex of groups over the poset Q„ of
faces of a regular Euclidean и-gon Ρ such that, in the local groups at the vertices of
P, the intersection of the two subgroups coming from the adjacent edges is equal
to the subgroup coming from the barycentre of P. Any such и-gon of groups is
non-positively curved if и > 4, hence is developable, in the light of (12.28).
We now turn to more explicit examples of non-positively curved spaces with
cocompact group actions.
12.30 Examples
(1) Hyperbolic Polyhedra. Let Ρ be a right-angled regular p-gon in H2 (such a
polygon exists if ρ > 4). We consider the following complex of groups G(Q) over
the poset Q of faces of P: the group associated to Ρ is trivial, the group associated
to each edge e, of Ρ is a cyclic group Z9 of order q (generated by an element j,·,
say) and at the vertex of Ρ adjacent to the edges e, and e,+ i, i e Ζ mod ρ, we have
the product Zq χ Zq generated by i, and j,-+i (the labelling of the generators defines
the inclusions of the edges groups). This complex of groups is developable, because
there is an obvious simple morphism of G(Q) to the product Т\1 ^ mapping i, to a
generator of the ith factor. The group Ypq := G(Q) is generated by the elements i,
subject to the relations if = landi,i,+) = ii+ii,fori e Ζ mod p. The link of the local
development at each vertex is the complete bipartite graph obtained by taking the
join of two copies of Z9; its girth is 4, therefore the M_i -complex lpA := D(P, ι) on
which Yp q acts properly and cocompactly is CAT(— 1). Marc Bourdon [Bou97a] has
determined the conformal dimension (in the sense of Pansu [Pan90]) of the boundary
at infinity of lpq. This boundary is homeomorphic to a Menger curve [Ben92].
(2) Graph Products of Groups. Let L be a simplicial graph with vertex set S and let
(Gs)sSs be a family of groups indexed by the set of vertices of L. The graph product Gi
of the family of groups Gs is the abstract group generated by the elements of the groups
Gs subject to the relations in Gs and the relations gsg, = g,gs if gs e Gs, g, e G,
and s and t are joined by an edge in L. Let G be the direct product Y[ssS Gs of the
groups Gs. There is a unique homomorphism φ : Gi —> G such that the natural
homomorphism of Gs into Gi composed with φ then the projection onto the factor
Gs is the identity.

Constructions Using Coxeter Groups 391
Following [Da98] and [Mei96], we construct a CAT(O) piecewise Euclidean
cubical complex X on which GL acts by isometries (preserving the cell structure)
with a strict fundamental domain.
Let К be the flag simplicial complex whose 1-skeleton is the graph L. let Q
be the poset of those subsets of S (the vertex set of K) which span a simplex of К
together with the empty set, ordered by reverse-inclusion, σ < τ if τ с σ. To each
σ = {si,... ,Sk] e Q, one associates the product Ga = f]*=1 GSi (define G0 = {1}),
and for τ с σ we denote by ψστ the natural inclusion Gt <^-> Ga. This defines on Q
a simple complex of groups G(Q). It is strictly developable thanks to the existence
of the homomorphism φ, and G(Q) is the graph product of the groups Gs. The affine
realization of the poset Q is the simplicial cone over the barycentric subdivision of
К (see 12.4(3)) which, in a natural way, can be considered as a cubical complex; let
F denote this cubical complex with its natural piecewise Euclidean metric.
Let X be the cubical complex D(F, ι), where t : G(Q) -> GL is the canonical
simple morphism. As F is simply connected, X is also simply connected. One can
check that the link of each vertex of X is a flag complex, hence X is CAT(O) by
Gromov's criterion (5.20) and the Cartan-Hadamard theorem.
(3) We should mention that Tits has constructed triangles of finite groups
([Tits86b], [Ron89, p.48-49]) whose simply connected developments provide
remarkable examples of Euclidean buildings which have the same type and are locally
isomorphic, but enjoy quite different properties.
We should also mention the striking work of Floyd and Parry [FP97]. They give
explicit calculations of the (rational) growth functions of the direct limits of certain
non-positively curved triangles of finite groups. They also exhibit examples of pairs
of such triangles such that all of the local groups are the same in each case, but the
direct limit of one triangle contains Z2 while the other does not (it is 5-hyperbolic
in the sense of (ΠΙ.Γ.2.1)). Remarkably, these two groups have the same growth
function.
Constructions Using Coxeter Groups
12.31 Definition. A Coxeter group is a group W with a specified finite generating set
S and defining relations of the form (st)m«, one for each pair s,t eS, where mss = 1
and if s ф t then mst is an integer > 2 or mst = oo (which this means no relation
between s and t is given).
The pair (W, S) is called a Coxeter system.
The matrix Μ = (ms,)SJSs describing the defining relations can be reconstructed
from a weighted graph, called the Coxeter graph. The nodes (vertices) of this graph
are labelled by the elements of S. The vertices labelled s and t are joined by an
unweighted edge if ms, = 3 and by an edge weighted (i.e. labelled) by mst if mst > 3
or mst = oo. (If s and t are not joined by an edge, this means that mst = 2, i.e. the
generators s and t commute.)

392 Chapter 11.12 Simple Complexes of Groups
Coxeter Groups
D2 = Z2 x Z2
D3 = 5(3)
Doo = Z2 • Z2
5(4)
Group of isometries of E2 preserving a tessellation
by equilateral triangles (see 12.17(3))
Fig. 12.12 Some Coxeter graphs and the corresponding Coxeter groups.
12.32 Basic Property of Coxeter Systems. Let (W, S) be a Coxeter system. Given a
subset Τ ofS, let Wj· be the Coxeter group with generating set Τ and relations (tf)"1"'
for t,feT (where the exponents mn· are as in the defining relations for (W, S)). Then
the natural homomorphism Wj —> W induced by the inclusion Τ <^-> Sis injective.
For a proof, see Bourbaki [B0118I].
Note that in particular this result says that each element s of S generates a cyclic
subgroup of order two in W, that two distinct elements s, s' e S with mss> = 00
generate an infinite dihedral group, and if mss> is the integer m > 2, then s and
s' generate a dihedral group of order 2m (denoted Dm). The subgroups of order 2
generated by different elements of S are all distinct.
12.33 Complexes of Groups Associated to Coxeter Systems. In our language,
12.32 can be reformulated as follows. Let S be the poset of subsets of S ordered by
reverse-inclusion, that is T\ < T2 if T2 C. Τι. We define a simple complex of groups
W(S) over the poset S: to Τ с S we associate the group WT; if Γ2 ί Г| с S, then
■ψτ,τ2 is tne inclusion WTl <^-> WTi induced by T\ ^-» Γ2. For Τ = 0 the associated
group W0 is the trivial group, and Ws = W. Note that W is the direct limit of the
groups WT, so we have a canonical morphism ι : W(S) —> W (see 12.12).
The affine realization of S is a simplicial subdivision of an и-cube, where η is the
cardinality of S (if Τ has cardinality k, the stratum indexed by Γ is an (и—fc)-cube, (see
12.4(2))). Let Τ be a subposet of S and let W(T) be the simple complex of groups
obtained from W(S) by restriction to T. The restriction to W(T) of the morphism ι
(which is injective on the local groups) will also be denoted ι : W(T) —> W. Let \T\
be the affine realization of Τ with its natural stratification and let Σ(Ψ, Τ) be the
stratified space D(\T\, i) given by the basic construction (12.18). The group W acts
on Σ(Ψ, Τ) with strict fundamental domain |T|. We consider various special cases
of this construction.
Coxeter Graphs
Si S2
Si
·-
·
S2
m
S2
OO
·
Si S2
• · ·
Si S2 S3
6
• · ·
Si S2 S3

Constructions Using Coxeter Groups 393
12.34 Examples
(1) If Τ is the poset of proper subsets of S, then Σ(Ψ, Τ) is sometimes called
the Coxeter complex of (W, S) (cf. [Bro88]). If S has cardinality n, this complex has
dimension η — 1. If W is finite, then Σ(Ψ, Τ) is an (n — l)-sphere tesselated by the
translates of the simplex \T\.
(2) M. Davis [Da98] introduced a different (inequivalent) notion of Coxeter
complex. He did so by considering the subposet & of S consisting of those subsets Τ С S
such that Wj- is a finite group. The geometric realization |<S^| is a cubical complex,
and the stratified space Σ(Ψ, S?) given by the Basic Construction is a simply
connected cubical complex on which W acts properly. Indeed the geometrical realization
of & is always contractible, because it is a cone (the cone point corresponds to the
empty subset).
To obtain an explicit example, we consider a polygon Ρ in the hyperbolic plane
H2 such that the angle between each consecutive pair of sides s and s' is of the form
π/'mss>, where ms? > 2 is an integer. Let S be the set of sides of the polygon and define
mss = 1 and mss> = oo if the sides are distinct and not adjacent. Let W be the Coxeter
group determined by the Coxeter matrix Μ = {mss>). In this case |<S^| is isomorphic
to the barycentric subdivision of Ρ and Σ(Ψ, S?) is isomorphic to the barycentric
subdivision of a tesselation of H2 by polygons congruent to Ρ (see 12.19(5)).
More generally, if Ρ has some vertices at infinity (and we define msj = oo for the
corresponding consecutive sides), then & is the poset associated to the stratification
of Ρ in which the strata are P, the sides of P, and the vertices that are not at infinity.
In this case |<S^ | is a subcomplex of the barycentric subdivision of P. The stratified
space D(P, i) given by the basic construction is isometric to a tesselation of H2 by
polygons congruent to P, while Σ(Ψ, S?) is isomorphic to a proper subspace of H2.
For instance, if all the vertices of Ρ are at infinity, then Σ(Ψ, &) is the barycentric
subdivision of a regular tree; the number of edges issuing from a vertex is equal to
the number of sides of P.
If (W, S) is a right angled Coxeter system (meaning that ms, = 2 or oo for all
s φ t), then Σ(Ψ, S?) is the universal covering of the complex constructed in 5.22.
In his thesis, Gabor Moussong [Mou88] proved that, for any Coxeter system
(W, S), one can metrize the cells of |<S^ | as Euclidean polyhedra so that in the induced
piecewise Euclidean polyhedral structure, Σ(Ψ, S?) is CAT(O).
(3) Let L be a finite simplicial graph with vertex set S and suppose that each edge
is assigned an integer mss> > 2 (where s and s' are the endpoints of the edge). There
is a natural way to associate to such a labelled graph a two-dimensional polyhedron
AT and a group W acting simply transitively on the set of vertices of K, so that the link
of each vertex of К is isomorphic to Ц in this isomorphism the edge of L joining s
to s' represents the corner of a polygonal face with 2mSj sides that is incident to the
vertex (see Benakli [Ben91 a] and Haglund [Hag91 ]). To construct К and W, consider
Μ = (mss>) to be a Coxeter matrix with mss> = oo if s and s/ are not joined by an edge
in L, and let W be the associated Coxeter group. Let Tc^be those subsets with
cardinality < 2. The geometric realization of \T\ is the cone over the barycentric
subdivision of L (see fig. 12.13, where L is the union of two edges having a vertex

394 Chapter П. 12 Simple Complexes of Groups
in common). The first barycentric subdivision of К will be the complex E(W, T). A
strict fundamental domain for the action of W on E(W, Τ) is the star of the vertex
corresponding to the empty set.
(4) Haglund's Construction of Gromov Polyhedra. Recall that a simplicial
graph is said to be complete if every pair of vertices is joined by an edge. In this
paragraph we descibe how to construct simply connected polyhedra in dimension
two where the faces are fc-gons and the link of each vertex is a complete graph with I
vertices. The existence of such polyhedra was asserted by Gromov [Gro83]. In fact,
as shown by Haglund [Hag91] and independently by Ballmann-Brin [BaBr94], there
is an uncountable number of isomorphisms classes of such polyhedra when к > 6
and I > 4. When 1 = 3, such a polyhedron is a simply connected surface tesselated
by £-gons: if к < 5 this is a sphere, if к = 6 the Euclidean plane, and if к > 6 the
hyperbolic plane.
The construction that we shall describe gives a polyhedron Xki with an action of
a group W which is transitive on triples (vertex, edge, face), where, of course, the
vertex is in the edge, and the edge is in the face. A strict fundamental domain for this
action will be a 2-simplex Τ of the barycentric subdivision of Xkj.
We start from the Coxeter system (W, S) withS = {ίο. si,..., si^i}andmSoSl = k,
mS(Sl+l = 3 for i = 1,..., I — 2. The other non-diagonal entries are all equal to 2. In
other words, the Coxeter graph is:
к
• · · · ·
SO Si S2 S(-i
Let S be the poset of subsets of S ordered by reverse-inclusion. Let Τ be the sub-
poset of S consisting of the proper subsets of S containing ρ = {s$,..., ί;_ ι}. The
geometric realization of Τ is the barycentric subdivision of a triangle Τ whose three
vertices το, Τι and %i correspond to the subsets of S of cardinality (/ — 1) not
containing respectively sq, si and S2- The barycentre ρ of Τ corresponds to the intersection
of these subsets and the barycentre σ, of the side joining Г/, r^. corresponds to the
intersection of the subsets Tj and r^ (where {i,j, к} = {0, 1, 2}). (See figure 12.14
where each group is represented by the corresponding Coxeter graph).
The second barycentric subdivision of the polyhedron Xkj will be Σ(Ψ, Τ), on
which we have the usual action of W with Τ as strict fundamental domain. The
vertices To, τι, Xj. of Τ are, respectively, a vertex of X^i, a barycentre of a side of X^;,
and a barycentre of a face ofX^j. These data define Xk j uniquely.
Let us check that the faces of X^i are fc-gons. The isotropy subgroup of Тг is
WTl = И^^0^|,1з...л_,), which is the direct product of a dihedral group of order 2k
(generated by ίο and s\) and the subgroup generated by the i, with i > 2. This second
factor acts trivially on Τ and the orbit of Τ under the dihedral group is a union of 2k
triangles forming the barycentric subdivision of a k-gon P.
To check that the link in X^ j of the vertex To is the complete graph on Ζ vertices, we
note that the isotropy subgroup of To is WTo = W^t i(_,), which can be represented
as the group of permutations of the set {1,..., /}, where i, acts as the transposition
(г, i + 1) (see 12.9(3) and 12.23(2)). This action extends to a simplicial action on the

Constructions Using Coxeter Groups 395
Fig. 12.13 The fundamental domain \T\ from 12.34(3) in the case S = {s], s2, S3}, mS]S2 = 3,
'%;A, = 2. "V3 = °0.
• · · <- · · -> · · ·
s0 Si S3 ' v S0 S3 ' v S0 S2 S3
Fig. 12.14 The triangle of groups W(T) for I = 4.
complete graph K(l) with vertices 1,... ,1. A fundamental domain for this action is
the edge of the barycentric subdivision K{1)' joining the vertex 1 to the barycentre
of the simplex joining 1 to 2. The natural action of WTo on the cone over the second
barycentric subdivision of K(l) has the cone over this edge as strict fundamental
domain, and the associated complex of groups is just the restriction of W{x) to the
star of the vertex r0 in T. We leave to the reader to check that the stars of the other
vertices of the fundamental domain Τ are isomorphic to the stars of the vertices in

396 Chapter П. 12 Simple Complexes of Groups
the second barycentric subdivision of any polyhedron satisfying the required local
conditions.
It is natural to identify Τ with a Euclidean triangle whose angles at the vertices
το, τι, x2 are, respectively, ^я, jn, ^π. In this way the 2-cells of Xkj become
regular Euclidean fc-gons. If к > 6, then the Euclidean polyhedron Xkj is CAT(O). If
к > 6, then we can metrize the 2-cells of К as regular hyperbolic fc-gons with vertex
angles 7Г/3. The resulting piecewise hyperbolic structure on Xkj is CAT(— 1).
Nadia Benakli [Ben92] has shown that, if к is even and at least 8, the boundary at
infinity of Xkj is homeomorphic to the Sierpinski curve if I = 4, and to the Menger
curve if I > 4.
(5) For/ = 4, Haglund has constructed a twisted variant Mkj of the above complex
Xkj. This arises from a triangle of groups obtained from the one considered above
by replacing the local group at the vertex τ2 by the dihedral group Dk generated by
the elements ίο and si, and then mapping i3 to the central element Сад)* € At·
The local developments for this triangle of groups are the same as in the previous
one, hence if к > 6 and we metrize Г as a Euclidean triangle as above, the triangle of
groups is non-positively curved, therefore strictly developable by 12.28. Its simply
connected development is a complex Mk,4 satisfying the same local properties as Xkj.
The direct limit of the triangle of groups is the group G generated by the elements
so, · · ·, S3 subject to the relations (s,·)2 = (sis2)3 = (s2s3)3 = Сад)2* = С*1^з)2 =
(даг)2 = Свд)*4 = 1- In fact Haglund showed directly that this triangle of groups
is strictly developable for к > 4. For к = 3 it is not developable.
To see that the complexes Xkj and Mkj are not isomorphic, let us fix a polygonal
2-cell Ρ in one of these complexes. Let e\,..., e* be the sides of Ρ in order. Let P\
be a 2-cell adjacent to Ρ along the edge e\ (there are two such cells). Let P2 be the
2-cell adjacent to Pi and to Ρ along e2. By induction we define P, to be the 2-cell
adjacent to P,_i and to Ρ along e,. We claim that the polygon Pk+\ adjacent to Pk
and to Ρ along e\ is equal to Pi in Xkj and is different from Pi in Mkj.
To see this, we fix our attention on the 2-cell Ρ which contains the fundamental
domain T. It is a k- gon which is the union of the 2fc triangles (J*ro r1. (Γ U ί0 · Γ), where
r = s\s0 acts as a rotation of angle 2n/k on P. Let e\ be the side of Ρ containing the
vertices το and T] of T. The к sides of Ρ are (in order) e\, e2 = r.e\,..., e* = r*-1 .e\.
The element s\ fixes To and exchanges e\ and e2. The element S3 fixes e\ and e2
and permutes the two other edges issuing from To. The element s2 fixes e\ and
maps e2 to an edge s2.e2 φ e\, hence Pi := s2.P is adjacent to Ρ along e\. As ίο
preserves P, reverses the orientation of e\, and commutes with s2, we have ίο-Pi = Pi
and hence the two other edges of Pi adjacent to e\ are s2.e2 and (soS2).e2. The
face P2 := (ί]ίο)·Ρι = r.P\ is adjacent to Ρ along e2 and to Pi along the side
(rs0s2).e2 = s2.e2. It follows that P,- := r'"1 .Pi is adjacent to Ρ along e, and to P,_i
for i < k. The face P^+i = r*Pi is adjacent to Ρ along e\ and to Pk- In the complex
Xk,A this face is equal to Pi, while in Mk,A it is equal to (s3s2).P = s^.P2 φ Ρ\.

Part III. Aspects of the Geometry of Group Actions
The unifying theme in this part of the book is the analysis of group actions, particularly
actions by isometries on CAT(O) spaces. The material presented here is of a more
specialized nature than that in Parts I and II and the reader may find the style a little
less pedestrian.
Part III is divided into four lengthy chapters, the contents of which were explained
briefly in the Introduction. Each chapter begins with an overview. Chapters III.C
and lll.G are independent of Chapters III.H and ΙΙΙ.Γ. The main arguments in the
chapter on groupoids, lll.G, rely only on material from Part II, but the reader who
digests the chapter on complexes of groups first, III.C, will have access to a greater
range of examples in lll.G and may find the ideas more transparent. Likewise, an
understanding of the main points in the chapter on hyperbolic metric spaces, III.H,
is required in order to appreciate certain sections in the chapter on non-positive
curvature in group theory, ΙΙΙ.Γ.

Chapter III.H (5-Hyperbolic Spaces and Area
In Part II we explored the geometry of spaces whose curvature is bounded above in
a strict, local, sense by means of the САТ(лт) inequality. In the non-positively curved
case, the Cartan-Hadamard Theorem (П.4.1) allowed us to use this local information
to make deductions about the global geometry of the universal coverings of the
spaces under consideration. In this way we were able to generalize classical results
concerning the global geometry of complete, simply connected manifolds of negative
and non-positive curvature.
M. Gromov's theory of δ-hyperbolic spaces, as set out in [Gro87], is based on a
completely different method of generalization. Ignoring the local structure, Gromov
identified a robust condition that encapsulates many of the global features of the
geometry of complete, simply connected manifolds of negative curvature. He then
showed that the geodesic spaces which satisfy this condition (5-hyperbolic spaces)
display many of the elegant features that one associates with the large-scale geometry
of such manifolds. Moreover, the robustness of this condition makes it an invariant
of quasi-isometry among geodesic spaces.
In this chapter we shall present the foundations of the theory of 5-hyperbolic
spaces, and in sections 2 and 3 of the next chapter we shall study the class of finitely
generated groups whose Cayley graphs are 5-hyperbolic.
5-Hyperbolic spaces form a natural context in which to explore the dichotomy
in the large-scale geometry of CAT(O) spaces exposed by the Flat Plane Theorem
(II.9.33): a proper cocompact CAT(O) space is hyperbolic if and only if it does
not contain an isometrically embedded copy of the Euclidean plane (Theorem 1.5).
An important aspect of this dichotomy concerns the behaviour of quasi-geodesics.
In 5-hyperbolic spaces the large-scale geometry of quasi-geodesics mimics that of
geodesies rather closely (Theorem 1.7), whereas this is not the case in spaces that
contain a flat plane (1.8.23). The stability properties of quasi-geodesics and local
geodesies in hyperbolic spaces play an important role throughout this chapter. In
particular this is true in section 3, where we describe the Gromov boundary at infinity
for hyperbolic spaces. If the space is С AT(0), the Gromov boundary coincides with the
visual boundary introduced in (II.8), and our construction of the Gromov boundary
is modelled on our earlier treatment of this special case.
In the second section of this chapter we discuss a coarse notion of filling-area for
loops in geodesic spaces. Our purpose in doing so is to characterize hyperbolicity in
terms of isoperimetric inequalities (2.7 and 2.9).

1. Hyperbolic Metric Spaces 399
1. Hyperbolic Metric Spaces
The Slim Triangles Condition
1.1 Definition (Slim Triangles). Let 5 > 0. A geodesic triangle in a metric space is
said to be δ-slim if each of its sides is contained in the 5-neighbourhood of the union
of the other two sides. A geodesic space X is said to be 5-hyperbolic if every triangle
in X is 5-slim. (If X is 5-hyperbolic for some 5 > 0, one often says simply that X is
hyperbolic.)
There are a number of equivalent ways of formulating the hyperbolicity
condition (see (1.17) and (1.22)). Gromov [Gro87] attributes the above (slim triangles)
formulation to Rips. A geodesic space is 0-hyperbolic if and only if it is an K-tree,
and it is often useful to think of 5-hyperbolic spaces as thickened versions of metric
trees.
Fig. H.l A 5-slim triangle
We shall deal with several aspects of the theory of 5-hyperbolic spaces in this
chapter, and in the next chapter we shall study the class of groups whose Cayley
graphs are 5-hyperbolic, but our account of this active field is by no means
comprehensive. It also owes much to the earlier works [CDP90], [GhH90] and [Sho91].
We begin by relating hyperbolicity to the ideas of negative curvature considered
in earlier sections.
1.2 Proposition. If к < 0 then every CAT(/c) space is 5-hyperbolic, where 5 depends
only on к.
Proof. This follows immediately from the CAT(/c) inequality and the fact that the
hyperbolic plane is a 5-hyperbolic space for a suitable 5. To see that H2 is hyperbolic,

400 Chapter III.H 5-Hyperbolic Spaces and Area
note that since the area of geodesic triangles in H2 is bounded by π, there is a bound
on the radius of semicircles that can be inscribed in a geodesic triangle. D
1.3 Exercise. ШР is hyperbolic — find the best 5.
1.4 Proposition. Let X be a proper С АТ(0) space. X is hyperbolic if and only if it is
uniformly visible (see II.9.30).
Proof. Suppose that X is 5-hyperbolic. Consider a geodesic triangle in X with vertices
p, x, y, and suppose that the distance from ρ to [x, y] is greater than ρ > 35. Because
X is 5-hyperbolic, we can find ζ on [x, у], и on [p,x], and υ on [p, y] such that
d(z, u), d(z, υ) < 5. Thus d(u, υ) < 25 and d(p, u), d(p, ν) > ρ — 5. Applying
the CAT(O) inequality for comparison angles (11.1.7(4)) to A(p, u, v), we see that
d(u, v) > (p — 5) s'm(Zp(x, y)/2). Thus
Zp(x, y) < 2 arcsin[25/(p - 5)],
which is independent of ρ and tends to 0 as ρ —»■ oo. Therefore X is uniformly visible.
Conversely, if X is uniformly visible then X is 5-hyperbolic with 5 = R(n/2)
(where R is as in definition (II.9.30)). To see this, consider a geodesic triangle with
vertices x, y, ζ and a point ρ on the side [x, y]. The angles Z.p{x, z) and Zp(y, z) must
sum to at least π, so one of these angles, Z.p(x, z) say, must be at least 7Г/2. But then,
by the definition of uniform visibility, d(p, [x, z]) < R{tt/2). D
As a corollary of (II.9.32 and 33) and (1.4), we get the following theorem.
1.5 Flat Plane Theorem. A proper cocompact CAT(O) space X is hyperbolic if and
only if it does not contain a subspace isometric to E2.
Quasi-Geodesics in Hyperbolic Spaces
One of the most important facets of the dichotomy highlighted by the Rat Plane
Theorem concerns the lengths of paths which are not close to geodesies. For example,
the following result is blatantly false in any space that contains an isometrically
embedded plane.
1.6 Proposition. Let X be a δ-hyperbolic geodesic space. Let с be a continuous
rectifiable path in X. If [p, q] is a geodesic segment connecting the endpoints of c,
then for every χ e [p, q]
d(x,im(c))<S |log2Z(c)| + 1.
(Here, as usual, 1(c) denotes the length ofc.)

Quasi-Geodesics in Hyperbolic Spaces 401
Proof. If 1(c) < 1, the result is trivial. Suppose that 1(c) > 1. Without loss of
generality we may assume that с is a map [0,1] —*■ X that parameterizes its image
proportional to arc length. Thus ρ = c(0) and q = c(\). Let N denote the positive
integer such that l(c)/2N+l < 1 < 1(c)/2N.
Let Δ, = A([c(0), c(l/2)], [c(l/2), c(l)], [c(0), c(l)]) be a geodesic triangle in
X containing the given geodesic [c(0), c(l)]. Given χ e [c(0), c(l)], we choose yi e
[c(0), c(l/2)] U [c(l/2), c(l)] with rf(jc, yi) < 5. If y{ e [c(0), c(l/2)] then we
consider a geodesic triangle A([c(0), c(l/2)], [c(l/4), c(l/2)], [c(0), c(l/4)]), which
has the edge [c(0), c(l/2)] in common with Δι and call this triangle Δ2. If on the
other handyi e [c(l/2), c(l)], then we consider Д([с(1/2), с(3/4)], [c(3/4), c(l)],
[c(l/2), c(l)]) and call this triangle Δ2. In either case we can choosey € Δ2 χ Δι
such that й?(уь y2) < 5.
We proceed inductively: at the (n + 1 )-st stage we consider a geodesic triangle
Δ„+ι which has in common with Δ„ the side [c(tn), c(^,)] containing y,„ and which
has as its third vertex c(t„+i), where tn+\ = (t„ + Q/2. We choose yn+i e A„+l \
[c(t„), c(Q] with d(y„,yn+l) < δ.
At the iV-th stage of this construction we obtain a point у ν which is a distance at
most SN from x, and which lies on an interval of length l(C)/2N with endpoints in
the image of c. If we define у to be the closest endpoint of this interval, then since
l(Q/2N+l < 1 and 2N < 1(c) we have d(x, y) < δ \ log2 l(c)\ + 1. D
c(0)=p x c(l)=g
Fig. H.2 Geodesies stay close to short curves (1.6)
We shall develop the theme of the preceding proposition by studying quasi-
geodesics in hyperbolic spaces. Quasi-geodesics were defined in (1.8.22). In
arbitrary CAT(O) spaces quasi-geodesics can be fairly wild; in particular the image of a
quasi-geodesic need not be Hausdorff close to any geodesic (cf. 1.8.23). In contrast,
the large-scale behaviour of quasi-geodesics in 5-hyperbolic spaces mimics that of
geodesies rather closely:
1.7 Theorem (Stability of Quasi-Geodesics). For all δ > 0, λ > 1, ε > 0 there
exists a constant R = /?(S, λ, ε) with the following property:
IfX is a δ-hyperbolic geodesic space, с is α (λ, ε)-quasi-geodesic in X and [p, q]
is a geodesic segment joining the endpoints ofc, then the Hausdorff distance between
\p, q] and the image ofc is less than R.

402 Chapter ΠΙ.Η 5-Hyperbolic Spaces and Area
We defer the proof of this theorem for a moment while we consider some of its
consequences.
By definition, a (λ, e)-quasi-geodesic triangle in a metric space X consists of three
(λ, e)-quasi-geodesics (its sides) p,- : [0, Γ,] -> X, i = 1, 2, 3 withρ,^Γ,-) = p,+i(0)
(indices mod 3). Such a triangle is said to be k-slim (where к > 0) if for each
г € {1, 2, 3} every point χ e im(p,) lies in the ^-neighbourhood of im(p,_i)Uim(p,+i)
(indices mod 3).
As an immediate consequence of (1.7) we have:
1.8 Corollary. A geodesic metric space X is hyperbolic if and only if, for every
λ > 1 and every ε > 0, there exists a constant Μ such that every (λ, e)-quasi-
geodesic triangle in X is M-slim. (IfX is δ-hyperbolic, then Μ depends only on δ, λ
and ε.)
The following result should be regarded as the analogue in coarse geometry of
the fact that a convex subspace of а С AT(— 1) space is а С AT(— 1) space. It shows in
particular that hyperbolicity is an invariant of quasi-isometry.
1.9 Theorem. LetX andX' be geodesic metric spaces and letf : X' —*■ X be a quasi-
isometric embedding. IfX is hyperbolic then X' is hyperbolic. (IfX is δ-hyperbolic
and f : X' —»■ X is α (λ, s)-quasi-isometric embedding, then X' is δ'-hyperbolic,
where δ' is a function of δ, λ and ε.)
Proof. Let Δ be a geodesic triangle in X' with sides р\,рг,ръ- According to (1.8),
there is a constant Μ = Μ(δ, λ, ε) such that the (λ, e)-quasi-geodesic triangle in
X with sides/ oph/o p2,f о p3 is M-slim, i.e. for all χ e im(j>i) there exists
у e im(p2) U im(p3) such that d(f(x),f(y)) < M. Since/ is a (λ, e)-quasi-isometric
embedding,
d(x, y) < λ d(f(x),f(y)) + ελ<λΜ + λε.
Repeating this argument with p2 and ръ in place of p\, we see that Δ is S'-slim,
where δ' = λΜ + λε. □
Before proving (1.7) we mention one other consequence of it. The spiral described
in (1.8.23) shows that in general quasi-geodesic rays in CAT(O) spaces do not tend
to a definite point at infinity, whereas (1.7) implies that in 5-hyperbolic spaces they
do. (This observation will play an important role in section 3.)
1.10 Proposition. Let X\ andX2 be complete CAT(0) spaces that are δ-hyperbolic.
Consider X; = X, U ЭХ, with the cone topology (II.8.6).
(1) If с : [0, oo) —»■ X\ is a quasi-geodesic, then there exists a point of dX\, denoted
c(oo), such that c(t) —»■ c(oo) as t —»■ oo.
(2) Iff :X\ —>- X2 is a quasi-isometry, then the map fa : dX\ —*■ dX2, which sends
the equivalence class of each geodesic ray с : [0, oo) —»■ X\ to the endpoint
if о c)(oo) of the quasi-geodesic rayf о с, is a homeomorphism.

Quasi-Geodesics in Hyperbolic Spaces 403
Proof. Exercise. (Stronger results will be proved in section 3 of this chapter.) D
Remark. If a finitely generated group Γ acts properly and cocompactly by isometries
on geodesic spaces X\ and X2, then Γ, X\ and X2 are all quasi-isometric (1.8.19). In
particular, if X\ and X2 are CAT(O) spaces that are hyperbolic, then dXi and ЗХ2 will
be homeomorphic, by (1.10). Chris Croke and Bruce Kleiner have recently shown50
that in the non-hyperbolic case dXi need not be homeomorphic to ЗХ2·
We now turn to the deferred proof of Theorem 1.7. The following technical result
allows one to circumvent the difficulties posed by some of the more pathological traits
of quasi-geodesics.
1.11 Lemma (Taming Quasi-Geodesics). LetX be a geodesic space. Given any (λ, ε)
quasi-geodesic с : [a, b] —*■ X, one can find a continuous (λ, ε') quasi-geodesic
с' : [a, b] —*■ X such that:
(1) c'(a) = c(a) and c'(b) = c(b);
(2) ε'=2(λ + ε);
(3) /(c'|[,,,<i) < ki d(c'(t), c'(O) + k2, for all tj e [a, b], where kx = λ (λ + ε) and
k2 = (λε' + 3)(λ + ε);
(4) the Hausdorff distance between the images of с and d is less than (λ + ε).
Proof. Define c' to agree with с on Σ = {a, 6}U(Zi1 (a, b)). Then choose geodesic
segments joining the images of successive points in Σ and define d by concatenating
linear reparameterizations of these geodesic segments. Note that the length of each
of the geodesic segments is at most (λ + ε). Every point of im(c) U im(c') lies in the
(λ + ε)/2 neighbourhood of c(E), thus (4) holds.
Let [t] denote the point of Σ closest to t e [a, b]. Because с is a (λ, г)ч}иа81-
geodesic and c([i]) = c'([i]) for all t e [a, b],
d(c'(t), c'(O) < d(d([t]), c'([fl)) + (λ + ε)
<λ|[ί]-[ί']|+ε + (λ + ε)
< λ(|ί - ί'| + 1) + (λ + 2ε);
and
Lt _ ί'| _ 2(λ + ε) < U\t - <Ί - 1) - (λ + 2ε)
Α Λ
<||М-[Щ-а + 2е)
Α
< d(d([t]), с'Ц/])) - (λ + ε)
< d(d(t), c'{t')).
This proves that d is a (λ, ε') quasi-geodesic with ε' as in (2).
50 "Boundaries of spaces with non-positive curvature", to appear in Topology.

404 Chapter Ш.Н 5-Hyperbolic Spaces and Area
For all integers n,m e [a, b],
m-\
Kc'\{n,m\) = Σ<ϊ(ο(ί), c(i + 1)) < (λ + s)\m - n\,
and similarly l{c'\{a,m\) — (λ + eXm ~ a+ 1) and /(c'|[«,ai) < (λ + e)(b - η + 1).
Thus for all t, ΐ e [a, b] we have:
/(с'|[(,а><^ + £Х1М-['']1 + 2),
and
d(c'(t), c'(t')) > l-\t - t'\ - ε' > i(|[f] - [i']| - 1) - ε'.
By combining these inequalities and noting that ε < ε' we obtain (3). D
Remark. A slight modification of the preceding argument allows one to extend the
result to length spaces.
Proof of Theorem 1.7. First one tames c, in other words one replaces it by d as in
the preceding lemma. We write im(c') for the image of c' and [ρ, q] for a choice of
geodesic segment joining its endpoints. Let D = sup{u?(jc, im(c')) \ x € \p, #]} and
let jco be a point on [p, q\ at which this supremum is attained. The open ball of radius
D with centre jco does not meet im(c')· We shall use (1.6) to bound D.
Let у be the point of \p,xo] С [ρ, q] that is a distance 2D fromxo (if ^(*ο,ί0 < 2D
then let у — ρ). Choose ζ e [x0, q] similarly. We fix /, z' e im(c') with d(y, y1) <D
and d(z, z') < D and choose geodesic segments \y, y'] and [z1, z]. (See figure H.3.)
Consider the path γ from у to ζ that traverses \y, y'] then follows c' from / to z', then
traverses [z!, z]. This path lies outside B(xq, D).
Ρ У х0 ζ q
Fig. H.3 Quasi-geodesics are close to geodesies
Since d(y, z') < d(y', y) + d(y, z) + d(z, z') < 6D, from 1.11(3) we have 1(γ) <
6Dkl+k2 + 2D. Andfrom(1.6),as<f(jEo, im(y)) = DwehaveD-1 < 5 |log2/(y)|.
Thus
D - 1 < 5 log2(6D/t, +k2 + 2D),
whence an upper bound on D depending only on λ, ε and 5. Let Do be such a bound.
We claim that im(c/) is contained in the ^'-neighbourhood of [p, q], where R' =
D0(I + ki) + кг/2. Consider a maximal sub-interval [a', b'] с [α, b] such that
c'([a\ b']) lies outside the D0-neighbourhood VDo[p, q]. Every point of [p, q] lies in

Λ-Local Geodesies 405
VDo(im(c')), so by connectedness there exist w e \p,q],t e [a, a'] and t' e [b', b]
such that d(w, c'{t)) < D0 and d(w, c'(t')) < D0. In particular d(c'(t), c'(t')) <
2D0, so /(c'lfj/]) < 2klD0 + k2, by (1.11(3)). Hence im(c') is contained in the R'
neighbourhood of [p, q]. From this and (1.11(4)) it follows that R = R' + λ + ε
satisfies the statement of the theorem. D
&-Local Geodesies
The following result provides a useful companion to Theorem 1.7. It gives a local
criterion for recognizing quasi-geodesics.
1.12 Definition. Let X be a metric space and fix к > 0. A path с : [a, b] -»■ X is said
to be a k-local geodesic if d(c(t), c(f)) = \t - /Ί for all t, t' e [a, b] with \t - /Ί < fc.
1.13 Theorem (k-Local Geodesies are Quasi-Geodesics). Let X be a 8-hyperbolic
geodesic space and let с : [a,b]^-Xbea k-local geodesic, where к > 85. Then:
(1) im(c) is contained in the 25-neighbourhood of any geodesic segment [c(a), c(b)]
connecting its endpoints,
(2) [c(a), c(b)] is contained in the 35 -neighbourhood oj?im(c), and
(3) с is α (λ, e)-quasi-geodesic, where ε = 25 and λ = (к + 48)/(к — 45).
Proof. First we prove (1). Let χ = c{t) be a point of im(c) that is at maximal distance
from [c(a), c(b)]. First suppose that both (t - a) and (b — f) are greater than 45. Then
we may suppose that there is a subarc of с centred at χ of length strictly greater than
85 but less than k. Let у and ζ be the endpoints of this arc and let / and г' be points on
[c(a), c(b)] that are closest to у and ζ respectively. Consider a geodesic quadrilateral
with vertices y, z, y', z' such that the sides \y, z] and [/, г'] are the obvious subarcs
of с and [c(a), c(b)]. By dividing this quadrilateral with a diagonal and applying the
5-hyperbolic criterion to each of the resulting triangles, we see that there exists w
on one of the sides other than \y, z] such that d(w, x) < 25. If w e \y, y'] then there
would be a path through w joining χ to / that was shorter than d(y, /):
d(x, y') - d(y, y) < [d(x, w) + d(w, /)] - [d(y, w) + d(w, /)]
= d(x, w) — d(y, w)
< d(x, w) - [d(y, x) - d(x, w)]
= 2d(x, w) — d(x,y)
< 45 - 45 = 0,
and this contradicts the choice of x. If w e [z, z7] then we obtain a similar
contradiction. Thus w e [y, z'] and hence the distance from χ to [c(a), c(b)] is at most 25. A
similar argument applies in the cases where at least one of (t — a) or (b — t) is less
than 45. (In fact, in those cases χ lies in the 5-neighbourhood of [c(a), c(b)].)

406 Chapter Ш.Н 5-Hyperbolic Spaces and Area
We now prove (2). Suppose that/? e [c(a), c(b)]. Every point of im(c) lies in one
of the two open sets that are the 25-neighbourhoods of [c(a), p] and [p, c(b)]. Since
im(c) is connected, some χ e im(c) lies in both; choose q e [c(a),p], r e \p, c(b)]
such that d(x, q) < 25 and d(x, r) < 25. By applying the 5-hyperbolic condition to a
geodesic triangle with vertices x, q, r, we see that/? e [q, r] is in the 5-neighbourhood
of [x, q] U [χ, r] and hence d(p, x) < 35.
For (3), note first that d(c(t), c(t')) < \ί-ίΊ for alii, t' e [a, b]. In order to bound
d(c(t), c(O) below by a linear function of \t — )?\, we shall argue that if one divides
с into subpaths of length Id = k/2 + 25 and projects the endpoints of these subarcs
onto [c(a), c(b)], then the points of projection form a monotone sequence. To this
end, we consider a subarc of с with length 2ld. Let χ and у be the endpoints of this
arc and let m be the midpoint of the arc; let x1, y1 and m' be points of [c(a), c(b)] that
are a distance at most 25 from χ and у and m respectively. We must show that m' lies
between x1 and /.
Let x0 (resp. yo) be the point on the image of с that is a distance 25 from χ
(resp. y) in the direction of m. By 5-hyperbolicity, any geodesic triangle A(x, x1, x0)
is contained in the 35-neighbourhood of χ and therefore (since d(x, m) = Id > 65)
any such triangle lies outside the 35-neighbourhood of m. Similarly, there exists a
geodesic triangle A(y, У, yo) outside the 35-neighbourhood of m. By applying the
5-hyperbolic condition to the (obvious) geodesic quadrilateral (x/, xq, y0, y') (divided
into two triangles by a diagonal), we deduce that m lies within a distance 25 of some
point m" e [Y, /] С [c(a), c(b)]. Any point between m' and m" is a distance at most
35 from m (by the hyperbolicity of A(m, m\ m")). In particular, neitherx1 nor/ lie
between m! and m", which means that m' e [xf, /], as we wished to prove.
We express с as a concatenation of Μ < (b — a)/Id geodesies of length Ы with a
smaller piece, of length η say, at the end. By the preceding argument, the projections
(choice of closest points) of the endpoints of these geodesies on [c(a), c(b)] form a
monotone sequence. By (I), the distance between successive projections is at least
kf — 45. And by the triangle inequality, the distance from the last projection point to
c(b) is at least η — 25.
Thus we have
b - a = Mk' + η,
and
d(c(a), c{b)) > M(k' - 45) + η - 25 = (b - a) - 45M - 25.
Since Μ <{b — a)/ld, we deduce that
k' - 45
d(c(a), c{b)) > —j^(b -a) -28.
This together with the (trivial) remark that a subpath of a k-local geodesic is again a
Ыоса1 geodesic proves (3). D
The above theorem shows that if к > 85 then there are no non-trivial closed
£-local geodesies in a 5-hyperbolic space:

Reformulations of the Hyperbolicity Condition 407
1.14 Corollary. IfX is a δ-hyperbolic geodesic metric space and с : [a, b] —*■ X is
a k-local geodesic for some к > 85, then either с is constant (i.e., a = b) or else
c(a) φ c{b).
Proof. If c(a) = c(b) then by the above theorem im(c) is contained in the ball of
radius 25 about c(a). Since с is an 85-local geodesic, this implies that с is constant.
D
Remark. By combining (1.7) and (1.13) in the obvious way one gets a criterion for
seeing that paths which are locally quasi-geodesic (with suitable parameters) are
actually quasi-geodesics.
Although quasi-geodesics stay Hausdorff close to geodesies in 5-hyperbolic
spaces, trivial examples show that they need not be uniformly close. On the other
hand, in any metric space, if geodesies with a common origin are Hausdorff close
then they must be uniformly close. In hyperbolic spaces one can say more:
1.15 Lemma. Let X be a geodesic space that is δ-hyperbolic and let c, c' : [0, T] —►
X be geodesies with c(0) = c'(0). If d(c(to), im(c')) < K, for some К > 0 and
t0 e [0, T], then d(c(t), c'{t)) < 25 for all t < t0 - К - 5.
Let c\ : [0,Ti]—*X and сг : [0, Гг] —»■ X be geodesies with ci(0) = сг(0). Let
Τ = тах{Гь Гг} and extend the shorter geodesic to [0, T] by the constant map. If
к = d(ci(T), c2(T)), then d(a(t), c2(0) < 2(fc + 25)/or all t e [0, T].
Proof. To prove the first assertion, we choose a geodesic cq joining c(io) to a closest
point c'(ii) on the image of c'. By the triangle inequality |i0 — t\\ <K. Note that c{t)
is not 5-close to any point on Co if t < ίο — К — 5. It follows from the 5-slimness of
the triangle with sides c0, c([0, t0]), c'([0, t{]) that d(c(t), d{f)) < 5 for some f. By
the triangle inequality \t - /Ί < 5. Therefore d(c(t), c'{t)) < 25.
To prove the second assertion we consider a geodesic triangle two of whose sides
are c\ and сг- If c\(t) is 5-close to a point сг(0> then as above \t — tf\ < 5 and hence
d(ci(t), ci{t)) < 25. If C\(t) is 5-close to a point on the third side of the triangle,
then it is {к + 5)-close to the endpoint of c% and, as in the first case, this implies that
</(ci(0.c2(0)<2(* + i). □
Reformulations of the Hyperbolicity Condition
In this section we shall describe some alternative ways of phrasing Gromov's
hyperbolicity condition for geodesic spaces. Each of the reformulations that we shall
discuss is mentioned in Gromov's original article [Gro87].

408 Chapter ΙΠ.Η 5-Hyperbolic Spaces and Area
Thin Triangles
It is often useful to think of 5-hyperbolic spaces as being fattened versions of trees5'.
With this in mind, we wish to compare triangles in arbitrary metric spaces to triangles
in metric trees.
Given any three positive numbers a, b, c, we can considerthe metric tree T(a, b, c)
that has three vertices of valence one, one vertex of valence three, and edges of length
a, b and c. Such a tree is called a tripod. For convenience, we extend the definition
of tripod in the obvious way to cover the cases where a, b and с are allowed to be
zero. Thus a tripod is a metric simplicial tree with at most three edges and at most
one vertex of valence greater than one.
Given any three points x, y, ζ in a metric space, the triangle equality tells us that
there exist unique non-negative numbers a, b, с such that d(x, y) = a+ b, d(x, z) =
a + с and d(y, z) — b + c; in the notation of (1.19), a = (y ■ z)x, b = (x ■ z)y and
с = {χ ■ y)z. There is an isometry from {x, y, z] to a subset of the vertices of T(a, b, c)
(the vertices of valence one in the non-degenerate case); we label these vertices
vx, vy, vz in the obvious way.
Given a geodesic triangle, Δ = A(x, y, z), we define Гд := T(a, b, c), and write
од to denote the central vertex of Гд (i.e., the point a distance a from vx). The
above map {x, y, z} —*■ {vx, vy, vz] extends uniquely to a map хд : Δ —»■ Гд whose
restriction to each side of Δ is an isometry.
A natural measure of thinness for Δ is the diameter of the fibres of хд. The
only fibre that may have more than two points is Хд'(од), which contains one point
on each of the sides of Δ. The points of Хд (од) are called the internal points of
Δ. Note that these internal points are the images in Δ of the points at which the
comparison triangle Δ С Ε2 meets its inscribed circle (figure H.4).
1.16 Definition (5-Thinness and Insize). Let Δ be a geodesic triangle in a metric
space X and consider the map хд : Δ —»■ Гд defined above. Let 5 > 0.
Δ is said to be δ-thin if p, q e Хд'(0 implies d{p, q) < 8, for all t е Гд.
The diameter of Хд'(од) is denoted insize Δ.
1.17 Proposition. LetX be a geodesic space. Thefollowing conditions are equivalent.
(1) There exists 5o > 0 such that every geodesic triangle in X is δο-slim (definition
1.1).
(2) There exists S\ > 0 such that every geodesic triangle in X is δ ι-thin (definition
1.16).
(3) There exists 82 > 0 such that insize Δ < bifor every geodesic triangle Δ in X
(definition 1.16).
51See[Bow9l], [CDP91], [Gro87] and [GhH90] for precise results regarding approximation
by trees.

Reformulations of the Hyperbolicity Condition 409
Fig. H.4 The internal points of Δ and the tripod Гд
Proof. It is obvious that (2) implies (1). To prove that (1) implies (3), we assume that
every triangle in X is So-slim and consider a geodesic triangle A(x, y, z) with internal
points ix, iy, iz (where ix e \y,z], etc.). By hypothesis, ix lies in the So-neighbourhood
of either \y, x] or [z, x]; say d(x, p) < So where ρ e \y, x]. By the triangle inequality,
\d(y,p) — d(y, ix)\ < S0. Since d(y, ix) = d(y, iz), it follows that d{iz,p) < So and
hence d(ix, iz) < 25o- Similarly, d(iy, {ix, iz}) < 28q and d(iz, {ix, iy}) < 28q. Hence
insize Δ < 450.
Now assume that (3) holds and let A(x, y, z) be a geodesic triangle in X with
internal points ix, iy, iz. If ρ e \y, z] is such that d(y, p) < d(y, ix), then the fibre of
Χδ containing ρ is {p, q], where q e \y,x] is the point with d(y,p) = d(y, q). We
shall bound dip, q) by constructing a geodesic triangle for which ρ and q are internal
points.
Letc : [0, 1] —»■ X be a monotone parameterization of \y, z]andforeachi e [0, 1]
consider a geodesic triangle Δ, = A(x, y, c(t)) two of whose sides are \y, x] с Δ
and c([0, t]). The internal point of Δ, on the side c([0, t]) varies continuously (though
perhaps not monotonically) as a function of t. At t = 0 this point is у and at t = 1 it
is ix, so for some ίο € [0, 1] it is p. And since d(y, p) = d(y, q), it follows that q is
also an internal point of Δ,0. D
1.18 Exercise. Prove that the above conditions are equivalent to: there exists S3 > 0
such that for every geodesic triangle A(x, y, z) in X,
inf{diam(x/, /, z1) \ x' e \y, z], y' e [x, z], z' e [x, y]} < 53.

410 Chapter ΠΙ.Η 5-Hyperbolic Spaces and Area
The Gromov Product and a 4-Point Condition
The Gromov product on a metric space (sometimes called the overlap function)
measures the lengths of the edges of the tripods described in the paragraph on thin
triangles.
1.19 Definition. Let X be a metric space and let χ e X. The Gromov product of
y,z £ X with respect to χ is defined to be
(y-z)x= -{d(y,x) + d(z,x) - d(y, z)}.
Equivalently, (y ■ z)x is the distance from the comparison point for χ to the adjacent
internal points in a comparison triangle A(x, y, z) С Ε2. (See figure H.4.)
Note that if Δ = A(x, y, z) is a geodesic triangle in any metric space X, then
d(x, [y, z])<(y- z)x + insize Δ. And if Δ is 5-thin, then \d(x, \y, z])-(y- z)x\ < δ.
In the following definition X is not required to be a geodesic space.
1.20 Definition. Let 5 > 0. A metric space X is said to be (S)-hyperbolic if
(x ■ y)w > min{(x · z)w, (У ■ z)w] - δ
for all w,x, y, ζ e
ΧΙ. 21 Remark. It is not difficult to show that if there exists some w e X such that the
above inequality holds for all x,y,z £ X, then X is (25)-hyperbolic, i.e. the inequality
holds for all x, y, z, w if one replaces 5 by 25. See [Gro87, 1. IB].
Henceforth, when we say that a metric space is hyperbolic we shall mean that it
is (S)-hyperbolic for some 5 > 0. Proposition 1.22 shows that this usage agrees with
our previous convention (1.1) in the case where X is a geodesic space.
There is a disturbing asymmetry about the respective roles of w, x, у and ζ in the
above definition. To understand this, we first unravel the definition of the Gromov
product, rewriting the above inequality as a 4-point condition:
(Q(5)) d(x, w) + d(y, z) < max{ф:, у) + d(z, w), d(x, z) + d(y, w)} + 25
for all w,x, y, zel
The geometry behind this equality becomes apparent if we think of w, x, y, ζ as
the vertices of a tetrahedron; d(x, y) + d(z, w), d(x, z) + d(y, w) and d(x, w) + d(y, z)
correspond to the sums of the lengths of the three opposite pairs of edges. With this
picture in mind, we call these three sums the pair sizes of {w, x, y, z}- The inequality
Q(5) states that if we list the pair sizes in increasing order, say S < Μ < L, then
L-M <2δ.
Suppose S = d(x, z) + d(y, w), Μ = d(x, y) + d(z, w) and L = d(x, w) + d(y, z)-
In terms of comparison triangles, the inequality S < Μ means that by choosing

Reformulations of the Hyperbolicity Condition 411
adjoining comparison triangles A(jc, w, y) and A(x, w, z) in E2, we obtain the
configuration shown in (H.5), with / > 0. (For convenience, we have omitted the overbars
in labelling the comparison points.) We shall examine the inequality L — Μ < 25 in
terms of this comparison figure.
Fig. H.5 The case d(x, y) + d(z, w) > d(x, z) + d(y, w)
1.22 Proposition. Let X be a geodesic space. X is hyperbolic in the sense of (1.17)
if and only if there is a constants > 0 such thatX is (S)-hyperbolic in the sense of
(1.20).
Proof. First we shall show that if insize Δ < 5 for all geodesic triangles Δ in X, then
X is (S)-hyperbolic.
Given w,*, y, ζ e X, we may assume without loss of generality that S := d(x, z)+
d(y, w) < Μ := d(x, y) + d(z, w) < L := d(x, w) + d(y, z). What we must show
is that L < Μ + 28. Let Δ = (χ, w, у) and Δ' = A(x, w, z) be geodesic triangles,
and denote their interior points by (ix, iw, iy) and (г^, i'w, i'z) respectively (see figure
H.5). By considering the path from у to ζ that proceeds via ix, iy, i'7 and i'w, we get
d(y, z) < d(y, ix) + d(ix, iy) + d(iy, i'z) + d(i'z, i'w) + d(i'w, z), which, in the notation
of the diagram, is (b + d(ix, iy) + 1 + d(i'z, i'w) + d). Also d(x, w) = a + c — I. Thus
L < (a + b + с + d) + 25 = Μ + 25.
Now we shall assume that X is (5)-hyperbolic and deduce that insize Δ < 65 for
all geodesic triangles Δ = A(x, y, z) in X. We first focus our attention on the internal
point ix e [)>, z] of Δ and apply the condition Q(8) to the four points {x, y, z, ix}-
Note that d(x, ix) + d(y, z) is the largest of the three pair sizes for {x, y, z, ix}. Indeed
2(d(x, ix) + d(y, z)) = [d(x, ix) + d(ix, z)] + [d(x, ix) + d(ix, y)] + d(y, z) is greater than
the perimeter Ρ of Δ, whereas the other pair sizes are (d(z, ix) + d(x, y)) = (d(y, ix) +
d(x, ζ)) = Ρ β. Therefore (d(x, ix) + d(y, ζ)) < Ρ β + 25. Since d(y, z) + d(x, iz) =
Ρβ, we deduce that \d(x, iz) - d(x, ix)\ < 25. Similarly, \d(z, ix) - d(z, iz)\ < 25.
Nowconsiderthefourpoints {x, z, ix, iz] (see figure H.6). In this case the three pair
sizes are d(x, z) = d(x, iz) + d(z, ix), d(ix, iz) + d(x, z), and d(x, ix) + d(z, iz). By the
inequalities above, the third of these terms is < d(x, iz) + d(z, ix) + 45 = d(x, z) + 45.
Thus we see that the last two of the three pair sizes listed are the largest. And applying
Q(S) we deduce that d(ix, iz) < 65. Similarly, ^(г^, iy) < 65 and d(iy, iz) < 65. Π

412 Chapter III.H <5-Hyperbolic Spaces and Area
Fig. H.6 Proving that (<5)-hyperbolic implies δ -hyperbolic
1.23 Exercise. Show that there exist quasi-geodesic rays in the Euclidean plane whose
images, equipped with the induced metric from E2, are not (S)-hyperbolic for any
5 > 0. (The spiral described in (1.8.23) provides one such example; the ray given in
rectangular coordinates by t м> (л/ί, t) is another.) Deduce that hyperbolicity is not
an invariant of quasi-isometry among arbitrary metric spaces (cf. 1.9).
Divergence of Geodesies
A key difference between Euclidean and hyperbolic geometry is that in the Euclidean
plane if two distinct geodesic rays C\ and c% have a common origin ρ then for all
t > 0 one can connect c\{t) to сг(?) by a path outside B(p, t) that has length at most
πί, whereas in the hyperbolic plane the length of the shortest such path increases as
an exponential function of t. In order to discuss such divergence properties in greater
generality we need the following definition.
1.24 Definition. Let X be a metric space. A map e : N —»■ R is said to be a divergence
function for X if the following condition holds for all R, r e N, all χ e X and all
geodesies c\ : [0, αϊ] -»■ X and c% : [0, a2] -> X with ci(0) = c2(0) = x.
If R + r < min{ai, a2] and d(ci(R), сг(/?)) > е(0), then any path connecting
a (R + r) to a{R + r) outside the ball B(x, R + r) must have length at least e(f).
1.25 Proposition. Let X be a geodesic space. If X is hyperbolic, then X has an
exponential divergence function.

Reformulations of the Hyperbolicity Condition 413
Proof. Suppose that triangles in X are 5-thin. We claim that e(n) = max{35, 2V}
is a divergence function for X. Let с ι and c2 be as in the preceding definition and
suppose that R, r e N are such that d{c\{R), c2{R)) > 35 and R + r < min{ab a2}.
Let yt = d(R + r) for i = 1,2, and consider a geodesic triangle Δ with sides
ci([0, R + r]), c2([0, R + r]) and \yi,y2]. Note that since d(ci(R), c2(R)) > 35, the
internal points of Δ on the sides c,([0, R + r]) are a distance less than R — δ from
x = c(0), and hence the internal point on \y\, y2] (which is actually the midpoint
m) is a distance less than R from x. Thus B(m, г) С B(x, R + r), and hence any path
joining у ι toy2 in the complement of B(x, R + r) lies outside B(m, r). According to
(1.6), such a path has length at least 2'~. D
Fig. H.7 Exponential divergence of geodesies
There is a (strong) converse to the preceding proposition:
1.26 Proposition. Let X be a geodesic space. IfX has a divergence function e such
that liminfn^oo e(n)/n = oo, then X is δ-hyperbolic for some 5 > 0.
1.27 Exercise (The Proof of (1.26)). We sketch the proof, leaving the details as
an exercise for the reader (cf. [Sho91, p. 37]). Consider a geodesic triangle Δ =
A(x, y, z). Let c\ : [0, a,] —»■ X, i = 1, 2 be the sides of Δ issuing from χ and define
Tx to be sup{i0 I d(c\(t), c^(i)) < e(0) Vi < t0}. Define Ty and Tz similarly. Show that
if Tx + Ty > d(x, y) then Δ is 5-slim, where 5 depends only on e. Now assume that if
we delete from Δ the images of cf([0, TW]), i = 1,2, w e {x, y, z}, then we are left
with three non-empty segments. Without loss of generality we may suppose that the
segment contained in [jc, y] is the longest of the three and that the one contained in
\y, z] is the second longest; let ρ be the midpoint of the former and let q be the point
of \y, z] that is a distance L := d(y,p) from y. Show that there is a path of length
<2L + 6e(0) from ρ to q outside the open ball of radius L about у and use the fact
that lim infn^oo e(n)/n = oo to bound L. Deduce from this bound that triangles in X
are uniformly thin.

414 Chapter III.H 5-Hyperbolic Spaces and Area
2. Area and Isoperimetric Inequalities
In this section we shall describe notions of area that are useful in the context of
geodesic metric spaces and consider the relationship between curvature and
isoperimetric inequalities, which relate the length of closed curves to the infimal area of
the discs which they bound. It is well-known that every closed loop of length I in
the Euclidean plane bounds a disc whose area is less than ί2/4π, and this bound is
optimal. Thus one has a quadratic isoperimetric inequality for loops in Euclidean
space. In contrast, loops in real hyperbolic space satisfy a linear isoperimetric
inequality: there is a constant С such that every closed loop of length I in H" bounds a
disc whose area is less than or equal to CI. The main goal of this section is to prove
that (with a suitable notion of area) a geodesic space X is 5-hyperbolic if and only
if loops in X satisfy a linear isoperimetric inequality. We shall also see that there is
a quadratic isoperimetric inequality for loops in arbitrary CAT(O) spaces (2.4), and
we shall see that the sharp distinction between linear and quadratic isoperimetric
inequalities is very general (2.13).
A Coarse Notion of Area
In order to begin a serious discussion of isoperimetric inequalities we must define a
notion of area that is sufficiently robust to withstand the lack of regularity in arbitrary
geodesic spaces.
2.1 Definitions (ε-Filling, Area£ and Isoperimetric Inequalities). Let D2 denote the
unit disc in the Euclidean plane (so dD2 = §'). A triangulation ofD2 is a homeo-
morphism Ρ from D2 to a combinatorial 2-complex in which every 2-cell is a 3-gon
(i.e. its attaching map has combinatorial length three — see I.8A for definitions).
We endow D2 with the induced cell structure and refer to the preimages under Ρ of
0-cells, 1-cells and 2-cells as, respectively, the vertices, edges and faces of P.
Let X be a metric space. Let с : §' —> X be a rectifiable loop in X. An ε-filling
(Ρ, Ф) of с consists of a triangulation Ρ of D2 and a (not necessarily continuous)
map Φ : D2 —>- X such that Φ^ι = с and the image under Φ of each face of Ρ is a
set of diameter at most ε. We write | Φ | to denote the number of faces of Ρ and refer
to this as the area of the filling. One says that с spans Ф. The ε-area of с is defined
to be:
Area£(c) := тт{|Ф| | Φ an ε- filling of c}.
(If there is no ε-filling then Area£(c) := oo.) An ε-filling Φ of с is called a least
ε-area filling if |Ф| = Area£(c).
A function/ : [0, oo) —>- [0, oo) is called a coarse isoperimetric bound for X
if there exists ε > 0 such that every rectifiable loop с in X has an ε-filling and
Area£(c)</(/(c)).
If/ is linear (resp. quadratic, polynomial, exponential, etc.) then we say that X
satisfies a (coarse) linear (resp. quadratic, polynomial, exponential etc.) isoperimetric
inequality.

2. Area and Isoperimetric Inequalities 415
Recall that, given two functions/, g : [0, oo) -»■ [0, oo), one writes/ < g if there
exists a constant К > 0 such that/O) < # g(Kx + K) + Kx + К for all χ e [0, oo).
One writes/ — g if, in addition, g -<f.
2.2 Proposition. LetX' and X be quasi-isometric length spaces. If there exists ε > 0
iwc/г that every loop in X has an ε-filling, and г/Атеа£(с) < f(l(c))for every rectifiable
loop с in X, then X' admits a coarse isoperimetric bound f ■< f.
Proof. Exercise: Follow the proof of (1.8.24) to construct fillings, and count the
number of faces. D
2.3 Remarks
(1) The triangulations Ρ in definition 2.1 are not required to be simplicial, for
instance the attaching map of a 2-cell need not be injective, and a pair of closed
2-cells may have two faces in common.
The number of vertices, edges and faces in any triangulation of a disc satisfy the
obvious relations V < 2E, Ε < 3F, F < 2E/3 and hence, by Euler characteristic,
V > 1 + 2£/3. Thus if one were to define area by counting the minimal number of
edges or vertices in a triangulation, instead of faces, the resulting notions would be
~ equivalent to the one that we have adopted.
(2) We shall use the following observation implicitly on a number of occasions.
Since ε-fillings need not be continuous, if one has specified a filling map Φ on the
vertices of a triangulation of the disc, then one can extend Φ across edges and triangles
in the interior of the disc by simply sending them to the image of a vertex in their
boundary. Thus, given a loop, if one wishes to exhibit the existence of an ε-filling
of a given area, then one need only specify a triangulation of the disc and specify a
map on the vertices of the triangulation, ensuring that the vertex set of each triangle
in the interior of the disc is sent to a set of diameter at most ε, and ensuring that if a
triangle has sides in the boundary circle then the image of these sides together with
the set of vertices has diameter at most ε.
(3) If ε' > ε then obviously Area£-(c) < Area£(c) for all rectifiable loops с in a
given space X. Thus as one increases ε, the function/*(Z) = sup{Area£(c) | /(c) < /}
(which need not be finite) may decrease. For example, consider the closed unit disc
Del2 and the loops c„ : §' —»■ X given in complex coordinates by c(z) = z". If
ε > 0 is small, then for a suitable constant к > 0 we get Area£(c„) > kn for all
η > 0, whereas if ε > 2 then Area£(c) = 1 for all loops in D.
(4) Instead of insisting that ε-fillings should be given by triangulations of the disc,
one could allow decompositions of the disc into combinatorial complexes for which
there is an integer к > 3 such that the boundary of each 2-cell has at most к faces.
This leads to an ~ equivalent notion of area, for if we write Area^ for the resulting
notion of area, then an obvious subdivision shows that Area£(c) < (k — 2)Area*)(c)
for all rectifiable loops, and trivially Area£(c) > Area^(c).
(5) Let X be a metric graph in which all edges have length one. Given a loop
с : §' —»■ X, proceeding along с from a vertex υι in the image we record the vertices

416 Chapter Ш.Н <5-Hyperbolic Spaces and Area
that с visits v\,V2- ■ -,v„ = vi, where we record υ, only if it is distinct from υ,·-ι (in
other words we do not record the occasions on which с returns to a vertex without
visiting any other vertices). Say υ, = c(i,).
Let d be the combinatorial loop in X obtained by concatenating the edges
[υ,_ι, ν;]. If ε > 2, then in order to construct an ε-filling of с : §' —*■ X we can
inscribe an и-gon in §' with vertices Ц, map this и-gon to X by d in the obvious
way, fill the и-gon with an ε-filling of d, and then extend the given triangulation of
the и-gon to a triangulation of the disc by introducing extra edges and vertices (as
necessary) in the sectors bounded by the arcs and chords [ί,_ι, ί,·]: the image of the
boundary of each of these sectors has diameter less than 2, so it suffices (see (2)) to
introduce a new vertex on the arc joining i, to tj+i and to introduce an edge joining
this vertex to each of the vertices of the filling of d that lie on [ί,_ι, ί,].
This construction bounds Area£(c) by a linear function of Area£(c') and /(c). Thus
for any metric graph with unit edge lengths, if ε > 2 then the function/* described
in (3) is — equivalent to the function sup{Area£(c') | 1(d) < I, d an edge-loop}.
2.4 Example. Every CAT(O) space satisfies a quadratic isoperimetric inequality.
Indeed ifX is a CAT(O) space, for every ε > 0 and every rectifiable loop с : §' —»■ Xone
hasAreaE(c) < 2(4/(c)^+l)2.Toseethis,divide§' into L arcs of equal length, where
L is the least integer such that 41(c) < sL; let θ\,..., θι+ι = θ\ be the endpoints of
these arcs. To define the 1-skeleton of an ε-filling of c, fori = 2,..., L one sends each
of the Euclidean segments σ, = [θ\, 0,·] to the geodesic segment [с(в\), с(0,·)] С X,
then one introduces equally spaced vertices θ\ = υ,,ο, υ,,ι,..., υ,,ζ. = #, along each
σ,, for i < L one connects each v,j to both vi+\j and υ,+ij+i by line segments, and
one connects each V2j and vij to θ\. The convexity of the metric on X ensures that
this is an ε-filling, and there are less than 2L2 triangular faces in the filling.
2.5 Exercises
(1) An ε-filling (Ρ, Φ) of a loop с in a metric space X is said to be piecewise
geodesic if the restriction of Φ to each edge in the interior of Ρ is a linear
parameterization of a geodesic segment in X. Suppose that X is a geodesic space and that с is
rectifiable. If с admits an ε-filling, then it admits a piecewise geodesic 3ε/2-Α11^
Φ with |Φ| = Area£(c).
(2) Let X be a geodesic space. Given ε > ε' > 0, if there exists a constant
Κ(ε, ε') > 0 such that Area£-(y) < К for all rectifiable loops γ with 1(γ) < 3ε, then
there is a constant K' such that Area£-(c) < K' Area£(c) for all rectifiable loops с in
X.
(Hint: Given c, consider a piecewise geodesic filling as in (1) and choose a least-
area ε'-filling of the boundary of each face. The union of these ε'-fillings gives a
combinatorial structure on the disc in which each 2-cell has (comfortably) less than
6K edges in its boundary — triangulate each of these 2-cells without introducing
new vertices.)
(3) Let X be а САТ(к) space. Prove that for all ε' < ε < 2π/^/κ there exists a
constant k = Ιί(ε, ε', κ) such that Area£'(c) < k Area£(c) for all rectifiable curves с
in X. (See (2) and (2.4). Compare with (2.17).)

The Linear Isoperimetric Inequality and Hyperbolicity 417
(4) Let X be a geodesic space. Suppose that there exists εο > 0 such that Area^ (c)
is finite for all rectifiable loops in X, and for all ε > εο there is a constant Κ(ε, εο)
satisfying (2). Use (2.2) to show that if X' is a metric space quasi-isometric to X, then
for all sufficiently large ε' and ε" the functions/Д' and/Д' (as defined in 2.3(2)) take
finite values and are ~ equivalent.
(5) Let X be the Cayley graph of a finitely presented group Γ. Prove that if N e N
is sufficiently large then/^(Z), as defined in (2.3(3)), is ~ equivalent to the Dehn
function of Γ, as defined in (I.8A).
The Linear Isoperimetric Inequality and Hyperbolicity
The purpose of this section is to characterize 5-hyperbolic geodesic spaces as those
which satisfy a linear isoperimetric inequality (in the coarse sense defined in (2.1)).
In the light of (2.2) and the quasi-isometry invariance of hyperbolicity (1.9), there
is no loss of generality in assuming that the spaces under consideration are metric
graphs with integer edge lengths. For hyperbolic graphs there is an efficient way of
filling edge-loops that is essentially due to Max Dehn52 [Dehnl2a]; the proof of (2.7)
is a straightforward implementation of this method.
2.6 Lemma. Let X be a metric graph whose edges all have integer lengths, and
suppose that X is Ь-hyperbolic where 5 > 0 is an integer. Given any non-trivial
locally-injective (hence rectifiable) loop с : [0, 1] —*■ X beginning at a vertex, one
canfinds, t € [0, 1] such that c(s)andc(t) are verticesofX, d(c(s), c{t)) < /(c|[Vj)— 1
andd(c(s), c{t)) + /(c|M) < 165.
Proof. Note that the difference in length between any two locally-injective paths with
common endpoints at vertices of X is an integer. According to (1.13), X contains no
closed loops which are Ыоса1 geodesies for к = 85 + \. Choose a non-geodesic
subarc c|[.Wo] of с that has length less than 85 + | and choose a geodesic connecting
c(sq) to c(io)· Define c(s) and c(t) to be the first and last vertices of X through which
this geodesic passes. (If c(so) is a vertex then s = sq, if not then the edge containing
c(so) is the image under с of an arc one of whose endpoints is s. Likewise for t.) D
2.7 Proposition. Let X be a geodesic space. IfX is Ь-hyperbolic, then it satisfies a
linear isoperimetric inequality.
Proof. According to (1.8.44), X is quasi-isometric to a metric graph X' with unit edge
lengths, and according to (1.9) this graph is 5'-hyperbolic for some 5' > 0. If we
In his foundational paper [Dehn 12a], Dehn applied this method to graphs which arise as the
1-skeleton of a periodic tesselation of the hyperbolic plane, and used it to solve the word
problem for surface groups (see Г.2.4).

418 Chapter ΙΠ.Η 5-Hyperbolic Spaces and Area
can show that X' satisfies a linear isoperimetric inequality, then it will follow from
(2.2) (where the roles of X and X' are reversed) that X satisfies a linear isoperimetric
inequality.
Assume that X is a metric graph with unit edge lengths. Assume that X is 5-
hyperbolic where 5 > 1 is an integer. In what follows it is convenient to use the term
edge-loop in a metric graph to describe loops which are the concatenation of a finite
number of paths each of which is either a constant speed parameterization of an edge
or a constant map at a vertex. As usual, we write /(c) for the length of such a loop с
We write /o(c) to denote the number of maximal non-trivial arcs where с : §' —»■ X is
constant. Note that/o(c) < /(c)+1. A standard ε-filling of an edge-loop is ane-filling
given by a triangulation of the disc such that all of the vertices on the boundary circle
are points of concatenation of the given edge-loop, and each edge of the triangulation
is either mapped to a concatenation of edges in X or else is sent to a vertex of X by
a constant map. We shall prove by induction on /(с) + /(со) that every edge-loop in
X admits a standard 165-filling of area (85 + 2)(/0(c) + /(c)). In the light of 2.3(5),
it will follow that X satisfies a linear isoperimetric inequality.
The first few steps of the induction are trivial. For the inductive step, given an
edge-loop с : §' —»■ X with /(c) > 2, we consider how to reduce /o(c) + /(c).
If /o(c) = 0, then с is either locally injective or else it contains a subpath which
backtracks (i.e. traverses an edge and then immediately returns along that edge). To
begin our construction of a filling for c, in the latter case we connect the endpoints s, t
of a backtracking subpath by a Euclidean segment in the disc and send this segment
to X by a constant map; in the former case we choose s and t as in Lemma 2.6 and
map the segment connecting s to t to a constant speed parameterization of a geodesic
segment joining c(s) to c(i) in X. If /o(c) > 1 then we focus on a subpath c|[J>(] which
is the concatenation of a maximal subpath of length zero (defined on a non-trivial
arc) and a subpath of length one. We again connect s to t by a Euclidean segment in
the disc and map this segment to X as a constant speed parameterization of a geodesic
segment [c(s), c(t)].
In each of these three situations, we have begun to fill с by dividing the disc
into two sectors, one (the "small" sector) whose boundary maps to an edge-loop
(containing c([s, t])) of length at most 165, and one (the "big" sector) whose boundary
map is an edge-loop c' with /o(c') + l(c') < k(c) + /(c). By induction, we may fill this
big sector with a standard 165-filling (Ρ, Φ) of c' that has at most (85+2)(/0(c')+/(c'))
faces. Subdividing and adding two extra faces if necessary, we may assume that s
and t are vertices of P. The restriction of Φ to the Euclidean segment [s, t] is a
concatenation of at most 85 edges and hence its interior contains fewer than 85
vertices from the triangulation of the filling. To complete the desired standard 165-
filling of c, we introduce edges connecting this set of vertices to a vertex introduced
on §' between s and t. D
2.8 Remarks
(1) By unraveling the various components of the above proof, one can show that
there exist universal constants ε, μ > 0 such that for every 5 > 0, every 5-hyperbolic
geodesic space X and every rectifiable curve с in X, one has Areae(i+ \){c)< μ(1(ο)+1).

The linear Isoperimetric Inequality and Hyperbolicity 419
The following result provides a converse to (2.7) and thereby completes the
characterization of hyperbolic spaces as those which satisfy a linear isoperimetric
inequality. Ourproof is based on that of Short [Sho91]. (See also [0191] and [Lys90].)
2.9 Theorem (Linear Isoperimetric Inequality Implies 5-Hyperbolic). Let X be a
geodesic metric space. If there exist constants K,N > 0 such that Area^(c) <
Kl{c) + К for every piecewise geodesic loop с in X, then X is δ-hyperbolic, where δ
depends only on К and N.
Proof. Again, in the light of (1.8.44), (1.9) and (2.2) it suffices to consider the case
where X is a metric graph with unit edge lengths. By increasing К and N if necessary,
we may assume that Areaw(c) < Kl(c) for every edge-loop с inX, and we may assume
that К and N are integers.
In order to show that there exists 5 > 0 such that X is 5-hyperbolic, we must
bound the size of integers η > 0 for which there is a geodesic triangle in X which is
not (n + 1) slim. To this end, we fix η and suppose that there is a geodesic triangle
Δ = Δ(ρ, q, r) in X and a point a e [p, q] so that the distance from a to the union
of the other two sides is greater than η + 1. We may assume that the sides of this
triangle are edge-paths. We may also assume that the length of the perimeter of Δ
is minimal. We replace a by an adjacent vertex v, which is a distance more than η
from [p, r] U [r, q].
Let к = 3KN2 and let m = 3KN. We suppose that η > 6k in order to avoid
degeneracies in the following argument.
Reversing the roles of ρ and q if necessary, there are only two cases to consider:
either [ρ, υ] is disjoint from the Ak neighbourhood of [r, q] and [υ, q] is disjoint from
the Ak neighbourhood of [p, r], or else there exists w e [v, q] and w' e [p, r] with
d(w, w') = Ak.
Fig. H.8 (i) The hexagonal case; (ii) The quadrilateral case
In the first case (Case (г) of figure H.8) we consider the minimal subarc [u,w] с
[ρ, q] that contains ν and has endpoints a distance exactly к from the union of the
other sides of Δ; let ύ e [p, r] and w' e [r, q] be the points closest to r with
d(u, u') = 2k and d(w, w') = 2k. Let [и', и"] с [и', г] and [u/, w"] с [и/, г]
be the maximal subarcs for which the к neighbourhoods of the interiors of these

420 Chapter III.H 5-Hyperbolic Spaces and Area
Fig. H.9 Obtaining a lower bound on area
arcs are disjoint. Hence d{u",w") = 2k. (Because d(u,w'),d(w,u') > Ak and
d(u, u'), d(w, w') = 2k, the points u', u", w', w" are all distinct.)
In the second case (Case (гг) of figure H. 8) we consider maximal subarcs [u, w] С
[ρ, q] and [u\ w'] С \р, r] for which the к neighbourhoods of the interiors of these
arcs are disjoint, d{u, u') = 2k, d(w, w') = Ak and υ e [u, w].
We deal with Case (г) in detail. Case (гг) is only slightly different.
We consider a geodesic hexagon Τι with vertices u, u', u", w", w'', w; the sides
[u, w], [и', и"] and [w", w'] are subarcs of the sides of Δ, and the other sides have
length 2k. Let (Ρ, Φ) be a least-area piecewise geodesic N-filling of Η (see 2.5(1)).
It is convenient to subdivide the edges of Ρ by adding vertices along the edges so
as to ensure that the restriction of Φ to each open edge is either a constant map or a
homeomorphism onto an open edge of X; since each edge in the original triangulation
was mapping to a path of length at most N, this subdivision can be performed in such
a way that the boundary of each 2-cell in the resulting combinatorial structure on
the disc contains at most 3N edges. We shall continue to refer to the disc with this
combinatorial structure as P. It is also convenient to identify points on Ti with their
preimages under Φ.
We wish to estimate the number of faces in P. By construction, the к
neighbourhoods of the segments [и, w], [u', w'] and [и", w"] are disjoint (in X and hence P).
We begin by estimating the number of faces in each of these neighbourhoods. Let
a = d(u, w), let β = d(u', u") and let γ = d(w', w").

The Linear Isoperimetric Inequality and Hyperbolicity 421
First we consider the union of those 2-cells in Ρ which intersect [и, w] (i.e.,
the star neighbourhood [u, w]); we take the minimal combinatorial subdisc D\ С Ρ
containing this union. We then iterate this process: take the star neighbourhood of D\,
'fill in the holes' to form the minimal subdisc D2 containing this star neighbourhood;
repeat m times. The boundary of Dm lies in the к = mN neighbourhood of [u, w].
(Recall that m = 3KN and к = 3KN2.)
We shall estimate how many 2-cells are added at each stage of the above process.
Because d(u, w) = a and the perimeter of each face of Ρ has combinatorial length
at most 3N, there are at least a/(3N) faces in D\. For i = 1,..., m — 1 there is a
(unique) injective edge-path connecting и to w in 3D, with no edge in [u, w]. This
edge path has combinatorial length at least a = d(u, w). Because m = k/N, if any
of the edges of the path lie in Ti = дР then they lie on [и, и'] or [w, w'], and hence
there are at most 2k of them. So since the boundary of each face in Ρ contains at
most 3N edges, there are at least (a — 2k)/(3N) faces in Ρ \ D, that abut this path,
and hence at least this many faces in Di+l \ D,. Summing, we get a lower bound of
m
—(a - 2k)
3NK '
on the number of faces in the ^-neighbourhood of [u, w] in P. Since m = KN, this
simplifies to K(a — 2k). Similarly, we get lower bounds of Κ(β — 2k) and Κ(γ — 2k) on
the number of faces in the ^-neighbourhoods of [и', и"] and [u/, w"] respectively. By
construction of Ti, these neighbourhoods are disjoint, so we may add these estimates
to obtain a lower bound on AteaN(Ti):
AreaN(H) > K(a + β + γ) - 6kK.
To improve this lower bound we note that dDm \ Ή intersects the £-neighbourhood
of иЬиЫ(и, Ti — [u,w])> n — 2k. It follows that there is an arc A+ of combinatorial
length at least η — 3k in dDm — Ti and hence at least (n — 3k)/(3N) faces of Ρ that
abut this arc but which do not lie in the neighbourhoods of [и, w], [u', u"], [w", w']
considered above.
Thus we obtain a lower bound
η — 3k
Area„(H) > K(a + β + γ) - 6kK + ——-.
3N
But Ti has length l(Ti) = a + β + γ + 6k, and according to the linear isoperimetric
inequality for X,
AreaN(Ti) < К 1(H).
Thus,
η-3k
< l2kK.
3N ~
К and N are constant and k = 3KN2, so this inequality is clearly nonsense if η is
large. Thus we have bounded η and therefore can deduce that X is 5-hyperbolic,
where 5 depends only on К and N. D

422 Chapter III.H 5-Hyperbolic Spaces and Area
Subquadratic Implies Linear
It follows easily from the Flat Plane Theorem and the characterization of hyperbolic
spaces by the linearity of their isoperimetric functions (2.7 and 2.9) that the optimal
isoperimetric inequality for any cocompact СAT(0) space is either linear or quadratic.
The following result shows that this dichotomy holds in much greater generality.
This insight is due to Gromov [Gro87], and was clarified by Bowditch [Bow95a],
Ol'shanskii and others (see [0191], [Paps95a] and references therein). Following
Bowditch, we consider notions of area that satisfy the following axiomatic scheme:
2.10 Definition. Let c\, сг, сз be rectifiable curves which have a common initial
point and a common terminal point. Let y,· denote the concatenation c,c,+ i, where the
overline denotes reversed orientation and indices are taken mod 3. Then, {γχ, γ2, уз}
is said to be a theta curve spanned by C\, Ci, C3. Under the same circumstances, we
say that γ\ and уг are obtained by surgering y3 along C2.
Let X be a geodesic space and let Ω be a set of rectifiable loops in X. We consider
area functionals A : Ω —> K+ such that:
(Al) (Triangle inequality) If yi, γι, Уз € Ω form a theta-curve, thenA(yi) < А(уг)+
МП)-
(A2) (Quadrangle inequality) There is a constant К > 0 such that if у e Ω is
the concatenation of four paths, γ = С\С2С-$сА, then Α(γ) > Kd\d2 where
d\ = u?(im(ci), гт(сз)) and di = d(im(c2), im(c4)).
The usual notion of infimal area for spanning discs of piecewise smooth curves in
complete Riemannian manifolds satisfies these axioms. A counting argument closely
analogous to the proof of (2.9) shows that for the set of edge-loops in a metric graph
with integer edge lengths, the notion of area defined in (2.1) satisfies axiom (A2). It
follows that the notion of area that one gets by counting vertices instead of faces also
satisfies this axiom (see 2.3(1)), and this latter notion also satisfies (Al), trivially.
Thus, in the light of (2.9), the following theorem implies in particular that if a metric
graph with integer edge lengths satisfies a subquadratic isoperimetric inequality, then
the graph is 5-hyperbolic.
A function/ : [0, 00) —> [0, 00) is said to be subquadratic if it is c^je2), that is
Hnv+oo/OiO/jE2 = 0. A class of loops Ω is said to satisfy a subquadratic isoperimetric
inequality with respect to an area functional A if
A,nW = sup{A(y) Ι γ e Ω, l(y) < x]
is a subquadratic function.
2.11 Theorem (Subquadratic Implies Linear). Let X be a metric space, let Ω be a
class of rectifiable loops in X that is closed under the operation of surgery along
geodesic arcs, and suppose that A : Ω —> R+ satisfies axioms (Al) and (A2). IfQ,

Subquadratic Implies Linear 423
satisfies a subquadratic isoperimetric inequality with respect to A then it satisfies a
linear isoperimetric inequality.
Again following Bowditch, we present the proof in the form of two technical
lemmas. There is no loss of generality in assuming that К = 1 in (A2), and we shall
do so in what follows. Let/(jc) =/a,q(x) be as above.
2.12 Lemma. For every χ e [0, oo) there exist ρ, q e [0, oo) such that:
fix) </(p) +/(<?),
3
p,q <-jc + 3v/W,
p + q <x + 6^/f(x).
Proof. In order to clarify the idea of the proof, we only consider the case where the
supremum in the definition of/(jc) is attained, by γ say. (The general case follows by
an obvious approximation argument.) Let Σ с Sl χ Sl be the set of points (t, u) such
that the restriction of γ to each of the two subarcs with endpoints {t, u} has length
is at least jc/4. Let L = rcnn{d(y(t), y(u)) | (t, u) e Σ} and (i0, m) е Σ be such
that d(y(t0), у(щ)) = L. Let γ+ and y_ be the paths obtained by restricting γ to the
connected components of Sl — {t, u}; these are chosen so that Z(y_) < l(y+) < 3x/A
and oriented so that each has initial point γ (to) and terminal point y(uq).
Consider the theta-curve spanned by γ+, γ~. and a choice of geodesic
segment [γ(ί0), у(и0)]; let β+ and /L be the loops obtained by surgering γ along
iy(to),Y(uo)]- Let a, b e [y(to), y(u0)] be such that ά(γ(ί0),α) = d(a, b) =
d(b, y(uq). Let a\, 5, аг be geodesies whose images are the subarcs [y(to), a], [a, b],
iy(to), Y(uo)] of [γ(to), y(u0)], respectively.
By construction d(imau 1та2) = L/3. We claim that d(im8, imy+) > L/3. Once
we have proved this claim, by Axiom A2 we will have Α(β+) > (L/3)2, whereas by
Axiom Al:
f(x) = Α(γ) < Α(β+) + A(/3_) < f(l(p+)) +Μβ-))·
It therefore suffices to let/? = 1(β+) and q = Z(/L).
It remains to prove the claim. Given ζ = y(v) e imy+ and i e im5, we
divide y+ into subarcs σ\ and σι with endpoint z. Without loss of generality we
may suppose that 1(σ\) < 1(аг). Note that 1(σ\) + 1(σι) + 1(γ~) = 1(γ) = χ while
ΚΥ~) ^ Κσ\) + Κσι)- Since 1(σ\) < 1(σι) and Z(y_) < χ/2, we see that (v, щ) е
Σ. Thus d(z, y(uq)) > L = d(y(to), y(uq)). Since i e [a, y(u0)] it follows that
d(z, z!) > d(z, Y(u0)) - d(y(u0), a)>L- 2L/3. D
2.13 Lemma. Let g : [0, oo) —> [0, oo) be an increasing function and suppose that
there exist constants к > 0 and λ e (0, 1) such that for every χ e [0, oo) one can
find ρ, q e [0, oo) with

424 Chapter ΠΙ.Η 5-Hyperbolic Spaces and Area
g(x) <g(p) + g(<lX
p,q <λχ + ky/g(x),
p + q <x + kyfg(x).
Ifg(x) = o(x2) then g(x) = 0{x).
Proof. By replacing g with a constant multiple we may assume that к = 1. Let
μ = (1 -\- λ)/2 and fix xq e [0, oo) such that if χ > xq then g(x) < (1 — μΫχ2. Note
that if χ > xq then p, q < λχ + (1 — μ)χ = μχ. Increasing xq if necessary, we may
assume that it is bigger than 1.
Let h(x) = g(x)/x. If x> xq then xh{x) < ph{p) + qh{q\ so
h(x) <-h(p) + - h(q).
χ χ
Without loss of generality we may assume that h(q) < h(p), so
cm) m<{p-±A т<(?±Ш\ m<li + JM\ m.
Thus if χ > xq, then there exists ρ e [1, μχ] with h(x) < (1 + -Jh{x)/x) h(p). We
want to show that it h{x) = o{x) ('little oh'), then h is bounded.
For this we fix ε > 0 and χι > 0 so that h(x) < ε1 χ for all χ > χχ. Let
В = max{h(x) | 1 < χ < jt|}. We have, for χ > x\, that h{x) < (1 + s)h(p) with
ρ < μχ. By using this inequality η < 1 + \og{x/x\)/ log(l/μ) times, we obtain:
h{x) < 5(1 + ε)" < 5(1 + ε){χ/χχΥ,
where r = log(l + ε)/ log(l^). We choose ε small enough to ensure that r < 1.
Fix s > 0 with r < 1 — 2s. Then, h(x) = o{xl~~ls), and so we may choose x2 with
h(x) < xl~~2s for all χ > x2 > x\. But then, by equation (13.1):
h(x)<(l+x~s)h(p).
Again we iterate our estimate, к times,
h(x) < C{\ + jTs)(1 + μ"Ξχ"Ξ) ···(!+ M~fc^~fa),
where С = m&x{h(y) \ 1 < у < x2] and к is the greatest integer such that μ1[χ > x2.
Therefore, since log(l + у) < у for all у > 0,
log й« < log С + jTs(1 + μ"'5 + ■ ■ ■ + μ"*'5)
(μ^)"1 - μ^"1
= logC +
< logC +
1 - μ1
1 -μ5
Thus /г is bounded. D

More Refined Notions of Area 425
More Refined Notions of Area
The notions of area considered above are rather crude but are well suited to our
purposes since they are technically simple and relatively stable under quasi-isometry
— they describe something of the large-scale geometry of spaces, which is the main
theme of our book. However, there is a striking body of work by the Russian school
which deals with area in a more sophisticated and local sense. In this paragraph
we describe some of the main concepts and results from this work, without giving
proofs. For a more complete introduction, see the survey article of Berestovskii and
Nikolaev [BerN93].
The following definition of area was given by Nikolaev [Ni79] and is adapted
from an earlier definition of Alexandrov [Ale57]. The basic idea is to define the
area of a surface to be the limiting area of approximating polyhedral surfaces built
out of Euclidean triangles. Thus, in particular, this coincides with Lebesgue area for
surfaces in Euclidean space.
2.14 Definitions. Let X be a geodesic space and let D2 denote the unit disc in the
Euclidean plane. By definition, a parameterized surface in X is a continuous map
/ : D2 —> X, and a non-parameterized surface is the image F of such a map.
Let/ be a parameterized surface. Intuitively speaking, a complex for the surface
f is a polyhedral approximation to /. More precisely, given a triangulation of D2
with vertices A[,... , A„, we choose points A,· e X such that A,· = Aj if and only if
/(A,·) = /(Ay); if the vertices A,, Aj are adjacent, we connect A,· to A,- by a geodesic
in X; thus we obtain a set of triangles in X. This set of triangles is called a complex
for the surface f and the A,· are called the vertices of the complex. Let Φ(/) denote
the set of sequences of complexes (Фт) which have the property that asm^ oo the
maximum of the distances ύ?(Α,·,/(Α,·)) tends to zero and the maximum of the lengths
of the sides of the triangles in Фт tends to zero.
The area of a complex Ф, denoted σ(Φ), is obtained by replacing the triangles
in Φ with Euclidean comparison triangles and summing the area of the latter. For a
parameterized surface/ one defines:
Areatf) := inf{lim infа(Фт) | (Фт) е Ф(/)}.
And for a non-parameterized surface F one defines:
Area(F) := inf{Area(/) \f(D2) = F}.
(Of course, these areas may be infinite.)
One has the following basic properties:
2.15 Lemma. Let X and Υ be geodesic spaces.
(1) (Semicontinuity) Iffn : D2 —> X is a sequence of parameterized surfaces in X
andfn —> / uniformly, then Area(/) < lim inf Area(/„).

426 Chapter ΠΙ.Η 5-Hyperbolic Spaces and Area
(2) (Kolmogorov's Principle) If ρ : X —> Υ does not increase distances then for
every parameterized surface in X one has:
Area(p of) < Area(/).
Let X be a CAT(/c) space and let с : Sl —» X be a rectifiable loop in X (of length
< 2π/*Jk if /c > 0). А ги/ес? surface bounded by с is a parameterized surface
/ : D2 —» X obtained by choosing a basepoint о e Sl and defining/ so that for each
θ e 51 the restriction of/ to the Euclidean segment [ο, Θ] is a linear parameterization
of the unique geodesic segment joining c(o) to c(#) in X.
The following result, which is due to Alexandrov [Ale57], follows from the Flat
Triangle Lemma (II.2.9) and Kolmogorov's principle.
2.16 Proposition. The area of any ruled surface bounded by a triangle Δ
(ofperimeter less then 2π/^/κ) in a CAT(/c) space is no greater than the area of the comparison
triangle Δ in M2, and is equal to it if and only if A and Δ' are isometric.
From this Alexandrov deduces:
2.17 Theorem. Let X be a CAT(/c) space and let с be a rectifiable loop in X. (If
к > 0 assume that 1(c) < 2π/«/κ.) Then, the area of any ruled surface bounded by
с is no greater than the area of a disc in M2 whose boundary is a circle of length
1(c). Moreover, equality holds only if the disc and the ruled surface are isometric.
Notice that by reference to the classical cases E2 and H2, this result gives a
quadratic isoperimetric inequality for loops in CAT(O) spaces and a linear isoperi-
metric inequality in CAT(— 1) spaces (with respect to the notion of area defined in
(2.14)). In the same vein, the following theorem of Reshetnyak [Resh68] can also be
viewed as a strong isoperimetric result (via Kolmogorov's principle).
A convex domain V С Μ2 is said to majorize a rectifiable loop с in a metric
space X if there is a non-expanding map V —> X which restriction to 3 У is an arc
length parameterization of c.
2.18 Theorem. For any rectifiable loop с in а САТ(к) space X (with 1(c) < 2π/^/κ
ifK>0) there exists a convex domain V С M2 which majorizes c.
2.19 Remark (Plateau's Problem). Plateau's problem asks about the existence of
minimal-area fillings for rectifiable loops in a given space (see [Alm66]). Nikolaev
[Ni79] solved Plateau's problem in the context of CAT(/c) spaces.

The Boundary ΘΧ as a Set of Rays 427
3. The Gromov Boundary of a (5-Hyperbolic Space
In this section we describe the Gromov boundary dX of a 5-hyperbolic space X. If
X is a proper geodesic space, then there is a natural topology on X U dX making it
a compact metrizable space and there is a natural family of "visual" metrics on dX.
The topological space dX is an invariant of quasi-isometry among geodesic spaces,
as is the quasi-conformal structure associated to its visual metrics. If X is a proper
CAT(— 1) space, or more generally a CAT(O) visibility space, the Gromov boundary
is the same as the visual boundary and the topology on X U dX is the cone topology
(II.8.6).
This section is organised as follows. First we shall consider the case where X is
a geodesic space, describing dX in terms of equivalence classes of geodesies rays
as we did for CAT(O) spaces in Chapter II.8. We shall then interpret dX in terms of
sequences of points that converge at infinity. As well as providing a definition of dX
in the case where X is not geodesic, the language of sequences provides a vocabulary
with which to discuss the extension of the Gromov product from X to X U dX. Taking
the case X = M" as motivation, we shall explain how this extended product can be
used to define metrics on dX.
The Boundary dX as a Set of Rays
Recall that two geodesic rays c, d : [0, oo) —> X in a metric space X are said to be
asymptotic if sup, d(c(t), c'{t)) is finite; this condition is equivalent to saying that the
Hausdorff distance between the images of с and d is finite. We define quasi-geodesic
rays to be asymptotic if the Hausdorff distance between their images is finite. Being
asymptotic is an equivalence relation on quasi-geodesic rays. We write dX to denote
the set of equivalence classes of geodesic rays in X and we write dqX to denote the set
of equivalence classes of quasi-geodesic rays. In each case we write c(oo) to denote
the equivalence class of c.
3.1 Lemma. IfX is a proper geodesic space that is δ-hyperbolic, then the natural
map from dX to dqX is a bijection.
For each ρ e X and ξ e dX there exists a geodesic ray с : [0, oo) —> X with
c(0) = ρ and c(oo) = ξ.
Proof. The natural map dX —> dqX is obviously injective. To prove the remaining
assertions, given ρ e X and a quasi-geodesic ray с : [0, oo) —> X, let cn be a
geodesic with c„(0) = ρ that joins ρ to c(n). Since X is proper, a subsequence of the
c„ converges to a geodesic ray c^ : [0, oo) -> X (by the Arzela-Ascoli Theorem
(1.3.10)). Theorem 1.7 provides a constant к such that the Hausdorff distance between
c([0, n\) and the image of c„ is less than k; thus we obtain a bound on the Hausdorff
distance between с and Coq. Π

428 Chapter ΙΠ.Η 5-Hyperbolic Spaces and Area
3.2 Lemma (Visibility of ЭХ). If the metric space X is proper, geodesic and 5-
hyperbolic, then for each pair of distinct points ξχ,ξι e ЭХ there exists a geodesic
line с : Μ. —> X with c(oo) = ξι and c(—00) = £2·
Proo/ Fix ρ e X and choose geodesic rays C\, c2 : [0, 00) —» X issuing from/? with
C\ (00) = ξι and Сг(оо) = ξ2. Let Γ be such that the distance from C\ (Γ) to the image
of C2 is greater than 5. For each η > Τ we choose a geodesic segment [ci (и), Сг(и)]
and consider the geodesic triangle with sides ci([0, n\), с2([0, и]) and [c\{n), c2(n)].
Since this triangle is 5-slim, [c\ (n), c2(n)] must intersect the closed (hence compact)
ball of radius 5 about C\{T\ at a point pn say. By the Arzela-Ascoli Theorem, as
η —> oo a subsequence of the geodesies [p„, Сг(и)] С [ci(«), c2(n)] will converge.
By passing to a further subsequence we may assume that the sequence [c\(n), c2(n)]
converges. The limit is a geodesic line which we call c.
Since each [c\ (и), Сг(и)] is contained in the 5-neighbourhood of the union of the
images of c\ and c2, the image of с is also contained in this neighbourhood. Thus the
endpoints of с are ξ\ and ξ2. Π
3.3 Lemma (Asymptotic Rays are Uniformly Close). LetXbeaproperb-hyperbolic
space and let c\,c2 '■ [0, 00) -^ X be geodesic rays with c\(00) = c2{oo).
(1) 7/ci(0) = c2(0) ί/геи rf(ci(i), c2(0) < T£ for all t > 0.
(2) /и general, there exist 7Ί, Г2 > 0 гас/г ί/ϊαί d{c\{T\ + t), C2(T2 + t)) < 58 for
all t > 0.
Prao/ (1) follows immediately from (1.15).
In order to prove (2), we apply Arzela-Ascoli to obtain a subsequence of the
geodesies cn = [ci(0), сг(и)] that converges to a geodesic ray c\ with c',(0) = ci(0).
Since the triangles Δ(οι(0), Сг(0), сг(и)) are 5-slim, all but a uniformly bounded
initial segment of each c„ is contained in the 5-neighbourhood of the image of C2,
and hence a terminal segment of c\ is also contained in this neighbourhood. In other
words, there exist T\, T2 > 0 with а(сг(Тг), c[{T\)) < 5 such that for all t > 0 one
can find f7 with d(c2(T2 +f), c\ {T\ + i)) < S. By the triangle inequality, ί and f7 differ
by at most 25. Thus for all t > 0 we have й?(с2(Г2 + 0. ^ (Γι +1)) < 35. And from
(1) we know that d(cx {T\ + t), c\ (Tx + t)) < 25 for all t > 0. D
3.4 Remark (Busemann Functions and Horospheres). Let X be a proper geodesic
space that is 5-hyperbolic and let с : [0, oo) —» X be a geodesic ray. The Busemann
function of с is £>c(je) : = lim sup^^ d{x, c(i))— t. (The triangle inequality ensures that
\bc{x)\ < d(c(0), x).) It follows from the preceding lemma that if d is a geodesic ray
with c(oo) = c'(oo), then |Ьс(х) — bc>(x)\ is a bounded function of χ e X. On the other
hand, if c(oo) φ c'(oo) then |^c(jc) — £ν(χ)| is obviously not a bounded function.
This observation allows one to construct ЭХ as a space of equivalence classes of
Busemann functions in analogy with (II.8.16) — see section 7.5 of [Gro87].
If we choose a basepoint ρ e X and alter each bc be an additive constant so that
bc(p) = 0, then the (modified) Busemann functions associated to asymptotic rays

The Topology on X U dX 429
differ by at most a fixed multiple of 5. (To remove this mild ambiguity one might take
the supremum or infimum over all rays с with c(oo) = ξ to define "the" Busemann
function of ξ.) Horospheres and horoballs about ξ (the level sets and sub-level sets
of Busemann functions) are well-defined in the same sense. See [CDP90], Chapter
8 in [GhH90] and [Gro87] for applications.
The Topology οηΐυθΧ
The description that we are about to give of the topology on X = X U dX is closely
analogous to (II.8.6). As was the case there, it is convenient to consider generalized
rays.
Notation. A generalized ray is a geodesic с : / —> X, where either / = [0, R] for
some R > 0 or else / = [0, oo). In the case / = [0, R] it is convenient to define
c(t) = c(R) for t e [R, oo]. Thus X := XU dX is the set (c(oo) | с a generalized ray}.
3.5 Definition (The Topology on X = XUdX). Let X be a proper geodesic space that
is 5-hyperbolic. Fix a basepoint ρ e X. We define convergence in X by: xn —> χ as
η —> oo if and only if there exist generalized rays c„ with c„(0) = /? and c„(oo) = xn
such that every subsequence of (c„) contains a subsequence that converges (uniformly
on compact subsets) to a generalized ray с with c(oo) = x. This defines a topology
on X: the closed subsets В с X are those which satisfy the condition [xn ей, V« >
0 and x„ —> jc] => jc e 5.
3.6 Lemma (Neighbourhoods at Infinity). Let X and ρ eX be as above. Let k > 25.
Let cq : [0, oo) —> Xbea geodesic ray with cq(0) = /? and for each positive integer n
let V„(co) be iAe sei of generalized rays с such that c(0) = /? α«ύ? d(c(n), co(n)) < k.
Then { V„(cq) | и е N} w a fundamental system of (not necessarily open)
neighbourhoods ofc(oo) in X.
Proof. Let d be a ray in X with c^O) = /?. It follows from (3.3) that c'(oo) = cq(oo)
if and only if c'(n) e V„(co) for all η > 0. And if c, is a sequence of generalized rays
in X with c,(0) = /? and c,· ^ V„(cq), then by the Arzela-Ascoli theorem there is a
subsequence сад that converges to some с ^ V„(co), hence c,- does not converge to
Co in X. Thus C; —> Co in X if and only if for every η > 0 there exists N„ > 0 such
that с ι e У„(со) for all i> Nn. D
3.7 Proposition. Lei X be a proper geodesic space that is b-hyperbolic.
(1) 77ге topology on X = X U dX described in (3.5) is independent of the choice of
basepoint, and
(2) ifX is a CAT(0) space this is the cone topology (II.8.6).

430 Chapter ΙΠ.Η 5-Hyperbolic Spaces and Area
(3) X <^> X is a homeomorphism onto its image and дХ С X is closed.
(4) X is compact.
Proof. (1) follows easily from Lemmas 3.6 and 3.3(2).
Suppose that X is CAT(O). In (II.8.6) we described a fundamental system of
neighbourhoods U(cq, r, ε) for cq(oo) e dX. Fix such a neighbourhood. Also fix
к > 25. If Ν > η > rand с : [0, Ν] -> X lies in Vn(co) (the neighbourhood of cq(oo)
defined in (3.6)), then d(c(r), Co(r)) < r/nd(c(n), Co(n)), because of the convexity
of the metric onX. It follows that if и is sufficiently large, then V„(co) С U(cq, r, ε).
Conversely, if ε < к then t/(co, r, ε) С Vr(co). This proves (2).
(3) Let χ e X and let cn : [0, Tn] —> X be a sequence of geodesies issuing
from a fixed point ρ e X. By the Arzela-Ascoli theorem, if c„(T„) —> jc as и —> oo
then every subsequence of (c„) contains a subsequence that converges to a geodesic
joining ρ to x. Conversely, if (c„(T„)) does not converge to χ as и —> oo, then some
subsequence of (c„) does not contain a subsequence that converges to a generalized
ray with terminal point x. Thus Χ <^-> Χ is a homeomorphism onto its image. X is
obviously open in X.
(4) The balls B(x, r), with r > 0 rational, form a fundamental system of
neighbourhoods about χ eX с X. This observation, together with the preceding lemma,
shows that the topology on X satisfies the first axiom of countability. Thus it suffices
to prove that X is sequentially compact, and this is obvious by Arzela-Ascoli. D
3.8 Exercise. Let X be a geodesic space that is proper and hyperbolic. Prove that the
natural map dX —> Ends(X) is continuous and that the fibres of this map are the
connected components of dX.
3.9 Theorem. Let X and X' be proper δ-hyperbolic geodesic spaces. Iff : X —> X'
is a quasi-isometric embedding, then c(oo) и (fo c)(oo) defines a topological
embedding/g : dX —> ЭХ'. Iff is a quasi-isometry, thenfs is a homeomorphism.
Proof. Suppose that/ : X —> X' is a (λ, ε)-quasi-isometric embedding, fix ρ e X
and let p' =f(p).lfc\, c2 : [0, oo) -> X are geodesic rays with c,(0) =p, then/oci
and/ о С2 are (λ, e)-quasi-geodesics with/ о с,(0) = p'. These quasi-geodesic rays
are asymptotic (i.e. their images are within finite Hausdorff distance of each other)
if and only if c\ and cj. are asymptotic. Thus, in the light of (3.1), /g is well-defined
and injective.
Fix к > 25. In order to see that/g is continuous, for i = 1, 2 we choose a
geodesic ray c\ with cj.(0) = p' and c{(oo) = / о с,(оо). The Hausdorff distance
between c'l and / о с,· is bounded by a constant К depending only on 5, λ and ε
(see 1.8). If C\{ri) lies in the ^-neighbourhood of the image of с2, then/o c\(n) lies
in the (Xk + e)-neighbourhood of the image of/ о сг and hence c',(io) lies in the
(2K + Xk + ε)-neighbourhood of the image of c'2 for some to > К — ε + η/Χ. Let
η' denote the least integer greater than η — (К — ε + η/Χ + 5). As in (1.15) we have
d(c[(t), c'2(t)) < 25 for all t < n'. Thus, in the notation of Lemma 3.6, c\ e У„(сг)

The Topology on X U ΘΧ 431
implies c\ e Vn>(c'2), and hence/3 is continuous. A similar argument shows that
/gc(oo) ы> c(oo) is also continuous.
If/ is a quasi-isometry then it has a quasi-inverse/' : X' —» X (see 1.8.16), and
since d(f'f(x), x) is bounded, (//)a = //a is the identity on dX. D
3.10 Corollary. The real hyperbolic spaces Ш" and Mm are quasi-isometric if and
only ifm — n.
Proof. ЗИ" is an (n — l)-dimensional sphere. D
dX as Classes of Sequences
Let X be a CAT(— 1) space with basepoint ρ and consider a sequence (xn) in X. In
order for there to exist a point ξ in the visual boundary of X such that xn —> £, it
is necessary and sufficient that the distance from ρ to the geodesic [x,, xj] should
tend to infinity as i,j —> oo (see II.9.30). And as we indicated after (1.19), this is
equivalent to saying that (jc,· · xj)p —> oo as i,j —> oo. Using the description of the
basic neighbourhoods of points at infinity given in (3.6) it is easy to generalize this
observation:
3.11 Exercise. Let X be a geodesic space that is proper and 5-hyperbolic. Fix ρ e X.
Show that a sequence (x„) in X converges to a point of 31 С X if and only if
(jc, · xj)p —> oo as i,j —> oo.
Motivated by this exercise, we define:
3.12 Definition. Let X be an arbitrary (S)-hyperbolic space. A sequence (x„) in X
converges at infinity if (jc, · xj)p —> oo as ij —> oo. Two such sequences (xn) and
(y„) are said to be equivalent if (x-, ■ yj)p —> oo as i,j —> oo. The equivalence class
of (jc„) is denoted limjc„ and the set of equivalence classes is denoted dsX. (These
definitions are independent of the choice of basepoint p.)
Remark. The "equivalence" of sequences defined above is obviously a reflexive and
symmetric relation, but for arbitrary metric spaces it is not transitive (consider E2
for example). However, it follows immediately from (1.20) that in hyperbolic spaces
this is an equivalence relation.
3.13 Lemma. IfX is a proper geodesic space that is δ-hyperbolic, then there is a
natural bijection dsX —> dX.
Proof. It follows from exercise 3.11 that (x„) ы> lim„ xn induces a well-defined map
ЭД —> dX that is injective. To see that this map is surjective, note that given any
geodesic ray с in X, the point lim c(n) e dsX maps to c(oo). D

432 Chapter III.H 5-Hyperbolic Spaces and Area
3.14 Example. In general dsX is not a quasi-isometry invariant among (non-geodesic)
hyperbolic spaces. For example, consider the spiral с : [0, oo) —> E2 given in polar
coordinates by c(t) = (t, log(l +i))· In (1-23) we noted that the image of c, with the
induced metric from E2, is not 5-hyperbolic. However, the intersection of im(c) with
any line through the origin in E2 is hyperbolic; indeed, since it is a subset of a line,
it is (O)-hyperbolic. Let S denote such an intersection.
We claim that although c~l(S) and S are quasi-isometric, their boundaries are
different. Indeed, since the spiral с crosses any half-line infinitely often, dsS has two
elements, whereas c~l(S) С [0, oo) has only one element.
Metrizing dX
In order to discuss metrics on theboundary of a hyperbolic spaceX, we need to extend
the Gromov product to dX. For this it is convenient to consider the boundary in the
guise of ЗД, and the symbol dX shall have that meaning throughout this section. If
X is a CAT(— 1) space, the product on the boundary can be defined by the simple
formula (limx, · lim^)p := Пт,,Дх, · yj)p (see 3.18). But for hyperbolic spaces in
general this limit may not exist (3.16).
3.15 Definition. Let X be a (S)-hyperbolic space with basepoint p. We extend the
Gromov product to X = X U dX by:
(x -y)p:= sup lim infOt, · y})p,
i,j—>-со
where the supremum is taken over all sequences (x,) and (yj) in X such that χ = lim xt
and у = lim yj.
The following example illustrates the need to take lim inf and sup rather than just
lim in the above definition.
3.16 Example. Let X be the Cayley graph of Ζ χ Ζ2 = (a, b \ [a, b] = b2 = 1).
Note that X is hyperbolic and that dX consists of two points (which we call a_ and
a+). Consider the following sequences: xn := ba~", yn = a", zn = ba". Define
wn to be equal yn if η is even and zn if η is odd. As η —> oo, the sequences yn, zn
and wn converge to a+ while xn converges to a_. For all positive integers i, j we
have (xj ■ yj)\ = 0 and (xt ■ zj)i = 1. Thus lim inf,^^(χί ■ yj)\ is not equal to
lim inf,·.;_».j»(jc,- ■ φι and in particular lim,J_+ooC':! ■ Щ)\ does not exist.
3.17 Remarks. Let X be a (S)-hyperbolic space and fix ρ e X. Let X = X U dX.
(1) (· )p is continuous on X (not X).
(2) (x ■ y)p = oo if and only if χ = у е дХ.
(3) For all x,y e X there exist sequences (xn) and (yn) in X such that χ =
limjc,,, у = limy,, and (x ■ y)p = lim„(xn ■ y„)p. If χ e X then one can take (xn) to be

Metrizing дХ 433
the constant sequence, and in the cases where jc or у belongs to dX, one can construct
(jc„) and (yn) by a diagonal sequence argument.
(4) For all x,y,zelwe have (jc ■ y)p > min{(jc · z)P, (z ■ y)p] - 25. To see this,
choose sequences in X with χ = liiruc,,, у = limy,, and ζ = Нтгл = Нтг^, such
that lim„(jc„ · zn)P = (x ■ z)p and \imn{z'n · yn)p = (z ■ y)p, then take liminf,j through
(xt ■ У])Р > min{(jc, · zdp, (Zi ■ Zj)p, (Zj ■ yj)p} - 25, noting that liminf,v(z, · zj)p = oo.
(5) For all x, у e dX and all sequences (jcJ.) and Ы) in X with χ = limxj and
у = limy/,
(x ■ y)p -25 < lim inf(jc; · y()p < (x ■ y)p.
The second inequality is true by definition, and the first is obtained by taking lim inf,-j
in(xj-;ypp > min{(;c|-jc,)p, (xt-yOp, (yj-yj)P}-2S, where (jc^andO^are as in remark
(3). (Note that (xj ■ x\)p and (>>, · yj)p tend to infinity as i,j -^> oo.)
(6) If X is proper and geodesic, and if (£„) is a sequence in dX (identified with
dsX as in (3.13)), then ξη —» ξ in the sense of (3.5) if and only if (ξη · £) —> oo as
и —> oo.
To see this, choose geodesies c„ issuing from /j with c„(oo) = ξη and define
xn = c„{n). Note that ξη -^> ξ as и —» oo if and only if jc„ —» ^, which is equivalent
to (xn ■ ξ) -> oo. As и -> oo, (^„ · jc„)p -> oo, and from (4) we have (ξη ■ ξ) >
min{(^„ · x„)p, (xn ■ ξ)ρ} - 25 and (xn ■ ξ) > min{(^„ · jc„)p, (£„ ■ $)p} - 25.
(1) Let X be a (5)-hyperbolic space and fix ρ e X. Show that there exists a
constant τ? such that if (jc,) and (yj) are sequences in X that converge in X, then
J? > ΙΟ · УД - (^i' · У/)р\ for all г, i'JJ' sufficiently large.
(2) Show that if X is a CAT(-l) space, ρ e X, and (jc,), (yf) are sequences in
X converging to ξ, ξ', then (ξ · £')ρ = lim,(jc, ■ y,)p. Show further that in the case
X = Ш" this limit is the distance from ρ to two of the internal points of the ideal
triangle A(p, ξ, ξ'), i.e. the points at which the inscribed circle meets the rays [ρ, ξ]
and [ρ, ξ']. (The third internal point of Δ(ρ, ξ, ξ') is the foot of the perpendicular
ftomp to [£,£']■)
(3) Fix 5 > 0. Show that there is a constant к such that if X is a proper geodesic
space that is 5-hyperbolic, then\d(p, im(c))—(ξ·ξ')ρ\ < Horallp eX, all£,£' e 3X
and every geodesic line с in X with c(—oo) = ξ and c(oo) = ξ'.
(4) (Uniform Structure on X). The purpose of this exercise is to indicate that,
for rather general reasons, if X is a proper 5-hyperbolic geodesic space, then X is
a compact metrizable space (cf. chapter 7 of [GhH90]). To see this, recall that if
X is a set then a family В of symmetric subsets В с Χ χ Χ form a base ("are the
basic entourages") for a uniform structure on X if each В е В contains the diagonal
Δ с X x X, for each В e В there exists Ε e В such that (jc, y), (y,z) e £ implies
(jc, г) e 5, and for all 5b 52 e В there exists i3e6 such that 53 с β,ηβ2.
The uniform structure is said to be separated if Ρ|β Β = Δ. A separated uniform
structure with a countable base gives rise to a metric by a well-known construction
(see sections II. 1 and IX.4 of [Bour53] or [Jam89]).

434 Chapter III.H 5-Hyperbolic Spaces and Area
Show that if X is a proper 5-hyperbolic geodesic space and ρ e X, then the sets
Br = {(ξ,ξ') Ι (ξ·ξ')ρ > r},withr > 0 rational, form a base for a separated uniform
structure on dX whose associated topology is that defined in (3.5). Describe a base
for a separated uniform structure on X.
Visual Metrics on dX
In this subsection we shall construct explicit metrics on the boundary of proper
5-hyperbolic spaces X. In order to motivate the construction we first consider the
exampleX = H".
3.19 Constructing Visual Metrics on Ш". If we fix ρ e H", then (ξ,ξ') м> Ζρ(ξ, ξ')
defines a metric on ЗИ" making it isometric to S"~'; since Zp describes the geometry
of ЗИ" as seen from p, it is called a visual metric on 3H". We wish to interpret Zp in
terms of the Gromov product and the distances between points and geodesies in Ш".
To this end, we apply the basic construction of (1.1.24) to the function ρ on
ЭИ" χ ЭИ" defined by
ρ(ξ,ξ') = ε-°^\
where ϋρ(ξ, ξ') is the distance from/? to the geodesic line in Ш" with endpoints ξ
and ξ'. Associated to ρ we have the (pseudo)metric
n-l
4,($,$'):=inf]rp&.&+i)>
i=0
where the infimum is taken over all chains (ξ = £0,..., ξη = ξ'), no bound on n.
According to (1.6.19(3)), tan(Z(£, £')/4) = e-D»^'\ \ia,b e [0, тг/4] then
tan(a + b) > tana + tanfr. And lim,_+o 7 tan г = 1. It follows from (1.3.6) that dp
is the length metric associated to άρ(ξ, ξ') := |Ζρ(£, ξ'). Since dp is itself a length
metric, we have dp = dp. Thus the visual metric Zp on ЗИ" can be constructed from
the function Dp. Moreover, since tan θ e [0,20]if0 e [0,7r/4],andtan|Zp(^, ξ') =
e~Di>^£\ we have
Le-тл·) < Ζρ(ξ, ξ') < Ae-0^
forall$,$' еЭИ".
According to exercise (3.18(3)), there is a universal bound on \Dp(i-, ξ')— (ξ -ξ')ρ |.
Hence there exist constants k\, &2 independent of ρ such that
k{e-m')P < Ζρ(ξ,ξ') < кге-1**'*
for all ξ, ξ' e ЭИ".
3.20 Definition. Let X be a hyperbolic space with basepoint p. A metric d on dX is
called a visual metric with parameter a if there exist constants k\,ki > 0 such that
kia-M* < ά(ξ,ξ') < k2a-<-^')"
ίοτΛΙξ,ξ' eX.

Metrizing дХ 435
In [Bou96] Marc Bourdon shows that if X is a proper CAT(—fe2) space, then for
each number a e (1, e6] and each ρ e X, the formula ^р(£, ξ') := a~^^">" defines
a metric on dX. However one cannot construct visual metrics on the boundary of
arbitrary hyperbolic spaces in such a direct manner. To circumvent this difficulty we
mimic the construction of (3.19).
The following discussion follows section 7.3 of [GhH90].
Constructing Visual Metrics. Let X be a hyperbolic space with basepoint p. Let
ε > 0 and consider the following measure of separation for points in ЭХ:
It is clear that ρε(ξ, ξ') = ρε(ξ', ξ). From remark 3.17(2) we see that ρε(ξ, ξ') = 0
if and only if ξ = ξ', and from remark 3.17(4) we see that
(#) Ρε(ξ,ξ') <(1 +ε')τηΆχ{ρε(ξ,ξ"),Ρε(ξ'\ξ')}
for all ξ, ξ', ξ" e дХ, where ε' = ε2δε - 1.
It may be that ρε does not satisfy the triangle inequality, but we can apply the
general construction of (1.1.24) with ρ = ρε to obtain a pseudo-metric; since we
wish to retain ε in the notation, we write dE rather than dp for this pseudo-metric.
Thus on dX we consider
η
4($,П = и*][>(&-1.6),
i=l
where the infimum is taken over all chains (ξ = ξο, ..., ξη = ξ'), no bound on п.
3.21 Proposition. Let X be a (S)-hyperbolic space. Let ε > 0 and let ε' = e2Ss — 1.
If ε' < ~J2 — 1, then dE is a visual metric on dX, indeed
(1-2ε')ρε(ξ,ξ')<άε(ξ,ξ')<ρε(ξ,ξ')
for all ξ, ξ' e дХ.
If X is a proper geodesic space, then the topology which de induces on dX is the
same as the topology described in (3.5).
Proof. The inequality d£ < ρε is obvious. The proof of the other inequality between
ρε and d£ is based on the standard technique for constructing a metric from a uniform
structure. Following [Bour53] (IX. 1.4) and [GhH90], we proceed by induction on n,
the size of chains (£o> · · ■, ξη), to prove that
η
(1-20α(£ο,&,)<Σα&-ι,&)-
i=l
For brevity we write S(p) = Yfi=l ρε(ξί-ι, ξι). The inequality is obvious if η = 1 or
S(n) > 1 — 2ε', so we suppose that η > 2 and S(n) < 1 — 2ε'. Let ρ be the greatest
integer such that S(p) < S(n)/2. (So S(n) - S(p + 1) < S(n)/2.) By our inductive

436 Chapter III.H 5-Hyperbolic Spaces and Area
hypothesis, both ρε(ξο, ξΡ) and ρε(ξρ+ι, £«) are no greater than S(n)/(2 - 4ε'). Also
Ρε{ζΡ> ζρ+ι) £ S(n)- And applying inequality (#) twice we have
Ρε(ξθ, ξη) < (1 + S'f ΠΙΆχ{ρε(ξ0, ξρ), ρε(ξρ, ξρ+ι), ρε(ξρ+ι, ξ,,)}.
Hence (1 - 2ε')ρε(ξ0, ξ„) < (1 + s'fS(n)max{l - 2ε', 1/2}. In order to complete
the induction, it only remains to note that (1 + ε')2(1 — 2ε') < 1 for all ε' > 0 and
(1 + ε')2 < 2 for all positive ε' < V2 - 1.
The topology associated to ds is the same as the topology denned in (3.5), because
by definition ρε(ξ, ξι) —> 0 as i —> oo if and only if (£,- ■ ξ)ρ —» οο, and by (3.17(6))
this is equivalent to convergence in the sense of (3.5). D
Finer Structure
We shall briefly describe, without giving proofs, the natural quasi-conformal structure
on the boundary of a (S)-hyperbolic space.
Let(X, d) be a metric space and let к > 1. A k- ring of d in X is a pair of concentric
balls (B(x, г), В(х, кг)). Following Pansu, we say that two (pseudo)metrics d\, dt on
X are quasi-conformally equivalent if there exist functions/i ,/2 : [1, 00) —» [1, 00)
such that for every fc-ring {B\,B2)otd\ (resp. di) there is an/i(fc)-ring of d% (resp. an
/2(A:)-ring of dx) (B\, S2) such that
ΒΊ с В\ с B2 с B'2.
A quasi-conformal structure on X is a class of metrics which are equivalent in
this sense, and a map between metric spaces φ : (X, d) —» (X', d') is called quasi-
conformal if d is equivalent to ^(x, y) := d'(tp(x), ф(у)).
Let X be a (5)-hyperbolic space. Note that all visual metrics on dX are equivalent.
Thus X has a canonical quasi-conformal structure, namely that associated to the
metrics ds described in Proposition 3.21. The following is a generalization of a
classical result of Margulis (see [GrPan91 ] section 3.10). It was first stated by Gromov
and a detailed proof was given by Bourdon [Bou96b].
3.22 Theorem. Let X and Υ be proper geodesic spaces that are δ-hyperbolic and
equip dX and 3 Υ with their canonical quasi-conformal structures. Iff : Υ —> X is
a quasi-isometric embedding, then the associated topological embedding ЗУ —> dX
(see 1.9) is a quasi-conformal map. ,
3.23 Corollary. 77ге canonical quasi-conformal structure on dX is an invariant of
quasi-isometry among hyperbolic spaces that are proper and geodesic.
3.24 Remark. In [Pau96] Paulin proves that under certain additional hypotheses
(which are satisfied if X is the Cayley graph of a finitely generated group for example),
the quasi-conformal structure on ЭХ uniquely determines the space X up to quasi-
isometry.

Metrizing дХ 437
There are many interesting aspects of the boundary of hyperbolic spaces that
we have not discussed at all. In particular we have not said anything about the
Hausdorff and conformal dimensions of the boundary, nor the action of Isom(X) on
dX and the structure of limit sets, nor the construction of measures at infinity and
their relationship to rigidity results. We refer the reader to [CDP90] and [GhH90] for
basic facts in this direction and [Gro87], [Gro93], [Pan89,90], [Bou95] and [BuM96]
for further reading.
Bowditch has given a topological characterization of groups whose Cayley graphs
X are 5-hyperbolic by examining the nature of the action of the group on ЭХ, and he
and S warup have shown that the structure of local cutpoints in dX tell one a great deal
about the graph of groups decompositions of the group (see [Bow98a], [Bow98b],
[S w96] and references therein). Also in the case where X is a Cayley graph, Bestvina
and Mess [BesM91] have explained how the dimension of dX is related to the virtual
cohomological dimension of the group. Various other notions of dimension for dX
are discussed at length in [Gro93].

Chapter ΙΙΙ.Γ Non-Positive Curvature
and Group Theory
We have already seen that one can say a good deal about the structure of groups which
act properly by isometries on С AT(0) spaces, particularly if the action is cocompact.
One of the main goals of this chapter is to add further properties to the list of things
that we know about such groups. In particular, in Section 1, we shall show that if a
group acts properly and cocompactly by isometries on а С AT(0) space, then the group
has a solvable word problem and a solvable conjugacy problem. Decision problems
also form the focus of much of Section 5, the main purpose of which is to demonstrate
that the class of groups which act properly by semi-simple isometries on complete
CAT(O) spaces is much larger and more diverse than the class of groups which act
properly and cocompactly by isometries on CAT(O) spaces; this diversity can already
be seen among the finitely presented subgroups of groups that act cocompactly.
The natural framework in which to address the issues that shall concern us in this
chapter is that of geometric and combinatorial group theory. We shall be concerned
mostly with the geometric side of the subject, regarding finitely generated groups
as geometric objects, as in Chapter 1.8. Within this framework, we shall examine
the extent to which the geometry of CAT(O) spaces is reflected in the large-scale
geometry of the groups which act properly and cocompactly on them by isometries,
and the extent to which the basic properties of such groups can be deduced directly
from features of the large-scale geometry of their Cayley graphs. Once the salient
features have been identified, results concerning such groups of isometries can be
extended to larger classes of groups whose Cayley graphs share these features.
This approach leads us to consider what it should mean for a finitely generated
group to be negatively curved, or non-positively curved, on the large scale. We saw
in the previous chapter that Gromov's notion of 5-hyperbolicity captures the essence
of negative curvature in a manner that is invariant under quasi-isometry. In Section
2 of the present chapter we shall discuss the algorithmic properties of groups whose
Cayley graphs are 5-hyperbolic (i.e. hyperbolic groups) and in Section 3 we shall
present further properties of these groups. Our presentation is phrased in such a way
as to emphasize the close parallels with results concerning groups that act properly
and cocompactly by isometries on С AT(— 1) spaces. The study of hyperbolic groups
is an active area of current research and our treatment if far from exhaustive.
In Section 4 we present the basic theory of semihyperbolic groups, following
[AloB95]. The theory of finitely generated groups which are, in a suitable sense,
non-positively curved is less developed than in the negatively curved (hyperbolic)

1. Isometries of CAT(O) Spaces 439
case. Nevertheless, we shall see that most of the results concerning groups which act
properly and cocompactly by isometries on CAT(O) spaces can be recovered in the
setting of semihyperbolic groups.
In Section 6 of this chapter we shall examine the circumstances in which the
gluing constructions from (II. 11 )can be used to amalgamate group actions on С AT(0)
spaces. In Section 7 we discuss the (non)existence of finite-sheeted covering spaces
of compact non-positively curved spaces.
1. Isometries of CAT(O) Spaces
Let us begin by compiling a list of what we already know about groups which act by
isometries on CAT(O) spaces.
A Summary of What We Already Know
1.1 Theorem. If a group Г acts properly and cocompactly by isometries on а С AT(0)
space X, then:
(1) Г is finitely presented.
(2) Г has only finitely many conjugacy classes of finite subgroups.
(3) Every solvable subgroup of Г is virtually abelian.
(4) Every abelian subgroup of Г is finitely generated.
(5) If Г is torsion-free, then it is the fundamental group of a compact cell complex
whose universal cover is contractible.
If Η is a finitely generated group that acts properly (but not necessarily
cocompactly) by semi-simple isometries on X, then:
(i) Every polycyclic subgroup of Η is virtually abelian.
(ii) Every finitely generated abelian subgroup of Η is quasi- is ometrically embedded
(with respect to any choice of word metrics).
(iii) Η does not contain subgroups of the form (a, 11 t~lapt = aq) with \p\ φ \q\.
(iv) If A = Z" is central in Η then there exists a subgroup of finite index in Η that
contains A as a direct factor.
(v) IfX is δ-hyperbolic then every element of infinite order γ e Η has finite index
in its centralizes
The class of groups which act properly and cocompactly by isometries on С AT(0)
spaces is closed under the following operations:
(a) direct products,
(b) free products with amalgamation and HNN extensions along finite subgroups,
(c) free products with amalgamation along virtually cyclic subgroups.

440 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
Proof. (1) is a special case of (1.8.11) and (2) was proved in (II.6.11). Parts (3), (4)
and (i) were proved in Chapter II.7, and (5) was proved in (II.5.13).
Part (ii) is a consequence of the Flat Torus Theorem (П.7.1), as we shall now
explain. By passing to a subgroup of finite index we can reduce to the case Z" =
Α <^-> Η. We choose a minimal generating set for A and extend this to a set of
generators for Η (thus ensuring that dH(a, a') < dA(a, a') for all a, a' e A). By the
Flat Torus Theorem, Min(A) contains an isometrically embedded subspace F = E"
on which A acts as a lattice of translations. Fix a basepoint xq e F. By applying the
Svarc-Milnor Lemma (1.8.19) to the action of A on F, we obtain a constant λ > 0
such that dx(a.x0, a'.xo) > XdA(a, α')ε for all a,a'eA (in fact one can take ε = 0).
And (1.8.20) yields a constant μ > 0 such that dn(a, a!) > μάχ(α.χο, a'.x0). Thus
ан{а, a!) > μλάΑ(α, α!) — με > μλαΉ(α, a') — με.
To prove (iii), first note that (a, t \ t~]apt = aq) is an HNN extension of Z, and
hence it is torsion free (6.3). In particular a has infinite order. The action of Η on
X is proper and semisimple, so if such a subgroup were to exist then a would be a
hyperbolic isometry. By (11.6.2(2)) we would have \ap\ = \aq\, and (11.6.8(1)) would
then imply that \p\ = \q\, contrary to hypothesis.
Part (iv) is (II.6.12). Statement (v) is a consequence of (iv) and (11.6.8(5)): from
(iv) weknow that Ся(у) has a subgroup of finite index of the form Κ χ (γ); according
to (11.6.8(5)), this acts on Min(y), which splits isometrically as Υ x M; because the
action is proper, if К were infinite then Υ would have to be unbounded; but X is
5-hyperbolic, so there is a bound on the width of flat strips in X.
Items (b) and (c) were proved in Chapter II. 11. To prove (a) one simply notes
that, given a proper cocompact action of a group Γ, by isometries on a space Xi: for
i = 1, 2, the induced action (yi, γι).{χ\ ,хг)'.= (γι -x\, γι-χι) of Γι χ Гг on X\ χ Χι
is also proper and cocompact. D
Decision Problems for Groups of Isometries
1.2 Dehn's Formulation of the Basic Decision Problems
Combinatorial Group Theory is the study of groups given by generators and defining
relations. This method of describing groups emerged at the end of the nineteenth
century. Much of the subject revolves around the three basic decision problems that
were first articulated by Max Dehn in 1912. Dehn was working on the basic problems
of recognition and classification for low-dimensional manifolds (see [Dehn87]). In
that setting, the key invariant for many purposes is the fundamental group of the space
at hand. When one is presented with the space in a concrete way, the fundamental
group often emerges in the form of a presentation. In the course of his attempts
to recover knowledge about fundamental groups (and hence manifolds) from such
presentations, Dehn came to realise that the problems he was wrestling with were
manifestations of fundamental problems in the theory of groups, which he formulated
as follows (see [Dehn 12b]).

Decision Problems for Groups of Isometries 441
"The general discontinuous group is given by η generators and m relations
between them, as defined by Dyck (Math. Ann., 20 and 22). The results of those
works, however, relate essentially to finite groups. The general theory of groups
defined in this way at present appears very undeveloped in the infinite case. Here
there are above all three fundamental problems whose solution is very difficult and
which will not be possible without a penetrating study of the subject.
1. The Identity [Word] Problem: An element of the group is given as a product
of generators. One is required to give a method whereby it may be decided in a finite
number of steps whether this element is the identity or not.
2. The Transformation [Conjugacy] Problem: Any two elements S and Τ of
the group are given. A method is sought for deciding the question whether S and Τ
can be transformed into each other, i.e. whether there is an element U of the group
satisfying the relation
S= UTU~l.
3. The Isomorphism Problem: Given two groups, one is to decide whether
they are isomorphic or not (and further, whether a given correspondence between
the generators of one group and elements of the other is an isomorphism or not).
These three problems have very different degrees of difficulty. [... ] One is already
led to them by necessity with work in topology. Each knotted space curve, in order
to be completely understood, demands the solution of the three above problems in a
special case."
1.3 Remark. Given the nature of our study, we should augment Dehn's last remark
with the observation that he was led to these problems while studying surfaces
and knots, and all surfaces of positive genus and all knot-complements (viewed as
compact manifolds with boundary) support metrics of non-positive curvature.
We should also remark that the word and conjugacy problems are closely related
to filling problems in Riemannian geometry (cf. [Gro93]).
Terminology. The explicit nature of the decision problems that we shall discuss here
is such that we shall not need to delve deeply into the question of what it means
for an algorithm or "decision process" to exist. (See [МШ71] for background.) For
completeness, though, we should mention that a group Γ with a finite generating set
A is said to have a solvable word problem if and only if the set of words w such
that w = 1 in Γ, and the set of words w such that w φ 1 in Γ, are both recursively
enumerable subsets of the free monoid опЛи А~'.
A finitely presented group has a solvable word problem if and only if its Dehn
function, as defined in (I.8A), is a computable function.
The main result of this section is:
1.4 Theorem. If a group Г acts properly and cocompactly by isometries on a CAT(O)
space, then its word and conjugacy problems are solvable.

442 Chapter Ш.Г Non-Positive Curvature and Group Theory
We shall treat the word and conjugacy problems separately, obtaining estimates
on complexity in each case. In the next section we shall see that the efficiency of the
solutions can be sharpened considerably if one assumes that the CAT(O) space on
which Г acts does not contain a flat plane. In Section 5 we shall see that the above
theorem does not remain valid if one replaces the hypothesis that the action of Г is
cocompact by the hypothesis that it is semi-simple.
The Word Problem
Let Г be a group acting properly and cocompactly by isometries on а С AT(0) space X.
Fixxo e XandletD > 0 be such that X is the union of the balls y.B(x0,D/3), γ e Γ.
1.5 Lemma. If Γ and D are as above, then A = {a e Γ | d(a.x0,x0) < D + 1}
generates Γ. And given γ e Γ, ifd(xo, γ.χο) <2D+l then γ = α^ατ,α^ for some
a, e Λ.
Proof. To say that Λ generates is a weak form of (1.8.10). We leave the proof of
the assertion concerning elements with d(xo, γ.χο) < 2D + 1 as an exercise for the
reader. D
Given a set Λ, to check if two words represent the same element of the free group
F(A) one simply looks at the reduced words obtained by deleting all adjacent pairs
of letters aaT'; the original words represent the same element of F(A) if and only if
the corresponding reduced words are identical. It follows that one can easily decide
if an arbitrary word belongs to a given finite subset of F(A). We shall solve the word
problem in Г by showing that in order to decide if a word represents 1 € Г, one
need only check if the word belongs to a certain finite subset of the free group on the
generators of Г.
We shall always write \w\ to denote the number of letters in a word w.
1.6 Proposition. Let Г and A be as in the preceding lemma, and let 1Z С F(A) be
the set of reduced words of length at most ten that represent the identity in Г. A word
w in the letters A±l represents the identity in Г if and only if in the free group F(A)
there is an equality
N
w = Y^xinx^,
/=1
where N < (D + l)|w|2, each rt e TZ, and \x,\ < (D + \)\w\. In particular Г has a
solvable word problem.
Proof. To each γ e Γ we associate a word σγ in the generators A as follows. Let
cY be the unique geodesic joining jco = cy (0) to y.jco in X. For each positive integer
i < d(x0, γ.χ0) let σγ{ϊ) e Γ be such that d(cy(i), ay(i).x0) < D/3, define ay(0) = 1
and σγ{ί) = γ for i > d(xo, γ.χο). Note that a, := ay(i — \)~ισγ{ϊ) e A. Define σγ

Decision Problems for Groups of Isometries 443
to be αϊ ... a„, where η is the least integer greater than d(xo, γ.χο). Note that у — σγ
in Г. We choose σ\ to be the empty word.
We fix у e Γ and b € A and compare σγ with σγ· where y' = yb. By appending
letters a,- that represent the identity if necessary, we may write σγ = a\ .. .a„ and
σγ< = a\ .. .a'n, wherew = n(y, γ') = max{|ay|, |ay-|}.Nowd(y.xo, γ'-xo) < D+ 1
(because b e A), so from the convexity of the metric onXwe have d(cY(i), Cy(i)) <
D+l for all/. Hence d(aY(i).xo, ау-(г).*о) < 2D+l.Foreachi wechoosea word ce(i)
of length at most 4 that is equal to σγ (г)-' σγ> (г). Note that a (i)~' α,·+1 a (i +1 )a -+,_ e
1Z. We choose а(и) to be b (the difference between γ and y') and a(0) to be the empty
word.
If we write pY{i) for the word a\ ... a, (the г'-th prefix of σγ), and similarly for
γ', and we define pY'(0) to be the empty word, then we have the following equality
in the free group on A (see figure Γ.1):
л(у,у')-1
(1.6.1) avba~} = Y[ pyii)\a(i)-lai+la(i+l)a',+ l-l\pY-(i)-1.
i'=0
Each of the words in square brackets belongs to TZ.
ρ (г) ~» .
σΊ(υ
->
η-φ
Fig. Γ.1 The Equality in Equation 1.6.1
Finally, we consider the given word w = b\ ... bm that represents the identity in
Γ, where each bt e A±x (so m = \w\). Let yo = 1 and let )} еГЬе the element
represented by έ>ι ...bj. Note that y„, = 1 and σ7ηι = σγα is the empty word. Trivially,
in F{A) we have the equality
(1.6.2)
By replacing each factor on the right hand side of this equality with the right hand
side of the previous equation (with у = у_\, b = bj, and у' = у,), we obtain the

444 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
desired expression for w as a product of conjugates of defining relations. We claim
that this product has the desired number of factors.
To see this, consider the image in X of Г under the map γ ы> γ.χ0. Under
this map the sequence of elements γ3 form a sequence of m points beginning and
ending at jco. Since each successive pair of elements differ by right multiplication by
a single element of Λ, their images in X are a distance at most D + 1 apart. Hence
d(Yj.xo, xq) < (D + l)\w\/2 forj= 1, ..., m. It follows that each of the integers
n(Yj, }j+\) in equation (1.6.1) is no greater than 1 + (£> + l)\w\/2. The final equality
for w is a product of m = \w\ terms, the 7th of which (coming from (1.6.1)) consists
of n()j, Yj+i) conjugates of relators. Thus we have less than (D + l)\w\2 factors in
total.
The conjugating elements in our final equality are the prefixes pYj (i) coming from
(1.6.1), and the lengths of these are bounded by the integers «(>?, >/+ι)- Π
1.7 Remarks
(1) The equality displayed in the statement of the proposition shows that w is in
the normal closure of 7£ с F{A) and hence (A | 72.) is a finite presentation for Γ.
(2) Proposition 1.6 states that the Dehn function for the presentation TZ с F(A),
as defined in (1.8 A), is bounded above by a quadratic function. It follows that the Dehn
function for any finite presentation of Г is bounded above by a quadratic function
(see [Alo90], H.2.5(5)).
Fig. Г.2 Constructing a van Kampen diagram with quadratic area
(3) The fact that one gets a quadratic bound on the number of factors on the right
hand side of the equality in the statement of this proposition is closely related to the
quadratic isoperimetric inequality for fillings in CAT(0) spaces (H.2.4).

Decision Problems for Groups of Isometries 445
Figure (Γ.2) illustrates the scheme by which relations were applied in the proof.
By removing faces whose boundaries map to the trivial path in Сд(Г) one obtains a
van Kampen diagram for w.
The Conjugacy Problem
The way in which we shall solve the conjugacy problem for groups that act properly
and cocompactly by isometries on CAT(O) spaces is motivated by the following
geometric observation.
1.8 Proposition. Let Υ be a compact non-positively curved space. If two closed
rectifiable loops cq, c\ : §' —> Υ are freely homotopic, then there is a homotopy
§' χ [0, 1] —> Υ from Co to c\ through loops c, such that l(ct) < max{Z(co), l{c\)}
(where I denotes length).
Proof. Parameterize both cq and c\ proportional to arc length. By hypothesis, there
is a homotopy Η : §' χ [0, 1] —> Υ from cq to c\. For each θ e §' we replace the
path t ы> Η(θ, t) by the unique constant-speed local geodesic pg : [0, 1] —> У in the
same homotopy class (rel. endpoints). Let ct(6) = pe(t).
We fix an arc [θ,θ'] in §' and in Y, the universal covering of Y, we
consider the geodesic rectangle obtained by lifting the concatenation of the four
paths Co|[6>,s']> Ρθ', Clips'] and ρθ, where the overline denotes reversed
orientation. By the convexity of the metric on Y, for sufficiently small |0 — θ'\ we have
d(c,(0), c,(0')) = d(pe(t),peit)) < (1 - f)d(co(0),co(0')) + fd(ci(0),ci(0')) (see
II.2.2andII.4.1). Hence l(ct) < (1 - t)l(c0) + tl(a). D
Free homotopy classes of loops in a connected space correspond to conjugacy
classes of elements in the fundamental group of the space. With this correspondence
in mind, we define an algebraic property motivated by the above proposition. In this
definition, the elements w~luWj e Γ play the role of the intermediate loops c, in
(1.8).
1.9 Definition (q.m.c). A group Γ with finite generating set Λ is said to have the
quasi-monotone conjugacy property (q.m.c.) if there is a constant К > 0 such that
whenever two words u, ν e F(A) are conjugate in Γ, one can find a word w =
a\.. .an with a, e A±l, such that w~luw = ν andu?(l, w~1uwj) < Kmax{\u\, \v\}
for i = 1,..., n, where w, = a\... at.
In this definition, the existence of К does not depend on the choice of generating
set A but its value does.
1.10 Lemma. If a group Г acts properly and cocompactly by isometries on а С AT(0)
space X, then Г has the q.m.c. property.

446 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
Proof. Fix xq e X. We choose generators Л for Г as in (1.5) and represent elements
γ e Γ by words σγ as in the proof of (1.6). We claim that there is a constant К > 0,
depending only on the parameters of the quasi-isometry γ м> γ.χο, such that if two
words и and υ are such that yuy~l = υ in Γ, then u> := σ7 satisfies the requirements
of (1.9).
To see this one uses the convexity of the metric on X and compares the geodesic
quadrilateral in X with sides [xq, γ.χο], [xo, u.xq], y.[xo-u.xo\ and [u.xq, yu.xo] to a
quadrilateral Q in the Cayley graph С^(Г). The vertices of Q are {1, у, и, у и = υγ},
two of its sides are labelled σγ and the other sides labelled и and v. We leave the
(instructive) verification of the details to the reader (cf. 4.9(3)). D
1.11 An Algorithm to Determine Conjugacy. Let Г be a group with finite
generating set A. Suppose that Г has a solvable word problem and also has the q.m.c.
property. Let the constant К be as in (1.9).
For each positive integer n, we consider the set B(n) of words in F(A) that have
length at most n. Because Г has a solvable word problem, given a pair of words
vi, i>2 € B(n) one can decide if there exists a e A±l such that a~lv\a = i>2 in Г; if
such an a exists we write ν ι ~ V2.
Consider the algorithmically constructed finite graph Q{ri) with vertex set B(n)
that has an edge joining ν ι to vi if and only if v\ ~ vi- The q.m.c. property says
precisely that two words и and υ are conjugate in Γ if and only if и and υ lie in the
same path connected component of Q{ri), where η = if тах{|и|, |υ|}. Thus we may
decide if и and υ represent conjugate elements of Γ.
Note that Q{n) has less than (2|.41)" vertices, and hence any injective edge-path in
G(n) has length less than (21A |)". Any path of length Ζ joining и to υ in Q(n) determines
a word of length I which conjugates и to υ in Γ, and therefore и is conjugate to υ in
Γ if and only if there is a word w of length < дтах11"1 Ml such that w~luw = υΐηΓ,
where μ = (2|_4|/.
1.12 Theorem. If Τ acts properly and cocompactly by isome tries on a CAT(O) space
X, then for any choice of finite generating set A there exists a constant μ > 0 such
that words u, ν e F(A) represent conjugate elements of Γ if and only if there is a
word w of length < дтах11"1>|] suc\l that w~luw = ν in Γ. In particular, Γ has a
solvable conjugacy problem.
Proof. Γ has a solvable word problem (1.6) and the q.m.c. property (1.10), so
algorithm (1.11) applies. Π

Decision Problems for Groups of Isometries 447
Conjugacy of Torsion Elements
It is unknown whether, in general, one can significantly improve the exponential
bound on the length of the conjugating element in (1.12). However one can do better
for elements of finite order.
1.13 Proposition. Let Υ be a group which acts properly and cocompactly by
isometries on a CAT(O) space X. If A is a finite generating set for Γ, there exists a finite
subset Σ of the free group F(A) and a constant К such that a word и e F(A)
represents an element of finite order in Г if and only if there exists σ e Σ and w e F(A),
with \u>\ < К \u\, such that w~luw = σ in Γ.
Proof. Choose a ball В = B(xq, D) big enough so that its translates by Г cover X. By
the Svarc-Milnor Lemma (1.8.19), there exist positive constants λ and ε such that
-d(xo, γ.χο) - ε < d(l, γ) < Xd{x0, γ.χ0) + ε,
λ.
for all у e Γ, where Γ is equipped with the word metric associated to A.
Let Σ be a subset of F(A) which maps bijectively under F(A) —> Γ to the union
of the stabilizers of the points of B. This set is finite because the action is proper.
Suppose that у e Γ is an element of finite order, represented by the word и е
F(A). We know that у fixes a point of X (II.2.8); choose a fixed point x\ closest to
jeo and let g e Γ be such that g~'.x\ e B. Note that g~lyg lies in the image of Σ. It
suffices to bound d{\, g) as a linear function of \u\.
Writing d(xo, x\) = p, we have:
d(g.x0, x0) < d(g.x0, x\) + d(xi,x0) <D + p,
and hence d{\, g) < λ(ρ + D) + ε. Thus it only remains to bound ρ by a linear
function of \u\ > d{\, γ).
Claim. There exist positive constants a and β such that for all torsion elements γ e Γ,
if d(xo, Fix(y)) = ρ > 1, then d(l, γ) > ар — β.
We write V\{y) to denote the set of points a distance exactly 1 from Fix(y).
Consider the (finite) set S С Σ consisting of elements σ such that V\{a) := V\{a) Π
B(xq,D + 1) is non-empty. Let La = inf{d(x, σ.χ) \ χ e V,(a)} > 0, and let
L = min{Z,CT | σ e S}.
Given a torsion element γ e Γ and a point χ e Vi (y), let у е Fix(y) be the point
closest to x, and fix h e Г such that h.y e B. Let σ = hyh~}; so h.x e V[(a). We
have d(x, γ.χ) = d(h.x, hyh~l.(h.x)) = d(h.x, a.(h.x)). Thus d(x, γ.χ) > L.
Now let ζ € X be any point such that ρ = d(z, Fix(y)) > 1, let у be the closest
point of Fix(y) and let χ be the point a distance 1 from у on the geodesic [y, z\- The
CAT(O) inequality for the triangle A(y, z, Y-z) gives d(z, γ.ζ) > ρ d(x, γ.χ) > pL.
To complete the proof of the claim, we cast jco in the role of г, recall that у ы> γ.χ0
is a (λ, ε) quasi-isometry, and let a = Ζ,/λ and β = ε. D

448 Chapter III.Γ Non-Positive Curvature and Group Theory
1.14 Corollary. Let Υ be a group with finite generating set A Suppose that Γ acts
properly and cocompactly by isometries on a CAT(O) space. Then:
(1) There is an algorithm to decide which words и е F(A) represent elements of
finite order in Г.
(2) There is a constant к such that in order to decide if two words и and υ in the
generators represent torsion elements of Γ that are conjugate, it suffices to
check if w~luw = ν in Γ for some word w with \w\ < km&x{\u\, \v\].
2. Hyperbolic Groups and Their Algorithmic Properties
We now set about the second main task of this chapter: we want to use the tools
of geometric group theory to study the structure of groups which act properly and
cocompactly by isometries on CAT(O) spaces and, more generally, groups which
in a strict metric sense resemble such groups of isometries. We begin with the 5-
hyperbolic case. In this section we shall use the results of the previous chapter (in
particular results concerning quasi-geodesics) to examine the algorithmic structure
of groups whose Cayley graphs are 5-hyperbolic (i.e. hyperbolic groups). We shall
describe solutions to the word and conjugacy problems for such groups — these are
considerably more efficient than the solutions described in the previous section —
and we shall explain a result of Jim Cannon which shows that geodesies in the Cayley
graphs of hyperbolic groups can be described in a remarkably simple algorithmic
manner (2.18).
Hyperbolic Groups
We saw in (H.1.10) that being hyperbolic is an invariant of quasi-isometry among
geodesic spaces. Hence the following definition does not depend on the choice of
generators (although the value of δ does).
2.1 Definition of a Hyperbolic Group. A finitely generated group is hyperbolic (in
the sense of Gromov) if its Cayley graph is a 5-hyperbolic metric space for some
5 >0.
2.2 Proposition. Every hyperbolic group is finitely presented.
Proof. This is a weak form of (2.6). We give an alternative (more direct) proof, the
scheme of which is portrayed in figures (Γ.1) and (Γ.2).
Fix a finite generating set A with respect to which Γ is 5-hyperbolic. Given a
word w e F(A) which represents 1 e Γ, consider the loop in the Cayley graph
С^(Г) that begins at 1 and is labelled w. Let w(i) be the j'th vertex which this loop
visits (i.e. the image in Г of the ith prefix of w). Let σ,- be a geodesic from 1 to w(i).
Suppose that w(i'+ 1) = w(i)b. Note that ύ?(σ,(ί), ai+ {(t)) < 2(5 + 1) for all integers

Dehn's Algorithm 449
i < \w\ and alii > 0 for which σ, (i) and σ,+i(i) are defined (H. 1.15). As inequation
(1.6.1), one can express σ&σ~^χ in F(A) as a product of conjugates of relations of
length at most (45 + 6). And from equation (1.6.2) it follows that w is also equal in
F(A) to a product of conjugates of relations of length at most (45 4- 6). D
Since much of the essence of a CAT(— 1) space is encoded in the geometry of its
geodesies, and since we know from the previous chapter that the geometry of quasi-
geodesics in hyperbolic spaces mimics the geometry of geodesies rather closely, a
great deal of the geometry of CAT(— 1) spaces ought to be transmitted to the (quasi-
isometric) groups which act properly and cocompactly on them by isometries. With
this in mind, when attempting to understand hyperbolic groups in general, it is often
useful to regard them as coarse versions of CAT(— 1) spaces. This viewpoint will be
the dominant one in Section 3.
An alternative perspective, which can be useful in motivating proofs, e.g. (2.8),
is to regard hyperbolic groups as fattened-up versions of free groups. After all, a
geodesic space is O-hyperbolic if and only if it is a metric tree, and if the Cayley
graph of a group is a tree then the group is free.
2.3 Remarks
(1) In a certain statistical sense (see [Gro87], [Cham93] and [0192]), almost all
finitely presented groups are hyperbolic.
(2) At the time of writing, it is unknown whether every hyperbolic group Γ acts
properly and cocompactly by isometries on some CAT(O) (or even CAT(— 1)) space.
In the torsion-free case, this would mean that Γ would be the fundamental group
of a compact non-positively curved space. In general Γ will not be the fundamental
group of a compact non-positively curved и-manifold with empty or locally convex
boundary, because such a fundamental group has cohomological dimension53 η (if
the boundary is empty) or η — 1, and there are hyperbolic groups of cohomological
dimension two, for example, that are not the fundamental group of any manifold of
dimension two or three — the construction of (II.5.45) gives many such groups.
Dehn's Algorithm
Dehn's algorithm is perhaps the most direct approach that one can hope for whereby
the information in a finite presentation is used directly to solve the word problem in
the group presented. It is the algorithm that Dehn used to solve the word problem in
Fuchsian groups [Dehn 12a].
2.4 Dehn's Algorithm for Solving the Word Problem. Given a finite set of
generators A for a group Γ, one would have a particularly efficient algorithm for solving the
word problem if one could construct a finite list of words u\,v\,U2,V2, . ■ ■ ,un,vn,
53 We refer to [Bro82] for basic facts concerning the cohomology of groups.

450 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
with щ — Vj in Г, whose lengths satisfy |υ,| < \щ\ and which have the property that
if a word w represents the identity in Г then at least one of the и, is a subword of w.
If such a list of words exists, then given an arbitrary word w one looks for a
subword of the form и,·; if there is no such subword, one stops and declares that w
does not represent the identity; if щ occurs as a subword then one replaces it with υ,
and repeats the search for subwords of the (shorter) word obtained from w — this
new word represents the same element of the group as w. After at most |io| steps
one will have either reduced to the empty word (in which case w = 1 in Γ) or else
verified that w does not represent the identity.
2.5 Definition. A finite presentation (A | 72) of a group Γ is called a Dehn
presentation if 72. = {wiuj-1,..., u„v~1}, where the words ui,vi,..., u„, vn satisfy the
conditions of Dehn's algorithm.
2.6 Theorem. A group is hyperbolic if and only if it admits a (finite) Dehn
presentation.
As we noted above, Dehn proved that Fuchsian groups admit Dehn presentations.
Jim Cannon extended Dehn's theorem to include the fundamental groups of all closed
negatively curved manifolds [Ca84]. Theorem 2.6 is due to Mikhael Gromov [Gro87].
(Cannon gave a proof in [Ca91] and alternative proofs can be found elsewhere,
e.g. [Sho91].)
Proof. Suppose that {A | 72) is a Dehn presentation for Γ and let ρ be the length
of the longest word in 72. Consider an edge-loop с of length η in the Cayley graph
Сд(Г). This is labelled by a word w in the generators and their inverses, and this
word represents the identity in Г. The Dehn algorithm of the presentation yields a
subpath (corresponding to a subword щ of w) which is not geodesic — there is a
shorter path with the same endpoints that is labelled щ.
Let c' be the edge-loop obtained by replacing the subpath of с labelled щ by
the path labelled i>;. Given a standard p-filling D2 —> Сд(Г) of c' (terminology of
H.2.7), one can obtain a standard p-filling of с by adding an extra polyhedral face to
the filling in the obvious way (figure Г.З) and adding extra edges to divide this face
into < ρ triangles. By induction on η = 1(c), we may suppose that there is a standard
p-filling whose p-area is < pl(c'). Hence АгеаДс) < pl(c). This implies thatC^^)
is hyperbolic (H.2.9).
Conversely, suppose that the Cayley graph of Г with respect to some finite
generating set A is 5-hyperbolic. Fix an integer к > 85. Every edge-loop in С^(Г)
contains a subpath ρ of length at most к that has its endpoints at vertices and is not
a geodesic (see H.2.6). If a word in the generators A±l represents 1 e Г then it is
the label on an edge-loop in С^(Г), and the non-geodesic subpath ρ described in the
preceding sentence is labelled by a subword и that is equal in Г to a shorter word ν
(the label on a geodesic in С^(Г) with the same endpoints as p). Thus we obtain a
Dehn presentation (A \ 72) for Г by defining 72 to be the set of words Μ,υ"1, where

The Conjugacy Problem 451
Fig. Γ.3 Applying a relation in Dehn's algorithm
щ varies over all words of length at most к in the generators and their inverses and
υ, is a word of minimal length that is equal to и, in Г. D
2.7 Remarks
(1) As a consequence of the above theorem, we see that if a group admits a Dehn
algorithm with respect to one set of generators then it admits such an algorithm with
respect to every set of generators. (This is not obvious a priori.)
(2) It follows from the above theorem and (H.2.7,9 and 11) that a group admits a
finite Dehn presentation if and only if its Dehn function (as defined in 1.8 A) is linear
(equivalently, sub-quadratic).
(3) The reader might like to prove directly that Z2 does not have a (potentially
obscure) Dehn presentation.
The Conjugacy Problem
2.8 Theorem. Every hyperbolic group has a solvable conjugacy problem.
Hyperbolic groups have the q.m.c. property (1.9), hence they have a solvable
conjugacy problem. Indeed it is easy to check that if two words и and υ in the
generators of a 5-hyperbolic group Γ are such that yuy~l = υ in Γ, and if w is a
geodesic word representing γ, then w satisfies the requirements of (1.9) (with respect
to a constant К that depends only on 5).
The bound on the length of the minimal conjugating word that one obtains from
the algorithm based on the q.m.c. property is an exponential function of \u\ and |u|
(see (1.12)). We shall describe a more efficient algorithm that gives a linear bound.
A still more efficient algorithm was recently discovered by David Epstein. In order
to motivate our algorithm, we reflect on our earlier remark that it is often worthwhile
to regard hyperbolic groups as fattened-up free groups.
In a free group one solves the conjugacy problem in the following manner. By
definition, a word w = uq ... a„ in the free group on Λ is cyclically reduced if

452 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
a~l φ ai+\ for 1 = 0,..., η — 1 and a^x φ a„. Given a word in the letters Λ and
their inverses, one can cyclically reduce it by repeatedly performing the following
operations: delete any subword of the form α," 'α,·, and delete the first and last letters
of the word if и = au'a~x. The cyclically reduced word that one obtains in this way
is unique up to cyclic permutation of its letters. (For example, bab~xbabb~x can be
reduced to baa or aab.) Given two words и and v, in order to check if they define
conjugate elements of F(A), one cyclically reduces both words and then looks to see
if one of the resulting words is a cyclic permutation of the other.
Motivated by the example of free groups, given a group Г with a fixed finite
generating set, we say that a word w in the generators and their inverses is fully
reduced if w and all of its cyclic permutations label geodesies54 will henceforth in
the Cayley graph of Г.
2.9 Lemma. Let Υ be a group that is 8-hyperbolic with respect to the finite generating
set A. If two fully reduced words u, ν e F(A) represent conjugate elements ο/Γ,
then
(1) тах{|и|, \v\] < 85 + 1, or else
(2) there exist cyclic permutations u' and υ' of и and υ and a word w e F(A) of
length at most 25 + 1 such thatwu'w~x = v' in Γ.
Proof. Let w be a geodesic word in F(A) such that wuw~x = v. Consider a geodesic
quadrilateral Q in the Cayley graph of Γ whose sides (read in order from a vertex)
trace out edge paths labelled w, u, w~x and υ-1 in that order. We shall refer to the
sides of Q that are labelled w±x as the vertical sides and we shall refer to the other
sides as the top and bottom.
Fig. Γ.4 Arranging the conjugacy diagram
By replacing и and υ with suitable cyclic permutations if necessary (see fig. Γ.4),
we may assume that each vertex on the top side of Q is a distance at least \w\ from
each vertex on the bottom. Consider the midpoint ρ of the top side. This is a distance
at most 25 from a point on one of the other three sides. If it were within 25 of a point
54 Such words are often called geodesic words. Likewise, it is common to speak of a word
as being a £-local geodesic, meaning that the path which it labels in the Cayley graph is a
£-local geodesic.

The Conjugacy Problem 453
p' on the bottom, then the vertices closest to ρ and p' would be a distance at most
25 + 1 apart; if |w| > 25 + 1, then this cannot happen.
Suppose now that ρ is within 25 of a point q on one of the vertical sides. Let χ
and у be, respectively, the top and bottom vertices of the side containing q. From the
inequalities |w| — (1/2) < d{p,y) < 25 + d(q, y) and d{q, y) = \w\ — d(x,q), we get
d(x, q) < 25 + (1/2), and therefore d(p, x) < d(p, q) + d(x, q) < 45 + (1/2). And
d{p, x) = \u\/2.
An entirely similar argument shows that if \w\ > 25 + 1, then the bottom side
of Q also has length at most 85 + 1. D
If applied naively, the preceding lemma would yield a solution to the conjugacy
problem involving an unacceptably large number of verifications to see if the words
considered were fully reduced. To circumvent this difficulty, instead of using (2.9)
directly we use (2.11), whose proof is essentially the same as that of (2.9) once one
has observed the following consequence of (H. 1.7) and (H.1.13).
2.10 Lemma. Let X be a δ-hyperbolic geodesic space. There is a constant C,
depending only on 5, such that if the sides of a quadrilateral Q с X are all (85 + l)-local
geodesies, then every side ofQ lies in the C-neighbourhood of the union of the other
three sides.
2.11 Lemma. Let Г be a group that is δ-hyperbolic with respect to the finite
generating set A. There is a positive constant K, depending only on 5, such that if
u, υ e F(A) represent conjugate elements ο/Γ, and if u, υ and all of their cyclic
permutations are (85 + l)-local geodesies, then
(1) тах{|и|, |υ|} < К, or else
(2) there exists a word w e F(A) of length at most К such that w~lu'w = v' in Г,
where u' and v' are cyclic permutations of и and v.
2.12 An Algorithm to Determine Conjugacy in Hyperbolic Groups. Let Г be
a group that is 5-hyperbolic with respect to the finite generating set A. Given two
words и and ν over the alphabet A±l, one looks in u, ν and their cyclic permutations
to find subwords of length at most 85 + 1 that are not geodesic. If one finds such a
subword, one replaces it with a geodesic word representing the same group element.
One continues in this manner until и and υ have been replaced by (conjugate) words
u' and v' all of whose cyclic permutations are (85 + 1)-local geodesies. (Working
with cyclic words, this requires the application of less than \u\ + \v\ relations from a
Dehn presentation of Γ.) Lemma 2.11 provides a finite set of words Σ such that и is
conjugate to ν in Γ if and only if w~lu'w = v' in Γ for some w e Σ. (One can take Σ
to be the set of words of length at most К together with a choice of one conjugating
element for each pair of conjugate elements uq, vq with max{ |и0|, |υο|} < К.) Using
Dehn's algorithm, one can decide whether any of the putative relations w~lu'w = v'
is valid in Г. D

454 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
2.13 Remark (Annular Diagrams). Let Ρ be a finite presentation of a group Γ, and
let К be the corresponding 2-complex (as described in 1.8A). Showing that a word
w in the generators represents 1 e Г is equivalent to showing that the edge-loop in
#(1) labelled w is homotopic to a constant map. We explained in (I.8A) how one can
construct a planar van Kampen diagram that portrays this homotopy, and the number
of 2-cells in a minimal such diagram is the minimal number of defining relations that
one must apply to show that w = 1 in Г.
Showing that two words и and ν in the generators represent conjugate elements
of Γ is equivalent to showing that the corresponding edge-loops in #(1) are freely
homotopic, in other words there is a continuous map from an annulus into К that
sends the boundary curves to these edge-loops. As in the case of a disc, one can
use cellular approximation and combinatorial arguments to deduce the existence of
annular van Kampen diagrams.
Such a diagram is a finite, planar, combinatorial 2-complex, homotopy equivalent
to an annulus, that portrays the homotopy between the curves representing и and v.
The 1 -cells of the complex are labelled by generators, the word labelling the boundary
cycle of each 2-cell is one of the defining relations, and the words labelling the two
boundary cycles of the diagram (suitably oriented) are и and v. The number of 2-
cells in a minimal such diagram is the minimal number N of factors among all free
equalities of the form
N
u~1(xovxq1) = Y[ х(г,хГ ',
where the jc, are any words and each r, is one of the defining relations (or its inverse).
The algorithm for solving the conjugacy problem in hyperbolic groups that is
described above yields the following analogue of the linear isoperimetric inequality
for the word problem. We leave the proof as an exercise for the interested reader.
2.14 Proposition. If Г is a hyperbolic group with finite presentation V = (A \ 1Z),
then there exists a constant Μ such that if two words u, υ in the letters A±l define
conjugate elements of Γ, then one can construct an annular van Kampen diagram
over V, with boundary labels и and v, that has at most Μ max {| и |, |υ|} 2-cells.
2.15 The Isomorphism Problem. Zlil Sela [Sel95] has shown that there is an
algorithm that decides isomorphism among torsion-free hyperbolic groups. More
precisely, there exists an algorithm which takes as input two finite group presentations
of torsion-free hyperbolic groups, and which (after a finite amount of time) will stop
and answer yes or no according to whether or not the groups being presented are
isomorphic. One requires no knowledge of 5: the simple fact that the groups being
presented are hyperbolic and torsion-free is enough to ensure that the algorithm will
terminate.
The proof of this result is beyond the scope of the techniques that we have
described.

Cone Types and Growth 455
Cone Types and Growth
The purpose of this subsection is to show that there is a simple algorithmic procedure
for recognizing geodesies in hyperbolic groups. More precisely, given a finite
generating set Λ for a hyperbolic group Γ, using the notion of cone type one can construct
a finite state automaton which accepts precisely those words in the free monoid on
A±l that label geodesies in the Cay ley graph Сд(Г). It follows that T,™=0pA(n)fl is
a rational function of t, where Дд (ή) is the number of vertices in the ball of radius η
about 1 e Сд(Г) (see 2.21). This important discovery is due to Jim Cannon [Ca84].
In [Gro87] Gromov discusses related matters under the heading "Markov properties"
(see also chapter 9 of [GhH90]).
Cannon's insights concerning the algorithmic structure of groups of isometries
of real hyperbolic space were the starting point for the theory of automatic groups
developed in the book by Epstein et al. [Ep+92].
2.16 Definition (Cone Types). Let Г be a group with finite generating set A and
corresponding word metric d. The cone type of an element γ e Γ is the set of words
ν e F(A) such that d{\, γν) = d(\, γ) + \v\. In other words, if γ is represented by
a geodesic word u, then the cone type of γ is the set of words ν such that uv is also
a geodesic.
2.17 Example. A free group of rank m has (2m + 1) cone types with respect to any
set of free generators, and so does a free abelian group of rank m.
2.18 Theorem. If a group Γ is hyperbolic, then it has only finitely many cone types
(with respect to any finite generating set).
Proof. We follow the proof of Cannon [Ca84] (see also [Ep+92, p.70]). The idea of
the proof is to show that in order to determine the cone type of γ e Γ, we only need
to know which vertices near γ in the Cayley graph are closer to 1 e Γ than γ is.
Thus, given к > 0, we consider the к-tail of γ e Γ, which is the set of elements
h e Γ such that d(\, yh) < d(\, γ) and d(\, h) < k.
Let С be the Cayley graph of Г with respect to a fixed finite generating set A;
suppose that it is 5-hyperbolic. Let к = 25 + 3. We claim that the fc-tail of each
element γ e Γ determines which words υ belong to the cone type of γ. We shall
prove this by induction on the length of v. (The first few steps of the induction are
trivial.) We fix an arbitrary γ' e Γ that has the same fc-tail as γ and choose a geodesic
word и representing γ'. In order to complete the induction we must show that if a
word ν is in the cone type of both γ and γ', and if a e A is such that va is in the
cone type of y, then va is also in the cone type of γ'.
If va were not in the cone type of γ', then there would exist a geodesic word
w e F(A) of length less than d(l, γ') + \v\ + 1 such that w = γ'να. We write w as
Wiw2, where \wi\ = d(l, γ') — 1 and |w2| < \v\ + 1.

456 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
Fig. Г.5 Cone types are determined by £-tails
The edge-paths determined by w and uv are geodesies in С that issue from the
identity and end a distance one apart. As such, these paths stay uniformly 2(5 + 1 )-
close (see (H.1.15)). In particular, since \w\\ = d{\, γ1) — 1 we have d(\i)\, γ') <
25+3.Thusy'~ W\ lies in the fc-tail of γ'. But then, by hypothesis, γ'~ W\ must lie in
the fc-tail of γ, which leads to the following contradiction: concatenate any geodesic
from 1 to γγ'~λυύλ with the edge-path from γγ'~ιινι that is labelled w2; the result is a
path from 1 to yva that has length strictly less than d(l, y)+|w2| < d(l, y)+|i>| + l,
contradicting the assumption that va is in the cone type of γ. Π
2.19 The Geodesic Automaton. Let Γ be a hyperbolic group with finite generating
set Λ and consider the finite graph whose vertices are the cone types of Γ and which
has a directed edge connecting the cone type of γ e Γ to the cone type of γα if and
only if a e Λ U A~l and a belongs to the cone type of γ.
Consider the set of edge-paths in this graph that begin at the cone type of the
identity and follow only positively directed edges; such paths have a natural labelling
by words in the letters A±l- It follows immediately from the definition of cone type
that the set of words that occur as such labels is precisely the set of words that label
the geodesic edge-paths in Сд(Г).
The graph described above is an example of a finite state automaton. A finite
state automaton over an alphabet В is a finite graph whose edges are directed and
labelled by elements of β; the vertices of the graph are divided into two sets "accept"
and "reject", and there is a distinguished vertex ίο, called the initial vertex. The set
of words which occur as labels on the directed edge paths that begin at ίο and end
at an accept vertex is called the accepted language of the automaton. In (2.19) all of
the vertices are accept vertices and the initial state is the cone type of 1 e Γ.

Cone Types and Growth 457
A subset of the free monoid on В is called a regular language if it is the accepted
language of some finite state automaton over B. We refer the reader to [Ep+92] for
a detailed account of the role which regular languages play in geometry and group
theory.
2.20 Corollary. Let Τ be a hyperbolic group with finite generating set A.
(1) The set of words in the free monoid (Λ U A~1)* that label geodesies in the
Cayley graph Сд(Г) is a regular language.
(2) Г is biautomatic.
Proof. A group with finite generating set A is biautomatic if and only if there is a
regular language £ in the free monoid (A U A~1)* that satisfies the following two
properties. First, every γ e Γ must be represented by at least one member of the
language, i.e. the restriction to £ of the natural map μ : (AU A~1)* -> Γ must
be surjective. Secondly, one requires the language to satisfy the "fellow-traveler
property": there must exist a constant К > 0 such that if и, υ e £ are such that
d(a.^(u), μ(υ)) < 1, where a e A U {1}, then the edge-path in С^(Г) that begins at
the identity and is labelled ν must remain uniformly if-close to the path that begins
at the vertex a and is labelled u.
We have just seen (2.19) that the language of geodesies in any hyperbolic group
is a regular language, and for the fellow-traveler property it suffices to take К =
2(25 + 1) (cf. H.1.15). D
For an alternative proof of (2.20), due to Bill Thurston, see [BGSS91].
In Chapter 1.8 we discussed the growth function of a group Г with respect to a
finite generating set: βΑ(η) is the number of elements γ e Γ such that d(l, γ) < η,
where d is the word metric associated to A. We now also wish to consider the function
σΛ(η) '■= {у е Г I d(l, y) = n]. It is convenient to encode the sequences Фа(п))
and (стд(и)) as the coefficients of formal power series:
σο σο
ίΑ<ί) = ΣβΑ(ηΥ and ^(ί) = Σ^(»)Λ
п=0 и=0
Note that, as formal power series,/^(i) = £4 (0/0 - 0·
Γ is said to have rational growth with respect to A if £4 (0 is a rational function,
i.e. it is the power series expansion of p(t)/q(t), where p(t) and q(t) are polynomials
with integer coefficients.
2.21 Theorem. If Γ is hyperbolic then the growth of Γ with respect to any finite
generating set is rational.
Proof. Let ibea finite generating set for Γ. We choose a linear ordering on the
alphabet A U A"1 and impose the associated lex-least ordering -< on finite words
over A U A_i, that is w -< ν if and only if |ш| < \v\ or \w\ = \v\ and w precedes ν
in the dictionary.

458 Chapter Ш.Г Non-Positive Curvature and Group Theory
Let £ be the set of words in the free monoid on A U A~' that occur as labels
on geodesic edge-paths in Сл(Г) (i.e. geodesic words). We have seen that £ is a
regular language. General considerations concerning regular languages imply that
£' = {w e £ | Vi> e £, w =r ν implies w < υ} is also a regular language (see page
57 of [Ep+92]). Note that the natural map (AU A~1)* -*■ Γ restricts to a bijection
from the set of words of length η in £' to the set of elements a distance η from the
identity in Γ.
Since £' is regular, it is the accepted language of a connected finite state automaton
over A U A~l. Let υ,0 be the initial vertex of the automaton and note that since every
prefix of a word in £' is again in £', every vertex of the automaton is an accept vertex.
We consider the transition matrix Μ of this automaton. The rows and columns of Μ
are indexed by the vertices υ,- of the automaton and the (i,j)-entry, which we denote
M(i, j), is the number of directed edges from υ,- to vj.
The number of words of length η in £' is the number of distinct edge-paths of
length η that begin at the initial vertex υ,·0, and this is the sum of the entries in the
i'oth row of M". Thus
σο
n=0 j
Let Co + C\ t Η l· Ck^ be the minimum polynomial of Μ over Ζ and let q(t) = Ck +
Ct-iiH h c0/*. For all integers η > 0, wehavec0M"+ciM"+1 Η VckM"+k = 0.
It follows that <2(0£д(0 1S a polynomial of degree at most к — 1, because for m > к
the coefficient of f is
Σ c0Mm-k(ioJ) + ■■■ + ckMm(i0,j),
j
which equals zero. Π
We close this section by noting a further consequence of (2.18).
2.22 Proposition. If a hyperbolic group is infinite then it contains an element of
infinite order. (More generally, this is true of any group Γ with finitely many cone
types).
Proof. Because Γ is infinite, there is a geodesic edge-path in its Cayley graph that
begins at the identity and has length greater than the number of cone types. Let w
be the word labelling such a geodesic and decompose w as щиъЩ, where u-i is the
label on a path that connects two vertices of the Cayley graph that have the same
cone type. By definition, щщ lies in the cone type of the first vertex, which is the
same as the cone type of the second vertex. Hence и\щ lies in the cone type of the
first vertex.
Iterating this argument we see that щи'{щ is a geodesic word for every positive
integer n. Since a subword of a geodesic word is geodesic, it follows that «j is a
geodesic word for every η > 0. In particular u'2 does not represent the identity, and
hence the image of u2 in Γ is an element of infinite order. D

Finite Subgroups 459
3. Further Properties of Hyperbolic Groups
In this section we shall present more of the basic properties of hyperbolic groups. We
shall see that our earlier results concerning isometries of С AT(— 1) spaces, including
parts (1) to (5) and (i) to (v) of Theorem 1.1, all remain valid in the context of
hyperbolic groups. Specifically, we shall prove that if a group Г is hyperbolic then it
contains only finitely many conjugacy classes of finite subgroups, the centralizer of
every element of infinite order contains a cyclic subgroup of finite index, and if Г is
torsion-free then it is the fundamental group of a finite cell complex whose universal
cover is contractible. We shall also describe an algebraic notion of translation number
and show that the translation numbers of elements of infinite order in hyperbolic
groups are positive and form a discrete subset of Μ (cf. 11.6.10(3)); in fact they are
rational numbers with bounded denominators (3.17).
The results in this section all originate from [Gro87].
Let's begin by noting the direct connection between hyperbolic groups and
isometries of CAT(O) spaces. This comes from the Flat Plane Theorem (H.1.5) and the
quasi-isometry invariance of hyperbolicity (H.1.10).
3.1 Theorem. If a group Γ acts properly and cocompactly by isometries on а С AT(0)
space X, then Г is hyperbolic if and only if X does not contain an isometrically
embedded copy of the Euclidean plane.
Finite Subgroups
The proof of the following theorem illustrates how arguments concerning groups of
isometries acting on CAT(/c) spaces, к < 0, can be transported into the world of
hyperbolic and related groups. The key to such adaptations is that one must find an
appropriate way to "quasify" the key role that negative curvature is playing in the
classical setting; one then attempts to encapsulate a robust form of the salient feature
of curvature in the more relaxed world of hyperbolic spaces 55. We apply this general
philosophy to the study of finite subgroups. The following result should be compared
with (II.2.8).
3.2 Theorem. If a finitely generated group Г is hyperbolic, then it contains only
finitely many conjugacy classes of finite subgroups.
As in (II.2.8), we shall deduce this result from the existence of an appropriate
notion of centre for bounded sets.
55 The first example of this is the very definition of a <5-hyperbolic space: one observes that
many of the global implications of the CAT(—1) inequality stem from the slim triangles
condition.

460 Chapter III .Γ Non-Positive Curvature and Group Theory
3.3 Lemma (Quasi-Centres). Let X be a δ-hyperbolic geodesic space. Let Υ cX be
a non-empty bounded subspace. Let rY = inf{p > 0 | Υ Я B(x, ρ), some χ e X].
For all ε > 0, the set C£(Y) = {x e Χ \ Υ с B(x, rY + ε)} has diameter less than
(45 + 2ε).
Proof. Given x, x1 e C£(Y), let m be the midpoint of a geodesic segment [x, jc']. For
each у e Υ we consider a geodesic triangle with vertices x, x', у and with [x, x']
as one of the sides. Because X is 5-hyperbolic, m is within a distance 5 of some
ρ e [x,y] U [χ',у]; suppose that/? e [x,y]. Then, since d(x,m) = d(x,x')/2 and
dip, x) > d(x, m) - 5, we have d(y, p) = d(y, x) - d[p, x) < d(y, χ) + 5 — d(x, x')/2,
and since d(x, y) < rY + ε,
d(y, m) < d(y,p) + d{p, m) < rY + ε + 25 - -d(x, x1).
But d(y, m) > rY for some у £ Y, hence ε + 25 — \d{x, x') > 0. Π
Proof of Theorem 3.2. Let Γ be a group whose Cayley graph С with respect to
some finite generating set is 5-hyperbolic. Let Η С Г be a finite subgroup, and
let С[(H) с С be as in the lemma. C\(H) contains at least one vertex and the
action of Η leaves C\ (H), and hence its vertex set, (set-wise) invariant. If χ is one
of the vertices of C\{H), then x~lHx leaves x~lC\(H) invariant. Since x~~lC\{H),
which is a set of diameter less than (45 + 2), contains the identity 1, it also contains
x~lHx = (лГ'Ях).1. Thus every finite subgroup of Г is conjugate to a subset of the
ball of radius (45 + 2) about the identity. D
The proof given above does not appear in the literature but the idea behind it was
known to a number of researchers in the field, in particular Brian Bowditch, Noel
Brady and Ilya Kapovich. Alternative proofs have been given by Ol'shanskii and by
Bogopolskii and Gerasimov [BoG95].
Quasiconvexity and Centralizers
Recall that a subspace С of a geodesic space X is said to be convex if for all x, у е С
each geodesic joining xloy is contained in C. Following Gromov [Gro87], wequasify
this notion.
3.4 Definition. A subspace С of a geodesic metric space X is said to be quasiconvex
if there exists a constant к > 0 such that for all x, у е С each geodesic joining χ to у
is contained in the ^-neighbourhood of C.
3.5 Lemma. Let G be a group with finite generating set Л and let Η с Г be a
subgroup. If Η is quasiconvex in the Cayley graph Сд (Г), then it is finitely generated
and Η ^-» Γ is a quasi-isometric embedding (with respect to any choice of word
metrics).

Quasiconvexity and Centralizers 461
Fig. Γ.6 A Quasiconvex Subgroup
Proof. Let к be as in (3.4). Given h e Я, we choose a geodesic from 1 to h in С^(Г);
suppose that it is labelled αϊ... α„. For i = 1,..., η we choose a word щ of length
at most к so that /г, := щ-хщщ е Я (where ио and un are defined to be the empty
word). (See figure Г.6.)
We have h = h\ ... hn. It follows that Я is generated by the (finite) set of elements
hj e Я that lie in the ball of radius (2k + 1) about 1 e Γ. It also follows that the
distance from 1 to h in the word metric associated to this generating set is at most
η = d^(\, h), and hence Я ^-» Γ is a quasi-isometric embedding. Π
3.6 Corollary. Let Γ be a hyperbolic group and let Η с Г be a finitely generated
subgroup.
(1) If Η is quasiconvex with respect to one finite generating set, then it is quasi-
convex with respect to all finite generating sets. (Thus we may unambiguously
say that Η is a quasiconvex subgroup ο/Γ.)
(2) Я с Г is quasiconvex if and only if it is quasi-isometrically embedded.
Proof. One direction of (2) is proved in the lemma. For the converse, we fix finite
generating sets Λ for Γ and В for Я and suppose that Я is quasi-isometrically embedded
in Г, which implies that there is a quasi-isometric embedding φ : Св(Н) —»■ Сд(Г).
Given two points h, h! e Я, we join them by a geodesic с in Св(Н) and consider the
quasi-geodesic φ о с joining h to h' in С^(Г). According to (H.1.7), any geodesic
joining h to h' in Сд(Г) is fc-close to this quasi-geodesic, where к depends only on
the hyperbolicity constant of Г and the parameters of the quasi-isometry φ. Thus Я
is quasiconvex in Сд(Г), and (2) is proved.
The statement "Я ^-» Г is a quasi-isometric embedding" does not depend on a
choice of generating sets, so (1) follows from (2). D
By combining (H.1.9) and (3.6) we get:
3.7 Proposition. The quasiconvex (equivalently, quasi-isometrically embedded)
subgroups of hyperbolic groups are hyperbolic.
3.8 Remarks
(1) The converse of (3.7) is not true: there exist pairs of hyperbolic groups Я С Г
such that Я is not quasi-isometrically embedded in Г (examples are given in (6.21)).

462 Chapter Ш.Г Non-Positive Curvature and Group Theory
(2) Proposition 3.7 shows that quasi-isometrically embedded subgroups of
hyperbolic groups are finitely presented; the same is not true of semihyperbolic groups,
for example the direct product of two finitely generated free groups (see 5.12(3)).
(3) The construction described in (II.5.45) yields examples of hyperbolic groups
with finitely generated subgroups that are not finitely presented and hence not
hyperbolic. It is more difficult to construct finitely presented subgroups that are not
hyperbolic, the only know examples are due to Noel Brady56.
The following results are due to Gromov [Gro87]. The ideas underlying them
have been used by other authors to obtain similar results in wider contexts (e.g.,
[Ep+92], [GeS91], [АЫВ95]). In the next section we shall provide proofs in a wider
context — see (4.13) and (4.14).
3.9 Proposition. Let Υ be a hyperbolic group.
(1) The centralizer C(y) of every γ e Γ is a quasiconvex subgroup.
(2) If the subgroups #i, #2 С Г are quasiconvex then so is H\ Π Щ.
3.10 Corollary. Suppose that Г is a hyperbolic group and that γ e Γ has infinite
order.
(1) The map Ζ —»■ Γ given by и м> γ" is a quasi-geodesic.
(2) (y) has finite index in C(y). In particular, Γ does not contain Ί?.
Proof of Corollary. C(y) is quasiconvex, hence finitely generated and hyperbolic. By
intersecting the centralizers of a finite generating set for C(y), we see that the centre
Z(C(y)) is also quasiconvex, hence finitely generated and hyperbolic. It is easy to
see that a finitely generated abelian group is hyperbolic if and only if it contains a
cyclic subgroup of finite index. Hence Z(C(y)) contains (y) as a subgroup of finite
index. Moreover, since Z(C(y)) С C(y) and C(y) С Г are quasiconvex, the maps
(у) ^ Z(C(y)) ^-» C(y) and C(y) ^-» Г are quasi-isometric embeddings, by (3.5),
and hence so is (у) <^-> Г. This proves (1).
Fix a finite generating set Л with respect to which Г is 5-hyperbolic and suppose
that у е Г has infinite order. It follows from (1) that if yp is conjugate to yq then
\p\ = \q\. For if t~lypt = yq, then t~myp"'f = y«"' for all integers m > 1, which
means d{\, y9"') < 2md{\,t) + \p\m d{\, y), and if \p\ were less than \q\ this would
contradict the fact that η м> у" is a quasi-geodesic.
Since the positive powers of у define distinct conjugacy classes, by replacing у
with a suitable power if necessary, we may assume that у is not conjugate to any
element a distance 45 or less from the identity. We claim that if g e Г commutes
with у then g lies within a distance К := 2d(l, y) + 45 of (y). Suppose that this
were not the case and suppose that d(g, (y)) = d(g, yr). Replacing g by y~rg, we
56 "Branched coverings of cubical complexes and subgroups of hyperbolic groups", J. London
Math. Soc, to appear.

Quasiconvexity and Centralizers 463
d(l,7)+2i d(l,7)+2J
Fig. Γ.7 Centralizers are virtually cyclic
may assume that d(g, (y)) = d(g, 1) > K. Consider a geodesic quadrilateral Q in
С^(Г) with sides {[1, y], [l,g], g.[l, γ], y.[l, g]}.Letg; denote the point a distance
ί from 1 on [1, g]. (See figure Γ.7.)
Because С^(Г) is 5-hyperbolic, gt lies a distance at most 25 from some point ρ on
one of the remaining three sides of Q. If we choose t so thatu?(l, g) — d(l, γ) — 2δ >
ί > d{\, γ) + 25, then ρ must belong to [y, yg], say ρ = yge. Since f7 = d(yg,>, (y))
and ί = ύ?(#,, (у)), we have \t — f\ < 25 and hence d(g„ ygt) < 45. But this means
^(11 #Г' /5') — 45, contrary to our hypothesis on у. This contradiction proves (2).
D
As a further illustration of the general approach of quasification, we adapt the
existence of projection maps in CAT(O) spaces (II.2.4).
3.11 Proposition (Quasi-Projection). IfX is α 8-hyperbolic geodesic space and
Υ с X is a quasiconvex subspace, then there exists a constant К > 0 with the
following property: given ε > 0, if π : X —»■ Υ assigns to each χ e X a point
π(χ) e Υ such that d(x, π(χ)) < d(x, Υ) + ε, then d(n(x), 7г(х/)) < d(x, χ?) + Κ + 2ε
for all x, x1 e X.
Proof Let к > 0 be as in definition (3.4). We shall show that it suffices to take
К = 85 + 2k.
Given x,xf e X, we choose geodesic segments joining each pair of the points
{χ, χ1, π{χ), тг(х/)} and consider the geodesic quadrilateral formed by the
triangles Δ, = Α([χ,π(χ)1 [π(χ),χΊ [χ,χ1]) and Δ2 = Α([π(χ),π(χ1)], [π(χ),χ!],
[χ1,7г(х/)]). There is a point ρ e [π(χ), χ1] which is within 5 of both [π(χ), ττ(χ/)] and
[χ1, тг(х')] (the 5-hyperbolic condition for Δ2), and ρ is also within 5 of a point on
[χ, π(χ)] U [χ, χ?] (the 5-hyperbolic condition for Δι). Each point on [n(x), nix1)] is
within к of Y. Hence the point q' e [x1', 7г(х/)]) closest to ρ is within 25 + к of Y.
Therefore, since d{x!, π{χ!)) = d(x!', q') + d(q', π(χ/)) and φ', π(χ/)) < d{x*, Υ) + ε,
we have d(q', ?r(x/)) < 25 + к + ε. For future reference we also note that
(#) d{p, nix1)) < ЗЗ + к + ε.
If ρ is within 5 of q e [χ, π(χ)], then, as above, d{q, π(χ)) <28+к + г. Hence

464 Chapter Ш.Г Non-Positive Curvature and Group Theory
d(n(x), π{χ')) < ά(π(χ), q) + d(q,p) + dip, q') + d(q', π(χ'))
<(2S+k + s) + 2S + (28+k + s)
<Κ + 2ε.
So we are done if ρ is a distance at most 5 from [χ, π(χ)]. If it is not, then instead
of considering Δι and Δ2 we consider Δ', — A([x, π(χ)], [π(χ), π(χ?)], [χ, π(χ/)])
and Δ2 = Α([χ, χ'], [χ, π(χ/)], [χ1, π{χ!)]). Let ρ' be a point of [x, 7г(х/)] that is a
distance at most 5 from both [χ, π(χ)] and [π(χ), π(χ1)]. As above, if d(p', [xf, nix1)])
is at most 5, then we are done.
It remains to consider the case where ρ e [χ1, π(χ)] is within 5 of a point r e
[x, xf], and p' e [χ, π(χ?)] is within 5 of some r' e [x, xf]. Since r and r1 both lie on
[x, x1], we have d(r, r') < d(x, xf). Thus, using (#) twice, we have:
d(n(x), π(χ')) < ά{ττ{χ), p') + d(p', r') + d(r', r) + dip, r) + dip, nix1))
< 2(35 + к + ε) + 25 + dix, χ).
О
In the special case Υ — Ζ, the following corollary answers a question raised by
Alonso et al. [Alo+98] in the course of their work on higher order Dehn functions.
3.12 Corollary. Let Η and Γ be hyperbolic groups. The image of every quasi-
isometric embedding f : Η —> Visa quasi-retract, i.e. there exists a map ρ : Γ —»■ Η
such that, given any choice of word metrics d-p anddf], there is a constant Μ such that
dHipfiy), y) < Μ for ally e Η and dHipix), pix1)) < Μ drix, xf)for all x, x1 e Γ.
Proof. The image of/ is quasiconvex in Γ. Let π : Γ —»■ /(Я) be a choice of closest
point and define ρ to be/' on, where/' is a quasi-inverse for/ in the sense of
(1.8.16). D
Translation Lengths
In Chapter Π.6 we saw that one can deduce a good deal about the structure of a group
Γ acting by isometries on a CAT(O) space X by looking at the translation numbers
\γ\ = mi{dix, γ.χ) \ χ e X}. We also noted (II.6.6) that \γ\ = Ηπι,,^οο j,dix, γ".χ)
and that if Γ acts properly and cocompactly then the set of numbers {\γ\ : γ e Γ} is
discrete (Π.6.10(3)).
3.13 Definition. Let Γ be a group with finite generating set A and associated word
metric d. The algebraic translation number of у е Г is defined to be
n—>oo η

Translation Lengths 465
(The function η м> d{\, у") is sub-additive and hence lim^oo ^f(x) exists — see
II.6.6.) Normally we shall write τρ or τ instead of Τγ,λ. mus suppressing the
dependence on A.
3.14 Remarks
(1) τ (γ) depends only on the conjugacy class of у.
(2) т(ут) = \m\ τ (γ) for every m e Z.
(3) If a finitely generated subgroup Я С Г is quasi-isometrically embedded, then
for any choice of word metrics there is a constant К such that ^тг(й) < г#(/г) <
ΑΓτΓ(Α) for all A e Я.
3.15 Proposition. If Г is a hyperbolic group with finite generating set A, then for
every R > 0 there exist only finitely many conjugacy classes [y] such that Τγ,α(υ) <
R.
Proof. Suppose that Сд(Г) is 5-hyperbolic. Let и be the shortest word among all those
which represent elements of the conjugacy class [γ]. If и has length at least (85 + 1),
then for every integer η > 0 the edge-paths in Сд(Г) labelled u" are (85 + l)-local
geodesies. It follows from (H.l. 13) that these edge-paths are (λ, e)-quasi-geodesics,
where λ and ε depend only on 5. Therefore т(и) = τ (γ) > \u\/X. D
3.16 Proposition. Let Γ be a hyperbolic group. If a subgroup ffcTis infinite and
quasiconvex, then it has finite index in its normalizer.
Proof. Fix finite generating sets for Я and Г, and let d be the associated word metric
on Г. We know from (2.22) that Я contains an element of infinite order, a say, and we
know from (3.10) that (a) has finite index in the centralizer Cr(a), i.e. there exists
a constant к > 0 such that if g e Cr(a) then d{g, (a)) < k.
Since Я is quasiconvex, there is also a constant К > 0 such that if γ~λαγ is in Я,
thenT#(y~'o!y) < Κττ{γ~λαγ) — ΚτΓ(α). By applying the preceding proposition
to Я, we see γ~ιαγ must belong to one of finitely many conjugacy classes in Я;
we choose representatives c~[lac\,..., c~xacn for these classes, where с, е Г (only
one of the c, is in Я).
Given an element γο e Γ in the normalizer of Я, there exists c, such that
y0_10!yo = h~xcjxacih for some /г е Я, which implies that уо/г_1с~' belongs to
Cr(a). Thus ^(уо/г-1^1, (α)) < к, and writing yo^_1 = /г'уо we have:
dfo, Я) = rf(A'y0, Я)
< d(h'y0, h'Yocj\ H) + d(tiYoc-})
= d(l,c,) + d(Yoh-lc;\H)
< тахй?(1, Cj) + k.
j
Thus d(yo, H) is uniformly bounded, i.e. Я has finite index in its normalizer. D

466 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
The following remarkable theorem is due to Gromov [Gro87].
3.17 Theorem (Translation Lengths are Discrete). If Г is 8-hyperbolic with respect
to a fixed generating set A, then {t(y) | γ e Γ} is a discrete set of rational numbers.
Indeed there is a positive integer N such that Ντ(γ) e Nfor every γ e Γ.
Proof We follow a proof of Thomas Delzant [Del96]. Given y, let и be the shortest
word among those which represent elements in the conjugacy class of γ. Ignoring
finitely many conjugacy classes, we may assume that и has length at least (85 + 1).
Consider the bi-infinite path pu in the Cay ley graph С^(Г) that begins at the identity
and is labelled by the powers of и. This is an (85 + l)-local geodesic, and hence it is a
quasi-geodesic. Let t/_, U+ e ЗГ = ЭС^(Г) be the endpoints of this quasi-geodesic.
(H.3.2) guarantees the existence of a geodesic line with endpoints £Л_ and U+, and
(H.1.7) implies that any such geodesic is contained in the ^-neighbourhood of pu,
where R depends only on 5.
We fix a linear ordering on the generators A±x and consider the induced lex-least
ordering -< on the finite geodesic edge-paths in С^(Г): one path is less than another
if it is shorter or else has the same length but its label comes before that of the other
path in the dictionary.
For each positive integer m, let Im be the geodesic that is lex-least among those
geodesic edge-paths that are contained in the if-neighbourhood of pu and have end-
points within a distance К of u~m and um respectively. For each η < m, let Imn be
the minimal sub-segment of Im that has endpoints within a distance К ofu~" and u"
respectively. For fixed n, there are only finitely many possibilities for /„,,„, so by a
diagonal sequence argument we may extract a subsequence of the Im such that /m„
remains constant for all nasw-> oo. The union of the geodesic segments in this
subsequence (Jn Im .„ gives a geodesic line joining U_ to U+. This (oriented) line has
the property that all of its subsegments are minimal in the lexicographical ordering
-< on segments in the Cayley graph that have the given length. (Delzant calls such
geodesies "special".)
Through each point of Сд(Г) there is at most one special geodesic with endpoints
i/_ and U+, because a segment of such a geodesic cannot have two extensions (both
forwards or both backwards) of a given length that are both lexicographically least.
Since any geodesic joining i/_ to U+ is contained in the R-neighbourhood of p„,
it follows that there are at most V special geodesies joining i/_ to U+, where V
is the cardinality of a ball of radius R in Г. The action of и by left multiplication
permutes this finite set. Therefore some power u\ where r divides V!, fixes a special
geodesic and acts by translations on it; let a e N be the translation distance of the
action of ur on this geodesic. If χ is a vertex on this line, then d(urn.x, x) — na for
all η > 0 and hence d((x~lumx), 1) = na. Dividing by η and taking the limit, we get
r τ(γ) = гт(и) = т{иг) = a e N. Thus we may take the integer N in the statement
of the theorem to be V!. D

Free Subgroups 467
Free Subgroups
The following result is due to Gromov [Gro87, 5.3B]. The first detailed proof was
given by Delzant [Del91]. One might think of this result as a strong form of the
assertion that two generic elements in a hyperbolic group generate a free subgroup.
3.18 Theorem. If Τ is a torsion-free hyperbolic group and Η is a two-generator
group that is not free, then up to conjugacy there are only finitely many embeddings
H^Y.
We recall the "ping-pong" construction of Fricke and Klein [FrK112] (cf. [Har83]).
3.19 Lemma (Ping-Pong Lemma). Leth\,..., hr be bijectionsofa set Ω and suppose
that there exist non-empty disjoint subsets Α\Λ,Α\,-ι,... ,Arj,Ar -\ С ilsuchthat
/^(Ω ν Α,,ε) с Α;,_ε for ε = ±1, i — 1,..., r. Then h\,... ,hr generate a free
subgroup of rank r in Ρειτη(Ω).
Proof. Exercise. D
3.20 Proposition. If Υ is hyperbolic, then for every finite set of elements h\,..., hr e
Γ there exists an integer η > 0 such that {h",..., h"} generates a free subgroup of
rank at most r in Y.
Proof. Let С be the Cay ley graph of Г with respect to a fixed finite generating set and
suppose that С is 5-hyperbolic. By replacing each hi with a sufficiently high power
and throwing away those which become trivial, we may assume that each of the /г,
has infinite order. Define r, = τ {hi).
As in the proof of (3.17), by raising the /г, to a further power we may assume
that there is a geodesic line с,- : К —> С which is invariant under the action of /гг; the
action is A,-.Cf(j) = ct{s + U). Define hf°° — c,-(±oo).
We claim that if c,(oo) = c,(oo), then some powers of/г, and hj generate a cyclic
subgroup of Г (and hence c,(—oo) = c,(—oo)). To see this, first note that according
to (H.3.3), we may parameterize the lines c, and q so that d{a{s), q{s)) < 55 for
all s > 0. For each integer r > 0 we shall estimate а{к~гЦЦ.с-,ф), q{tj)). For this
purpose it is convenient to use the notation α ~ε b to mean \a — b\ < ε; thus
d{y, Cj{s)) ~5i d{y, q{s)) for all s > 0 and all γ e Y.
а{К~гЩ.аф), Cj{tj)) = d(hj.c,(rtt), hTt.q(t,))
~5S d{hj.Ci{rti), Ц.Cj{tj))
~5S d{hj.Cj{rti), h-.Ci{tj))
= d{cj{tj + rtt), Ci(tj + rt-,)) < 55 .
Thus d{h~rhjhlci{0), 9(5)) < 15δ for a11 r > 0, which means that h~rЩ.ct{<3) =
h~shjhs;.Cj{0) for some integers r, s with 0 < r — s < v, where v is the number of

468 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
elements in a ball of radius 155 in Г. Since the action of Г on С is free, it follows
that hs~r lies in the centralizer hj, which according to (3.10) is virtually cyclic. This
proves the claim.
If hj and hj have powers that generate a cyclic subgroup, then we may replace
them with a generator of that cyclic subgroup. Thus it only remains to consider the
case where the 2r points /if00,..., hf°° e ЭГ are distinct.
For each χ e Γ, let jc, be a point on the image of c, that is closest to x. We
claim that if R > 0 is sufficiently large then the following 2r sets are disjoint:
Ajy-i — {x | x( e c,-(—oo, — /?]} and A,,| = {x \ Xj e d[R, oo)}, where i— 1,..., r.
It is clear that by replacing the /г, with sufficiently high powers we can ensure that
their action on the above sets satisfies the hypotheses of the ping-pong lemma, thus
it only remains to prove that the Α, ε really are disjoint.
To see this, we fix a constant ρ > 0 that is sufficiently large to ensure that the
closed ball В of radius ρ about 1 e Γ contains c,-(0) for г = 1,..., r, and every
geodesic segment of the form [xt, Xj] intersects B. Given χ e Γ, since A(x, xh Xj) is
5-slim, there is a point ρ on [x, xt] or [x, Xj] that is within a distance 5 of [xi,xj] Π
В. Suppose that ρ e [χ,χι]. Since d(Cj(0),p) > d{x(,p) (by definition of Xj), and
d(Ci(0),p) < d(a(0), 1) + d(l,p) < 2p + 5, we have d(a(0), x{) < d(c,(0),p) +
а{р,хд<2а(а(0),р)<4р + 28.
It follows that if R > Α ρ + 25, then each χ e Γ can belong to at most one of the
setsAie defined above. D
The Rips Complex
We continue the work of deciding which of the properties listed in (1.1) are
enjoyed by all hyperbolic groups. This paragraph concerns (1.1(5)). We describe a
construction due to E. Rips which shows that every torsion-free hyperbolic group
has a finite Eilenberg-MacLane space. It also shows that every hyperbolic group has
an Eilenberg-MacLane space with finitely many cells in each dimension (see 3.26).
3.21 Theorem. Every hyperbolic group Γ acts on a simplicial complex Ρ such that:
(1) Ρ is finite-dimensional, contractible and locally finite;
(2) Γ acts simplicially, with compact quotient and finite stabilizers;
(3) Γ acts freely and transitively on the vertex set of P.
In particular, if Γ is torsion-free then it has a finite Eilenberg-MacLane space
К(Г, \), namely Y\P.
Remark. The complex described in this theorem has no particular local geometric
features; in particular there is no reason to expect it to support a CAT(O) metric.
3.22 Definition. Let X be a metric space and let R > 0. The Rips complex Pr(X) is
the geometric realization of the simplicial complex with vertex set X whose n-
simplices are the (n + l)-element subsets {jco, ..., xn} С X of diameter at most
R.

The Rips Complex 469
One usually endows Pr(X) with the weak topology, which is the same as the
metric topology (1.7) if each vertex lies in only finitely many cells, i.e. B(x, R) is
finite for every χ e X. (In any case, the weak topology and the metric topology
define the same homotopy type [Dow52]). Note that Pr(X) is finite dimensional only
if there is a bound on \B(x, R)\ as χ varies over X.
Let r > 0. Recall that a subset X of a metric space Υ is said to be r-dense if for
every у e Υ there exists χ e X with d(x, y) < r.
3.23 Proposition. Let Υ be α geodesic space and let X be an r-dense subset. If Υ is
δ-hyperbolic then Pr(X) is contractible whenever R > 45 + 6r.
Proof. The complex Pr(X) is contractible if and only if all of its homotopy groups
are trivial (see [Spa66]). The image of any continuous map of a sphere into Pr(X)
(with the weak topology) lies in a finite subcomplex (because it is compact), so it
suffices to prove that any finite subcomplex L С Pr(X) can be contracted to a point
in Pr(X). We fix a basepoint x0 £ X.
Case 1: If the distance in X between jco and each vertex of L is at most R/2 then
L is contained in a face of a simplex of Pr(X), and hence it is contractible.
Case 2: Suppose that there is a vertex ν e L such that d(xo, v) > R/2, and choose
ν so that d(xo, v) is maximal. (Beware: the metric d is on X, not L.) The idea of the
proof is to homotop L by pushing υ towards jco while leaving the remaining vertices
of L alone. If we are able to do this, then by repeating a finite number of times we
can homotop L to a complex covered by Case 1.
We choose a point у on a geodesic [jco, v] С Υ with d(v, y) = R/2, and then
choose υ' e X with d(y, v')<r. Let p=d{v, v') and note that ρ e [25 + 2r, (R/2) +
r]. We claim that if и is a vertex of L and d(u, v) < R then d(u, v') < R.
Consider a geodesic triangle Δ(*ο, и, υ) с Υ with у e [xo, υ]. This is 5-slim, so
either d(y, u') < 5 for some u' £ [xq, u] or else d(y, w) < 5 for some w Ε [и, υ].
(Readers will find the ensuing inequalities easier to follow if they draw a picture of
each situation and label the lengths of arcs.)
In the first case, by hypothesis,
d(xo, v') + d(v', v) > d(xo, v) > d(x0, u) = d(xo, u) + d(u', u),
and d(x0, v') < d(x0, u') + d(u', v') < d(x0, u') + (5 + r). Thus ρ = d(v, υ') >
d(u', M)-(5 + r),andhenceu?(i<, v') < d(u, u')+d{u', v') < (p + S + r)+(S + r) < R.
In the second case, d(v', w) < r + 5 and we have: ρ = d(v, υ') < d(v, w) +
d(w, v'),sod(v, w) > p-(5+r).Whenced'(i<, w) = d(u, v)-d(v, w) < R-p+S+r,
and d(u, v') < d(u, w) + d(w, v') < R - ρ + 2(5 + r) < R.
Thus we have shown that if a vertex и of L is in the star of υ then it is also in
the star of υ'. Moreover, υ' is also in the link of v. Let L' be the complex obtained
from L by replacing each simplex of the form {v,x\,... ,xr} with {υ',x\,...,xr].
The obvious (affine) homotopies moving each {v,x\,... ,xr] to {υ',x\,... ,xr} in
the simplex {υ, υ', χ\, ..., xr], together with the identity map on L \ st(u), give a

470 Chapter Ш.Г Non-Positive Curvature and Group Theory
homotopy from L <^-> Pr(X) to L' ^-» Pr(X), and (as we noted above) by iterating
this operation we can contract L to jco- Π
Proof of Theorem (3.21). The inclusion of Γ as the vertex set of its Cayley graph is 1 -
dense, so we can apply the preceding proposition to deduce that Pr(T) is contractible
when R is large enough. Because there is a bound on the number of points in any ball
of a given radius in Γ, the complex Pr(T) is finite dimensional and locally finite.
One extends the action of Γ on itself by left multiplication to a simplicial action
on Pr(T) using the affine structure (1.7 A) in the obvious way. The action of Γ on
Pr(T), which is obviously free and transitive on vertices, is simplicial in the sense
that it takes simplices to simplices setwise, but if the group has torsion then certain
simplices will be sent to themselves without being fixed pointwise (and as a result
the quotient space will not be a simplicial complex).
Consider the stabilizer of a simplex σ = {x\,..., x„]. Because the action of Γ
is free on vertices, the representation of Stab(a) into the symmetric group on the
vertices is faithful, and hence |Stab(a)| < n\. D
3.24 Remarks
(1) The dimension of the complex Pr(T) is one less than the cardinality of the
largest set in Γ of diameter R. This is bounded (crudely) by the number of words of
length R in the generators and their inverses, which is less than (2|.4|)й.
(2) Let Г be a group equipped with the word metric associated to the finite
generating set Л, and let В be the set of non-trivial elements in the ball of radius R
about the identity. The 1-skeleton of the Rips complex Pr(T) is the Cayley graph of
Г with respect to B.
3.25 The Finiteness Conditions FPn,FLn and Fn. We remind the reader that an
Eilenberg-MacLane complex K(T, 1) for a group Г is a CW complex with
fundamental group Г and contractible universal cover. Such a space always exists and its
homotopy type depends only on Г.
If there exists a K(T, 1) with a finite и-skeleton, then one says that Г is of type
F„. Being of type Fq is an empty condition, being of type F\ is equivalent to being
finitely generated, and being of type Fi is equivalent to being finitely presented.
A group Г is said to be of type FP„ (resp. FLn) if Z, regarded as a trivial module
over the group ring ΖΓ, admits a projective (resp. free) resolution in which the first
(n + 1) resolving modules are finitely generated. Г is said to be of type FPoo if it is
FPn for every n. A finitely presented group Г is of type FP<x> if and only if it has a
K(T, 1) with finitely many cells in each dimension (see [Bro82] or [Bi76b]).
The cohomology of Г with coefficients in a ring R can be defined as H*(T, R) :=
Н*(К(Г, l),R).
3.26 Corollary. Let Υ be a hyperbolic group.
(1) If Τ is torsion-free then it has afinite K(T, 1).

4. Semihyperbolic Groups 471
(2) In general, Γ is of type FP^, and
(3) H"(T, Q) = Ofor all η sufficiently large.
Proof. (1) follows immediately from Theorem 3.21. Parts (2) and (3) are standard
spectral sequence arguments (see [Bro87], for example), we omit the details. D
3.27 Remark. We mentioned earlier that if a group Γ acts properly and cocompactly
by isometries on а С AT(0) space X then X is Г-equivariantly homotopy equivalent to
a finite-dimensional cell complex on which Г acts cellularly and cocompactly with
finite stabilizers (cf. II.5.13). As in the hyperbolic case, this implies that Г is finitely
presented and of type FP^, and H*(T, Q) is finite dimensional.
4. Semihyperbolic Groups
Let Г be a finitely generated group that acts properly and cocompactly by isometries
on a CAT(O) space X. By fixing a basepoint jeo e X we obtain a quasi-isometry
γ м- γ.χο from Γ to X. One of the main goals of this chapter is to understand how
the geometry that is transmitted from X to Γ by such quasi-isometries is reflected
in the structure of Γ. In the case where X is a visibility space, we have seen that
a great deal of the geometry of X is transmitted to Γ through the slim triangles
condition. In this section we consider the case where X is an arbitrary CAT(O) space.
We shall describe a weak convexity condition that Γ inherits from the convexity of
the metric on X. The groups that satisfy this condition are called semihyperbolic
groups. Following the treatment of [AloB95], we shall see that most of our previous
results concerning isometries of CAT(O) spaces can be extended to semihyperbolic
groups.
If Γ acts properly and cocompactly by isometries on a CAT(O) space X, then there
is a natural correspondence between the sets of quasi-geodesics in X and Γ. However
we have seen that in general the set of all quasi-geodesics in a CAT(O) space X is not a
very manageable object (cf. 1.8.23). With this is mind, given a quasi-isometry X —» Γ
associated to an action of Γ on X, we restrict our attention to those quasi-geodesics
in Γ that are the images of geodesies in X. In a coarse sense, the convexity properties
of the metric on X will be reflected in this set of quasi-geodesics. In order to describe
the resulting weak convexity properties, we need the following definitions.
Definition of a Semihyperbolic Group
4.1 Definitions. Let X be a metric space and let V(X) be the set of eventually constant
maps ρ : N —» X, thought of as finite discrete paths in X. Let Tp denote the integer
at which ρ becomes constant, i.e. the greatest integer such that p(Tp) φ p(Tp — 1). A
(discrete) bicombing of X consists of a choice of a path s^y) e V(X) joining each
pair of points x, у е X; in other words a bicombing is a map s:XxX^ V(X) such

472 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
that e о s is the identity on Χ χ X, where e : V{X) —> Χ χ Χ is the endpoints map
e{p) = (p(0), p(Tp)). The path j^>y) is called the combing line from χ to y.
A bicombing s is said to be quasi-geodesic if there exist constants λ and ε such
that, for all x, у e X, the restriction of S(X-y) to [0, TS(iV)] is a (λ, e)-quasi-geodesic.
A bicombing s is said to be bounded if there exist constants k\ > 1 ,ki > 0 such
that for all x, y, x?,y' eX and all t e N
(4.1.1) d(sfry)(t), V./)W) < ^imax {d(x, x1), d(y, y')} + k2.
If s is a bounded bicombing by (λ, e)-quasi-geodesics and the constants k\ and ki
are as above, then we say that s has parameters [λ, ε, k\, кг\.
Let Г be a finitely generated group. We use the term metric Y-space to mean a
metric space X together with an action of Г on X by isometries. X is said to be Г-
semihyperbolic if it admits a bounded quasi-geodesic bicombing which is equivariant
with respect to the action of Г, that is y.S(X,y)(t) = %*.уз,)0) for all x, у e X and t e N.
Fix a word metric on Г. The action of Г on itself by left multiplication makes
it a metric Γ-space. Г is said to be semihyperbolic if, when viewed in this way,
it is а Г-semihyperbolic metric space, with bicombing s, say. Let [λ, ε, k\, кг\ be
the parameters of s. Because s is equivariant, it is entirely determined by the map
σ : Γ —> Ρ(Γ) that sends у to J(i.y). One calls σ a semihyperbolic structure for Γ
with parameters [λ, ε, &ι, £2].
4.2 Lemma. 77ге definition of a semihyperbolic group does not depend on the choice
of generators (word metric).
Proof. Let Г be a finitely generated group. Any two word metrics d, ά on Γ are
Lipschitz equivalent, i.e. there exists a constant К > 1 such that j^d(x, y) < d'(x,y) <
К d(x, y) for all x, у Ε Γ. Therefore any semihyperbolic structure σ for (Γ, d), with
parameters [λ, ε, k\, £2], is a semihyperbolic structure for (Γ, d') with parameters
[KX,K8,K2kuKk2]. D
4.3 Example. If Γ acts by isometries on a CAT(O) space X, then X is Γ-semihyper-
bolic: for all x, у e X there is a unique geodesic cx-y : [0, d(x, y)] —> X joining χ to
y, and the desired Γ-equivariant bicombing s:Xxi-> Ρ(Χ) is s^y)(t) := cXiy(t)
for all t e N П [0, d(x, y)] and 5(*-3,)(ί) = у for all r > d(x, y).
4.4 Exercise. Let Г be a group with finite generating set Л. There is natural map
from the free monoid (A±l)* to V(T) that sends each word w to the path 11-» w(t),
where w(t) is the image in Г of the prefix of length tinw.
Show that if Г admits a semihyperbolic structure σ : Γ -> Ρ(Γ), then it admits a
semihyperbolic structure σ' : Γ —> Ρ(Γ) (close to σ) whose image lies in (A±l)* С
ИГ), (cf. [AloB95] Section 10.)
This exercise shows that one can express the definition of a semihyperbolic group -
in purely algebraic terms:

Basic Properties of Semihyperbolic Groups 473
4.5 Proposition. Let Υ be a group with finite generating set A. Then Γ is
semihyperbolic if and only if there exist positive constants λ, ε, к and a choice of words
{wY | у ε Г} С (Л±1)*, such that wY = γ in Γ, the discrete paths t t-» wY(t)
determined by the words wY are (λ, e)-quasi-geodesics, and
d(wY(t),a.wa-iYa-(t)) < k,
for all a, a' e A±{ U {1} and all г е N.
Proof. The displayed condition is illustrated in figure (Г.8). If there exist words wY as
described then the triangle inequality in Г implies that the paths S(.t,j,)(0 := x.wx-iy(t)
satisfy condition (4.1.1) with k = k\ and &2 = 0. Thus γ м- wY is a semihyperbolic
structure for Г. The converse is the content of the preceding exercise. D
/v a Sfc ffl,A
Fig. Г.8 A semihyperbolic structure for Г
4.6 Proposition. Every hyperbolic group is semihyperbolic.
Proof. Let Г be a hyperbolic group with finite generating set A. For each γ e Γ,
choose a geodesic word wY that represents it. In any 5-hyperbolic geodesic space,
geodesies that begin and end a distance at most one apart are uniformly (25 + 2)-
close. By applying this observation to the Cayley graph of Γ, we deduce γ м- wY is
a semihyperbolic structure for Г. D
Basic Properties of Semihyperbolic Groups
Since the definition of semihyperbolicity is given in terms of Γ-spaces and not just
metric spaces, the natural morphisms under which to expect this property to be
preserved are not quasi-isometries but rather Γ-equivariant quasi-isometries. Thus the
following result provides an analogue of the quasi-isometric invariance of hyper-
bolicity (H. 1.9). In contrast to the hyperbolic case, the following theorem does not
extend to the case of quasi-isometric embeddings (cf. 5.12(3)).

474 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
4.7 Theorem (Invariance Under Equivariant Quasi-Isometries). Let X\ and X2 be
metric Y-spaces and suppose that there exists a Y-equivariant quasi-isometry f :
X\ —> X2. If the action of Υ on X\ is free andX2 is Y-semihyperbolic, then X\ is also
Y-semihyperbolic. Conversely, if Υ acts freely on X2 and Χι is Y-semihyperbolic,
then X2 is Y-semihyperbolic.
The proof of this result is straightforward but involves some quite lengthy
verifications so we omit it — see [AloB95]. Our main interest lies with the following
corollaries.
4.8 Corollary.
(1) If Υ acts properly and cocompactly by isometries on а С AT(0) space X, then Y
is semihyperbolic.
(2) If G is semihyperbolic and Η С G is a subgroup of finite index, then Η is
semihyperbolic.
(3) Let 1 —» F —» Γ —» Q —» I be a short exact sequence. IfF is finite and Q is
semihyperbolic, then Υ is semihyperbolic.
Proof. (1) Fix xq e X. In the light of (4.3), we may apply the theorem with Χι =
Y,X2 = X andДу) = γ.χ0.
(2) If we fix a word metric on G, then left multiplication by Η is an action by
isometries. If G is G-semihyperbolic then it is a fortiori H- semihyperbolic. The
action of Я is free and Η <^> G is an Я-equivariant quasi-isometry, so we may apply
the theorem with H=Y, X\ = Η and X2 = G.
(3) π is a Γ-equivariant quasi-isometry, where Γ acts on Q by y.q = n(y)q.
Thus we may apply the theorem with X[ = Υ and X2 = Q. □
Finite Presentability and the Word and Conjugacy Problems
4.9 Theorem. Let Υ be a group with finite generating set A. If Υ is semihyperbolic
then there exist positive constants K\ and K2 with the following properties.
(1) If lb is the set of words in F(A) that have length at most K\ and represent 1 e Γ,
then (A I TV) is a presentation of Y.
(2) A word w in the letters A±l represents the identity in Υ if and only if in the free
group F(A) there is an equality
N
where N < K2\w\2, each r, e TZ, and \xj\ < K2\w\. In particular Υ has a
solvable word problem.
(3) Γ has the quasi-monotone conjugacy property {1.9). In particular Υ has a
solvable conjugacy problem (1.11); indeed there exists a constant μ > 0 such

Subgroups of Semihyperbolic Groups 475
that words u, υ e F(A) represent conjugate elements of Τ if and only if there
is a word w e F(A) of length < дтах<М-М] w-lXn w-luw = v inT.
Proof. We arranged the proofs in the special case of groups which act properly and
cocompactly on С AT(0) spaces precisely so that they would generalize immediately
to the present setting. Thus the proof of (1.6) remains valid with an arbitrary
semihyperbolic structure γ м- σγ in place of the explicit one constructed in that proof: the
distance between σγ(ή and σγα(ί) is now bounded by the parameters of the
semihyperbolic structure rather than in terms of D (notation of (1.6)), but modulo a change
of constant the proof is exactly the same. This proves (1) and (2).
The fact that Γ has the q.m.c. property follows immediately from the
definition of semihyperbolicity: if σ is a semihyperbolic structure for Γ with parameters
[λ, ε, k\, £2] and и and υ are words in the generators such that y~luy = υ for some
γ e Γ, then w — σγ satisfies the requirements of (1.9) with constant К = k\ + k%.
D
Subgroups of Semihyperbolic Groups
In this section we generalize the subgroup results in (1.1) to the class of
semihyperbolic groups. In particular we shall prove the following algebraic analogue of the Flat
Torus Theorem (II.7.1), and we shall also prove an analogue (4.21) of the Splitting
Theorem (II.6.21).
4.10 Algebraic Flat Torus Theorem. If Г is semihyperbolic and A is a finitely
generated abelian group, then every monomorphism φ : A ^-» Γ is a quasi-isometric
embedding (with respect to any choice of word metrics).
We defer the proof to (4.16).
When is a subgroup of a semihyperbolic group itself semihyperbolic? Motivated
by the hyperbolic case (3.7), one might ask if being quasi-isometrically embedded or
quasi-geodesic is a sufficient condition. However, being quasi-isometrically
embedded does not force a subgroup to be semihyperbolic; it does not even force it to be
finitely presented (5.12(3)). However, when suitably adapted to the semihyperbolic
structure, the notion of quasiconvexity does provide a useful criterion for showing
that subgroups of semihyperbolic groups are semihyperbolic. This was recognised by
Gersten and Short, who were working in the context of biautomatic groups [GeS91].
4.11 Definition. Let X be a metric space with a bicombing s. A subset С С Х
is said to be quasiconvex with respect to s if there is a constant к > 0 such that
d(s(xy)(t), C) <k for all x, у е С and t e N.
If Г is a group with semihyperbolic structure σ, then a subgroup Я с Г is said
to be σ-quasiconvex if it is quasiconvex with respect to S(.v,j,)(i) := x.ax-\y(t). This is

476 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
equivalent to requiring that there exist a constant к > 0 such that d(a^(t), Я) < к for
all h e Η and t e N.
As in the hyperbolic case (3.7) we have:
4.12 Proposition. Let Υ be a group with a semihyperbolic structure σ. If a subgroup
Я С Г is σ-quasiconvex, then it is finitely generated, quasi-isometrically embedded,
and semihyperbolic.
Proof. Suppose that with respect to a fixed word metric dA, each of the paths σγ, γ e
Γ, is a (λ, e)-quasi-geodesic. Let к be the constant of quasiconvexity for Я (as in
4.11) and let В be the intersection of Я with the closed ball of radius (2k + λ + ε)
centred at 1 e Г.
The idea of the proof is as shown in figure (Г.6). Given h e Η with combing line
σ/, : [0, Th\ —> Γ, for each integer t < T/, we choose a point p, e Я closest to σι,(ή.
Let bj = p~}xPi. Note that bt e Я and dA(\, bt) = dA(Pi,Pi-\) < dA(ph σή(ι)) +
dA(ah(i), σή(/ - 1)) + dA(ah(i - 1), pt-\) <k + (X + s)+k, therefore bt e B.
The word a;f = b\ ... bj,: is equal to h in H, therefore β is a finite generating set
for#. Moreover, йв(1, h) < T/, < XdA(l, h) + s, and hence Я is quasi-isometrically
embedded in Г. Finally, since dA(a^(t), ah(t)) < к for all h e Η and t e N, we see
that σΗ is a semihyperbolic structure for Η with parameters that depend only on к
and the parameters of σ. D
The following result is due to Hamish Short57.
4.13 Proposition. Let Υ be a group with a semihyperbolic structure σ. The
intersection of any two σ-quasiconvex subgroups of Υ is σ-quasiconvex (and hence
semihyperbolic).
Proof. See figure (Γ.9). Suppose that H\, Яг С Г are σ-quasiconvex. If До is a finite
generating set for Г, then for some λ > 1 and ε > 0 each of the combing lines σγ(ί)
is a (λ, e)-quasi-geodesic in the word metric associated to Aq. Let А С Г be the set
of elements with dAo(l, a) < λ + ε. Note that A is finite and generates Γ; we shall
work with the associated word metric d. Note that ά(σγ(ί), σγ(ί + 1)) < 1 for all
γ e Υ and t e N. Let к be a constant of quasiconvexity (as in 4.11) for both Hi and
Я2.
Fix h e Я[ Π Яг and consider the combing line σ/, : [0, 7)J —> Г. For each integer
to 5 Th and; = 1, 2 we choose a point hj(t0) e Hi such that d(Oh(to), hj(t0)) < k.
Note that d(hi(t0), h2(t0)) < 2k.
We claim that ά(σ^ο), Η\ Π Яг) < V2, where V is the number of points in the
ball of radius к about 1 e Г. To see why, we write γ = σ^ίο) and consider those
words w over the alphabet A±l which have the property that γw e Hi Π Яг and
for every prefix и of w we have d(yu, Hj) < к for j =1,2. Let Ω denote the set of
"Groups and combings", preprint, ENS Lyon, 1990

Subgroups of Semihyperbolic Groups 477
Fig. Γ.9 The intersection of σ-quasiconvex subgroups
such words. Ω is non-empty; indeed it contains the label on any geodesic edge-path
connecting the vertices of σι,φο, Τ]) с Сд(Г) (in order). Let w0 e Ω be a word of
minimal length. We will be done if we can show that |wq\ < V2.
For each prefix и of wq and/ = 1, 2 we choose a point и,- e Щ closest to yu.
Note there are at most V2 possibilities for (и~'иь и_1иг) £ Γ χ Γ. If |wo| > V
then there would exist prefixes u, u' of Wq with \u\ < \u'\ and (u_iui, и~'иг) =
(u'~ u\, u'~xu'2). If this were the case, then for all g e Г and i = 1, 2 we would have
d(ug, Hi) = d(g, u~lUiHi) = d(g, и'~'^#,) = d(u'g, #,). Thus if w0 = uv = u'v',
then the word w = uv' would be in Ω. But \w\ < \vuq\, contradicting the minimality
of|w0|. □
We shall no w present a series of results that reveal an intimate connection between
quasiconvexity and the structure of centralizers in semihyperbolic groups.
4.14 Proposition. Let Г be a finitely generated group and let σ be a semihyperbolic
structure for Γ. The centralizer of every element in Γ is σ-quasiconvex.
Proof. Let σ be a semihyperbolic structure with parameters [λ, ε, k\, k2]. Let the
constant μ be as in (4.9(3)).
If γ e Cr(g) then the combing lines σγ and g.aY begin and end a distance d(l, g)
apart, so d(ay(t), g.oY{t)) < k\ d{\, g) + k2 for all t e N. In particular, if we fix t0
and let h = aY(to)~lgaY(to) then d(l,h) < k\ d{\, g) + k2. The elements g and h are
obviously conjugate, so according to (4.9(3)) there is an element χ e Γ with hx = xg
and

478 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
9Ί=Ί9
Fig. Γ.10 Centralizers are σ-quasiconvex
We will be done if we can show that aY(to)x e Cr(g).
<rY(to)xg =aY(t0)hx
= σγ (ί0) (σγ (ί0Γ' 8σγ (ί0)) χ
= gcrY(to)x·
The idea of the proof is illustrated in figure (Γ. 10).
D
4.15 Proposition. Let Υ be a group with semihyperbolic structure σ.
(1) If S с Г is a finite subset, then Cr(S) '■= С^\х€$Сг(х) is σ-quasiconvex; in
particular it is finitely generated and semihyperbolic.
(2) If Η С Г is a finitely generated subgroup, then Cr(H) is σ-quasiconvex.
(3) The centre Ζ(Γ) of Υ is a finitely generated abelian group, and Z(Y) с Г is
σ-quasiconvex
Proof. Part (1) follows immediately from (4.12), (4.13) and (4.14). To prove (2),
apply (1) to a finite generating set S for H, and to prove (3) take Η = Υ. D
4.16 The Proof of Theorem (4.10). We must show that if Γ is semihyperbolic and
Л is a finitely generated abelian group, then every monomorphism φ : A ^-» Γ is a
quasi-isometric embedding. The map φ can be factored as follows.

Subgroups of Semihyperbolic Groups 479
A ^ Z(Cr(<«A))) ^ Сг(ф(А)) ^ Г.
In the light of (4.12) and (4.15) we know that the last two of these maps are quasi-
isometric embeddings, so since the composition of quasi-isometric embeddings is
a quasi-isometric embedding, it suffices to show that A <^> Z(Cr((p(A))) is a quasi-
isometric embedding. But Z(Cr((/)(A))) is a finitely generated abelian group, and it
is easy to see that any monomorphism from one finitely generated abelian group to
another is a quasi-isometric embedding. D
In Chapter II.7 we used the Flat Torus Theorem to see that a polycyclic group
cannot act properly by semisimple isometries on а С AT(0) space unless it is virtually
abelian (II.7.16). One can use the Algebraic Flat Torus Theorem in much the same
way.
4.17 Proposition. A polycyclic group Ρ is a subgroup of a semihyperbolic group if
and only if Ρ is virtually abelian.
Proof. If Ρ is virtually abelian then it acts properly and cocompactly by isometries
on a Euclidean space of some dimension (II.7.3), and hence it is semihyperbolic.
To show that if Ρ is not virtually abelian then it is not a subgroup of a
semihyperbolic group, we use an idea due to Gersten and Short [GeS91]. As in (II.7.16),
induction on the Hirsch length reduces the problem to that of showing that if φ e GL(«, Z)
has infinite order then Η = Ζ" χφ (t) cannot be a subgroup of any semihyperbolic
group.
Consider the £i-norm on GL(«, Z) с W . Since φ has infinite order, the
function к ы> || фк || grows at least linearly, and hence for some basis element a e Z"
the function к м> аг»(фк(ак), 1) grows at least quadratically. But in Я we have
t~kak^ = фк(ак), showing that к н> ан(фк(ак), 1) grows linearly. Thus Z" is not
quasi-isometrically embedded in Η (nor any group containing it, a fortiori), and
hence Η cannot be a subgroup of any semihyperbolic group. G
In (1.1) we showed that if \p\ φ \q\ then groups of the form (x, t | t~lxpt = x4)
cannot act properly by semisimple isometries on any CAT(O) space. We did so by
means of a simple calculation involving translation lengths. This is one of a number
of results that one can recover in the setting of semihyperbolic groups using algebraic
translation lengths (3.13) in place of geometric ones.
4.18 Lemma. If Υ is semihyperbolic and γ e Γ has infinite order then τ (γ) > 0.
Proof. The inclusion into Γ of the infinite cyclic subgroup generated by у is a quasi-
isometric embedding (4.10). G
4.19 Corollary. If \p\ φ \q\ then (x, t \ t^yPt = x4) cannot be a subgroup of a
semihyperbolic group.

480 Chapter Ш.Г Non-Positive Curvature and Group Theory
Proof. In any finitely generated group Г, the number т(у) depends only on the
conjugacy class of у e Γ and τ (γ") = \n\ τ(γ) for all η e Z. Thus if γρ is conjugate
to Yq in Γ, then
|p|r(y) = r(yO = r(y9)=|^|T(y).
Thus \p\ ψ \q\ implies т(у) = 0. Since* has infinite order in В = (χ, t \ t~lxPt = x4)
(as we explained in the proof of (1.1)), it follows from (4.18) that В С Г implies Г
is not semihyperbolic. D
Direct Products
In analogy with (1.1(a)) we have:
4.20 Exercise. The direct product of any two semihyperbolic groups is
semihyperbolic.
And in analogy with the Splitting Theorem (II.6.21) we have:
4.21 Theorem (Algebraic Splitting). If Γ = Γι χ Γ2 is semihyperbolic and the
centre 0/Г2 is finite, then T\ is semihyperbolic.
Proof. Since Г is finitely generated, so too are Γι and Γ2. Let A be a finite generating
set for Γ2. Let S - {(1, a) \ a e А} с Г and note that Cr(S) = Γ{ χ Ζ(Γ2). From
(4.15) we know that Cr(S) is semihyperbolic, and since Γ ι has finite index in Cr(S),
it too is semihyperbolic. D
4.22 Remarks. At this stage we have established that most but not all of the properties
listed in (1.1) remain valid in the context of semihyperbolic groups. We conclude by
saying what is known about the remaining points.
The class of semihyperbolic groups is closed under free products with
amalgamation and HNN extensions along finite subgroups (see [AloB95]). It is also known
that every semihyperbolic group Γ is of type FPoo, but it is unknown whether Γ has a
finite K(T, 1) if it is torsion-free. It is unknown if semihyperbolic groups can contain
solvable subgroups that are not virtually abelian or abelian subgroups that are not
finitely generated.
In contrast to (1.1 (iv)), if Γ is semihyperbolic and у е Г has infinite order, then in
general (y) will not be a direct factor of a subgroup of finite index in Cr(y). Indeed,
following a suggestion of Thurston and Gromov, Neumann and Reeves [NeR97]
showed that if G is a hyperbolic group and Л is a finitely generated abelian group,
then every central extension Г of the form 1 —> A —» Г —> G —» 1 is semihyperbolic
(indeed biautomatic) — in general such central extensions Г do not have subgroups
of finite index that are non-trivial direct products.
In the above construction, Г is quasi-isometric to G χ A (although in general
they are not commensurable, cf. 1.8.21). Using this fact one can show that there exist

Finiteness Properties 481
semihyperbolic groups Γι and Γ2 such that Γι and Γ2 are quasi-isometric and Γι
is the fundamental group of a closed non-positively curved manifold, yet Γ2 cannot
act properly by semi-simple isometries on any CAT(O) space. For example, if Σ is
a closed hyperbolic surface, then one can take Tt to be я 1Σ χ Ζ and Γ2 to be the
fundamental group of the unit tangent bundle to Σ. (This example is explained in
detail in various places, including [Ep+92] and Section 8 of [AloB95].)
5. Subgroups of Cocompact Groups of Isometries
We have said a great deal about groups which act properly and compactly by
isometries on CAT(O) spaces. In the case of some results it was sufficient to assume that
the groups concerned were acting properly and semisimply (see (1.1)), in other cases
we definitely used the fact that the groups concerned were acting cocompactly. The
main purpose of this section is to show that in the latter case the results concerned do
not remain valid if one assumes only that the group is acting properly by semisimple
isometries. Specifically, we shall prove that the finiteness properties that we
established for cocompact groups of isometries are not inherited by finitely presented
subgroups, and we shall show that the solutions to the word and conjugacy
problem established in Section 1 are not inherited by finitely presented subgroups. In
the course of our discussion of decision problems we shall also prove that there are
compact non-positively curved spaces X for which there is no algorithm to decide
isomorphism among the finitely presented subgroups of π\Χ.
The underlying theme throughout this section is that subgroups of groups which
act properly and cocompactly by isometries on CAT(O) spaces can be rather
complicated.
Our discussion of finiteness properties is motivated by examples of John Stallings
[St63] and Robert Bieri [Bi76b]. Our discussion of decision problems is based on that
in the papers of Baumslag, Bridson, Miller and Short ([BBMS98] and subsequent
preprints).
Finiteness Properties
The most common and useful finiteness properties of groups are finite generation
and finite presentability. In (3.25) we described three conditions that measure higher-
dimensional finiteness, F„, FP„ and FL,,. (See [Bro82] or [Bi76b] for a detailed
introduction to these conditions.)
If we write #„(Г) to denote the nth homology group of Г with coefficients in the
trivial Γ-module Z, then
Fn => FLn =► FPn =► H„(Df.g.

482 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
Bestvina and Brady [BesB97] recently proved that there exist finitely generated
groups which are of type FPi but not of type Fi-
We saw in (II.5.13) that if Г is the fundamental group of a compact non-positively
curved space, then Г is of type Fn for every η e N. Our purpose in this subsection is
to show that this property is not inherited by arbitrary finitely presented subgroups of
Γ. In the process of doing so we shall construct explicit examples of finitely presented
subgroups that cannot be made to act properly and cocompactly by isometries on
any CAT(O) space (cf. 5.10).
Any direct product of finitely generated free groups is the fundamental group of
a compact non-positively curved space, namely the cube complex obtained by taking
the Cartesian product of graphs which have one vertex and the appropriate number of
loops. Such direct products have a surprising variety of finitely presented subgroups,
as the following theorem illustrates. We shall use this theorem as a convenient focus
for our discussion of finiteness properties, but we should offset this by emphasizing
that results of this type apply to many groups other than direct products of free groups
(see [BesB97], [MeiMV97], [Bi98]).
Given a group G, a subgroup Η С G and a positive integer m, we write Am(G; H)
to denote the free product of m copies of G amalgamated along Η (see 6.5) and
refer to the m distinguished copies of G as the factors of Am(G; H). There is an
implicit isomorphism from each factor to G and these isomorphisms agree on H. It
is sometimes convenient to regard Am(G; H) as a subgroup of A„(G; H)\im < n,
namely the subgroup generated by the first m factors. Note that there is then a natural
retraction A„(G; H) —» A,„(G; H) — this retraction is the identity on the first m
factors and sends each of the remaining factors to the first factor by means of the
implicit isomorphisms with G.
5.1 Theorem. L be a free group of rank m > 2 and Ζ,(,ί) denote the direct product
of η copies of L. Let φ : L -» Ζ be an epimorphism and let Kn be the kernel of the
induced homomorphism L^ -» Z. For every η > 1,
(1) Kn+l = Am(Z,<">; Kn), and
(2) K„ is of type F„-\ but Hn(Kn) is not finitely generated.
5.2 Exercises. With the notation of (5.1):
(1) Show that up to isomorphism K„ is independent of the epimorphism φ.
(2) Let Φ : L(n) -> Ζ be a homomorphism. Show that if ker Φ is of type Fr but
not type Fr+\, then the restriction of Φ to exactly (n — r — 1) of the direct factors of
Z,(n) is trivial. (Hint: If Η has finite index in G, then G is of type Fr if and only if Я
is of type Fr·)
We shall present the proof of (5.1) as a series of lemmas. The following simple
lemma indicates how the lack of finiteness described in 5.1(2) begins.
5.3 Lemma. Let L be a finitely generated free group and let N с L be a non-trivial
normal subgroup. IfL/N is infinite then Hi(N) is not finitely generated.

Finiteness Properties 483
Proof. L is the fundamental group of a finite graph X with a single vertex. Consider
the covering space X = X/N corresponding to N. Because L/N acts freely and
transitively on the vertices of X, if L/N is infinite then X contains infinitely many
disjoint loops. Thus the lemma follows from the fact that the fundamental group of
a connected graph is freely generated by the set of edges in the complement of any
maximal tree. D
5.4 Example (The Groups of Bieri and Stallings).
Let L be a free group of rank two with basis {αϊ, йг}· Let Ζ,(,ί) denote the direct
product of η copies of L, and let h„ : Z,(n) —» (t) = Ъ be the homomorphism that
sends each of the generators (1,..., α,,..., 1) to t. Let SB„ = ker hn; in the notation
of (5.1), this is K„ in the case m = 2.
Lemma 5.3 says that SBi is not finitely generated. If η > 2 then SB„ is finitely
generated, for example SB2 is generated by
{(αια2'. 1), (1, сца.21), (аь aj~'), (a2, a^1)}.
Theorem 5.1 says that SB2 is not finitely presentable and that SB3 is finitely
presentable but #з(8Вз) is not finitely generated. SB3 was the first example of a group
with these properties; it was discovered by John Stallings [St63]. Robert Bieri [Bi76b]
recast Stallings's example in the terms we have used and proved that SB„ is of type
Fn-\ but not type F„; this was the first such sequence of groups to be discovered.
There is now a substantial theory of finiteness properties of groups — see [Bi98] for
a recent survey and references.
The first of the following lemmas provides a tool for showing that groups are not
of type F„, the second provides a tool for showing that groups are of type Fn.
5.5 Lemma. Let Г = A *c B. If A and В are of type Fn but Hn-\(C) is not finitely
generated, then #„(Г) is not finitely generated.
Proof. Consider the Mayer-Vietoris sequence for Г = Л *с В:
> Hn(A)®Hn(B) —► Я„(Г) —* Я„_,(С) —► #„_ι(Α)Θ#„-ι(β) —* ·..
D
5.6 Corollary. Let A and С be finitely generated groups, let m > 2 be an integer,
and let Dm = Δ,„(Λ; С). If A is of type Fn but #„_i(C) is not finitely generated, then
H„(Dm) is not finitely generated.
Proof. As we noted before stating (5.1), there is a retraction of Dm onto Do = A *cA.
Thus Η,,φχ) is a direct summand of H„(Dm), and we can apply the lemma. D
5.7 Lemma. If A is of type Fn and С is of type Fn-\, then Am(A; C) is of type F„.

484 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
Proof. Let Xa be an Eilenberg-MacLane complex for A that has a finite «-skeleton.
Let Xc be an Eilenberg-MacLane space for С that has a finite (« — l)-skeleton. Let
/ : Xc —> Xa be a cellular map that induces the inclusion С <^> A. One obtains an
Eilenberg-MacLane complex X for Am(A; C) by taking (m — 1) disjoint copies of
the double mapping cylinder for/ and joining them at one end:
UTJi {(Xa, 0, 0 U {Xc x [0, 1], i) U (*л, 1, θ)
(x, 0, 1) ~ (*, 0, 0, (y, 0, 0 ~ (f(y), 0, i) and (y, 1, i) ~ (fty), 1,0'
where jc e Xa, у eXc and i = 1,..., m — 1.
The open и-cells in X are of two kinds: there are the open и-cells in the images
of Xa, and there are the products ex (0, 1) where e is an open (« — l)-cell in Xc.
Thus X has only finitely many «-cells. Π
We need two more lemmas.
5.8 Lemma. IfN с G is normal then there is an embedding
Φ : Am(G; Ν) ^ {*tiG/N)xG.
Proof. Let0 : Am(G;N) -> *?=lG/N be the quotient by N, and let ψ : Am(G; N) -►
G be the natural retraction onto the first factor (copy of G). ker$ = N and the
restriction of ψ to N is injective. Thus Φ :=(φ, ψ) is injective. D
In the following lemma it is convenient to write the infinite cyclic group (τ)
additively.
5.9 Lemma. Let G be a group. Let h : G —> {τ) be an epimorphism to an infinite
cyclic group and let N = ker h. Choose a e G with h(a) = τ and let a denote the
image of a in G/N. Let h: {*™=i G/N) xG-» (τ) be the map that is given on each
of the free factors of*™=lG/N by h(a) = —τ and is such that h\c = h. Let Φ be as
in (5.8). Then ker h = im Φ.
Proof. Am(G; N) is generated by N and a\,...,am, where Φ(α,) = (α,, a) and a,-
generates the г-th free factor of *™=iG/N. For each g e N we have Φ(#) = (1, g),
and hence h<&{g) = h(g) — 0. For each a,- we have: Ш(а) — ft(a,·, a) = -τ + τ.
Thus im Φ с ker h.
Now suppose (и, ν) e ker h. Write и as a word a*^ . ■ ■ а^г, and let
/ si .Tr χ —h(v)
Obviously aj" υ lies in N = ker ft, so the first coordinate of Ф(у) is и. On the other
hand, because —h(u) = h(v) = h(v), we have J^s, = h(v), and therefore the second
coordinate of Ф(у) is v. G

Finiteness Properties 485
Proof of Theorem 5.1
In the notation of (5.1): Lemma 5.8, with G = Z,(n) gives an embedding Am(Z,(,,); Kn)
^ ( *ы 2) χ LM = Lin+l), and Lemma 5.9 shows that (modulo the natural
identifications) the image of this embedding is K„+\. This proves 5.1(1).
We shall prove 5.1(2) by induction on n, using 5.1(1). In the induction we shall
use the fact that Z,(n) is of type Fr for every r e N. (It is the fundamental group of
the compact non-positively curved cube complex obtained by taking the product of
η metric graphs each of which has one vertex and m edges.)
The case η — 1 is covered by Lemma 5.3. Thus we may suppose that η > 2 and
that K„ С L(n) is of type Fn-\ but that Hn(Kn) is not finitely generated. According to
(5.6), the (n + l)st integral homology of Am0n); Kn) is not finitely generated. And
according to (5.7), Am(L"; Kn) is of type Fn. This completes the induction. D
The first part of Theorem 5.1 enables one to write down a simple presentation
of Kn. We give this description in the case m = 2, in which case K„ = SB,,, the и-th
Stallings-Bieri group.
5.10 Proposition (A Presentation for SB,,). If η > 2 then SB„+i is generated by Ъп
elements x\,..., x„, у ι, y\,..., y„, y'n subject to the relations
У~\ХУ) =У\~{Ур ta.J?] = fo.)';] = \xi,yfi = bi,yj\ = b'i,y'j\ = 1
for 1 < i <j < n.
Proof. Let Li be a free group with basis {x, y} and let Xj, yt, i = 1, ..., η be the
obvious generators for Z^ (that is, x, = (1,..., x,..., 1) etc.). The group SB„ is
the kernel of the map Φ : Εζ -» Ζ that sends each jc, to the identity and each у ι to a
fixed generator of Z.
The presentation displayed above describes Дг(/4 ; Я), where Η is the subgroup
generated by {χι, ■ ■ ■ ,хп,у[1У2, ■ ■ ■ ,y^lyn}· According to 5.1(1), in order to prove
the present proposition, it suffices to show that Η — ker Φ. It is clear that Η С ker Φ.
Conversely, suppose that γ = γ\...γη € ker Φ, where у, is a word in the generators
Xi andy,.
Each Yi is equal in the free group on {jc,, y-,} to a word of the form
For example, у\х^х] = (у\^2){у]хЪТЪ)у] ■_
If г > 2, then (y~[1yi)pxri(y~[iyiyp = У^х^У, р, and we also have
(yjХу\Ухгх(у2 1У\)~Р = /хАуТ- Hence all elements of the form y^uх\'чкy^Pi'k are
contained in H. Thus in order to show that ker Φ С Я, we need only show that Я
contains all elements of the form γ = yf ... y%" with Σ q^ = 0. But in this case, in
/4 we can write

486 Chapter Ш.Г Non-Positive Curvature and Group Theory
η
/=2
and since each yj~ ly,- is in H, we are done. D
5.11 Remarks
(1) The paper of Bestvina and Brady that we mentioned earlier [BesB97] clarified
the relationship between various finiteness properties of groups. Their main tool for
doing so was a Morse theory that they developed for CAT(O) cube complexes. If Г
is the fundamental group of a compact non-positively curved cube complex X and
h : Г —> Ζ is an epimorphism for which there is an h-equivariant Morse function58
μ : X —> К, then Bestvina and Brady are able to relate the finiteness properties of
ker h to the connectedness properties of the sub-level sets of μ.
The maps h„ : L(n) —> Ζ considered in (5.4) were the prototypes for this approach;
the complex X is a product of graphs each with one vertex and two edges; Z,(n) can
be identified with the vertex set of X and if one identifies L(n) with the zero-skeleton
of the universal covering X, then h„ extends uniquely to a Morse function X —> R.
(2) For и > 2, the Dehn functions of the groups K„ described in Theorem 5.1 are
quadratic [Bri99b]. Thus the existence of a quadratic isoperimetric inequality for a
group does not constrain the higher finiteness properties of a group, in contrast to
the subquadratic (i.e. hyperbolic) case (3.21).
5.12 Exercises
(1) Let L and Кг be as in (5.1). We have shown that Кг is not finitely presentable.
There is a theorem of Baumslag and Roseblade [BR84] which states that a subgroup
of Z,(2) is finitely presentable if and only of it contains a subgroup of finite index
that is either free or a direct product of free groups. By examining the structure of
centralizers, verify directly that Кг does not contain such a subgroup of finite index.
(2) (The HNN analogue of Lemma 5.8.) Let G be a group and let Л с G be a
normal subgroup. Show that the HNN extension (G, t | t~lat — a, Va e Л) can be
embedded in the direct product of G and (G/A) * Z.
(3) In the notation of (5.1): Show that K„ is an isometrically embedded subgroup
of LM if η > 2. (Note that in the case η — 2 this shows that an isometrically embedded
subgroup of a semihyperbolic group need not be finitely presented — cf. 3.7.)
(Hint: Follow the last part of the proof of (5.10).)
i.e. a map μ : X —> R that is affine on cells, constant only on 0-cells, and has the property
that the image of the 0-skeleton is discrete

The Word, Conjugacy and Membership Problems 487
The Word, Conjugacy and Membership Problems
We now turn our attention to decision problems for finitely presented subgroups of
groups which act properly and cocompactly by isometries on spaces of non-positive
curvature. We shall say something about each of the basic decision problems that
lie at the heart of combinatorial group theory: the word problem, the conjugacy
problem, and the isomorphism problem. We shall also say something about the
generalized word problem. At the end of the section we shall explain how these group
theoretic results impinge on the question of whether or not there exists an algorithm
to determine homeomorphism among compact non-positively curved manifolds.
The Word Problem
If a group has a solvable word problem then so do all of its finitely generated
subgroups: any word in the generators of the subgroup can be rewritten in terms of the
generators of the ambient group, and by applying the decision process in the ambient
group one can decide if the given word represents the identity.
The complexity of the word problem is a more complicated matter. The Dehn
function of a finitely presented group, as defined in (I.8A), estimates the number of
relations that one must apply in order to decide if a given word in the generators and
their inverse represents the identity in the group. If Γ acts properly and cocompactly
by isometries on a CAT(O) space, then the Dehn function for any finite presentation
of Γ is bounded above by a quadratic function (see 1.7). Since there is no obvious
way to bound the Dehn function of a finitely presented subgroup in terms of the Dehn
function of the ambient group, given a group that acts properly and cocompactly by
isometries on a CAT(O) space one might be able to use Dehn functions to identify
finitely presented subgroups that cannot themselves act properly and cocompactly
by isometries on any CAT(O) space. Here is an example of this phenomenon.
5.13 Theorem. There exist closed non-positively curved 5-dimensional manifolds
and finitely presented subgroups Η С π [Μ such that the Dehn function of Η is
exponential.
Proof. Let N be a closed hyperbolic 3-manifold that fibres over the circle with
compact fibre Σ (see [Ot96] for examples). Let S = π\Σ. The Dehn function of the
double A2(7T[N; S) is exponential (see 6.22). Lemma 5.8 gives an embedding of
Α2{π[Ν; S) into the direct product π\Ν χ F, where F is a free group of rank 2. This
direct product is a subgroup of any closed non-positively curved 5-manifold of the
form Μ = Ν χ Υ where У is a closed surface of genus at least two. Π
5.14 Remark. Examples such as (5.13) show that the difficulty of the word problem
in a group G is not completely described by the Dehn function: the Dehn function
measures the complexity of the challenge that one faces when trying to solve the word
problem using only the information apparent in a given presentation, but extrinsic

488 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
information, such as the existence of a nice embedding, might facilitate a more
efficient solution.
The Membership Problem
The generalized word problem (often called the Magnus problem or membership
problem) asks about the existence of an algorithm to decide whether words in the
generators of a group G define elements of a fixed subgroup H. If Η and G are finitely
generated and G has a solvable word problem, then the generalized word problem
for Η С G is solvable if and only if the distortion of Η in G, as defined in (6.17),
is not bounded above by any recursive function.
5.15 Proposition. There exist compact negatively curved 2-complexes К and finitely
generated subgroups N С π \ Κ for which there is no algorithm to decide membership
ofN (equivalently, the distortion ofN in π\Κ is non-recursive).
Proof. There exist finitely presented groups G with unsolvable word problem (see
[Rot95]). Let G be such a group and let (x\,..., xn \ r\,..., r,„) be a finite
presentation of G. The generalized Rips construction (II.5.45) associates to this presentation
a short exact sequence
1 -> Ν -» π{Κ^ G^ 1.
К is a compact, negatively curved, piecewise-hyperbolic 2-complex whose
fundamental group Tt\K has a finite presentation
{x\,...,xn,a\,...,aM \rk = vk, xfa-,xj~6 = uiJiS,
1 < i <M, 1 <j < n, 1 < k < m, ε = ±1),
where Vk and li[ : g Э.ГС words in the generators a — {a\, ...,ам] and N is the subgroup
generated by a.
The key thing to observe is that a word w in the letters χ = {x\,... ,xn] and
their inverses represents an element of N С Г if and only if w = 1 in G. We chose
G specifically so that there is no algorithm to decide if w = 1 in G, and therefore
there is no algorithm to decide whether words in the letters χ represent elements of
N сГ. D
The groups N considered in (5.15) are never finitely presented (II.5.47). In order
to construct finitely presented examples of a similar nature we shall use the following
theorem of Baumslag, Bridson, Miller and Short59. The name of this theorem comes
from the fact that the groups appearing in the short exact sequence are assumed to
be of types F\, Fj. and F3 respectively. (Type F„ was defined in (3.25).)
"Fibre products, non-positive curvature, and decision problems", Preprint 1998.

The Word, Conjugacy and Membership Problems 489
5.16 The 1-2-3 Theorem. Suppose that 1 —» N —» Γ —» G —» I is exact, and
consider the fibre product
P:=i(Yi,y2)\p(Yi)=p(yi)}CTxT.
Suppose that N is finitely generated, that Γ is finitely presented and that G of type
Fy Then Ρ is finitely presented.
It is easy to show that if N has finite generating set a and Γ is generated by (aUx),
then Ρ is generated by the elements a\ := (a,-, 1), af := (1, a,), and (xj,Xj), where
a, e a and x/ e x. The key to proving the theorem lies in understanding all of the
relations that hold in N С Г. The proof is long, technical, and purely algebraic; since
it would add little to the present discussion, we omit it.
5.17 Proposition. There exist compact non-positively curved 4-dimensional
complexes X and finitely presented subgroups Ρ С n\Xfor which there is no algorithm
to decide membership of Ρ (i.e., the distortion of Ρ in τΐ\Χ is non-recursive).
Proof. Collins and Miller [CoMi98] have constructed examples of finitely presented
groups of type F3 that are torsion-free and have an unsolvable word problem. By
applying the construction of (II.5.45) to a finite presentation of such a group G we
obtain a short exact sequence
1 ->N-> π{Κ Л- G^ 1,
where К and N are as in the proof of (5.15). LetX = Κ χ Κ.
Take generators χ/ and a, for it\K as in (5.15) and consider the associated
generators (xj, 1), (l,Xj), (a,, 1), (1, a;) for π\Χ. Theorem 5.16 tells us that the subgroup
P = {(уь γ2) Ι ρ(γ\) = ρ{γι)} С П[Х is finitely presented. Given a word w in the
generators (xj, 1), we ask if this word defines an element of P. The answer is "yes"
if and only if the word obtained from w by replacing each (jj, 1) by x- represents the
identity in G. Thus membership of Ρ is undecidable. D
The Conjugacy Problem
In general, the conjugacy problem is less robust than the word problem. For example,
there exist finitely presented groups which have a solvable conjugacy problem but
which contain subgroups of index two that have an unsolvable conjugacy problem
[CoMiW]. In (1.12) we showed that if a group Γ acts properly and cocompactly by
isometries on a CAT(O) space then it has a solvable conjugacy problem. Thus we
might use the (un)solvability of the conjugacy problem as an invariant for identifying
finitely presented subgroups of Γ that cannot act properly and cocompactly on any
CAT(O) space.
Notation. We shall continue our earlier practice of writing Cr(g) for the centralizer
of the element g in the group Γ.

490 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
5.18 Lemma. Let Η С Q С Г be finitely generated groups. Suppose that Η is
normal in Γ and that there exists ho e Η such that Сг(^о) С Q.
If there is no algorithm to decide membership of Q, then Q has an unsolvable
conjugacy problem.
Proof. We fix finite generating sets В for Г and Л for H, and for each a e A, b e В
and ε = ±1 we choose a word ua^,e m the generators A±l so that b6ab~~s = иа.ь,е
in H. Fix a word in the generators A that equals ho in Г.
Given an arbitrary word w in the generators B±l, we can use the relations
beab~e — иаЬе to write wh0w~~l as a word u/ in the generators A±l ■ (The length
of w' is bounded by an exponential function of the length of w and the process of
passing from w to w' is entirely algorithmic.)
Now ask if w' is conjugate to ho in Q. Well, if there exists q e Q such that
q~~[hoq — w', then gw e Сг(^о) С β, whence w e (λ Thus u/ is conjugate to ho in
Q if and only if w e Q. And there is no algorithm to decide membership of Q. D
5.19 Theorem. There exists a negatively curved 2-dimensional complex К and a
finitely presented subgroup Ρ С π ] (К х К) such that P has an unsolvable conjugacy
problem.
Proof. Let K,N and Ρ be as in the proof of (5.17). Fix one of the generators aoiN
(each generator is non-trivial, II.5.46). Let ho = (a, a) e π\Κ χ π\Κ. Because π\Κ
is torsion-free (II.4.13) and hyperbolic, the centralizer of a in it\K is cyclic (3.10).
Moreover, a is not a proper power, because the closed geodesic in К representing a
is the shortest homotopically non-trivial loop in К (II.5.46). Thus the centralizer of
α in 7Ti/ns (a) and the centralizer of h0 in 7τι (^ x K)is {(a", oi") \n,m e Z}, which
is contained in Ν χ Ν С P. Therefore we may apply Lemma 5.18 with Η = Ν χ Ν
and Q = P. D
5.20 Remarks
(1) There exists a negatively curved 2-dimensional complex К and a finitely
generated normal subgroup N с n\K such that N has an unsolvable conjugacy
problem. To see this, let К and N be as in the proof of (5.15) and apply Lemma 5.18
with N = Η = Q and one of the generators a e N in the role of ho.
(2) At the time of writing, there are only a few known examples of non-hyperbolic
finitely presented subgroups in hyperbolic groups. The known examples all have a
solvable conjugacy problem.

Isomorphism Problems 491
Isomorphism Problems
The main result of this subsection is:
5.21 Theorem. There exists a compact, non-positively curved 4-dimensional
polyhedral complex X and a recursive sequence of finitely presented groups Pn (n e N)
with explicit monomorphisms Ρn —> Tt\X, suchthat there is no algorithm to determine
if Ρ n is (abstractly) isomorphic toP\.
This theorem is due to Baumslag, Bridson, Miller and Short. The original proof
was algebraic. We shall give a shorter geometric proof based on the Flat Torus
Theorem (II.7.1).
Whenever one is trying to prove that the isomorphism problem is undecidable
in a certain class of groups, one is invariably faced with the difficulty of ruling
out "accidental" isomorphisms - one needs invariants that allow one to deduce that
if some obvious map is not an isomorphism then the groups in question are not
isomorphic. The invariants that we shall use for this purpose are the centralizers of
elements; our previous results concerning centralizers in groups that act by semi-
simple isometries on CAT(O) spaces will be useful in this regard.
In the following lemma we can ignore the obvious basepoint difficulties
associated to statements about Tt\ because we are only interested in specifying elements
and subgroups up to conjugacy.
5.22 Lemma. Let Η be a group acting freely and properly by semisimple isometries
on a complete CAT(— 1) space X. Let cq,c\ : S —> H\X be isometrically embedded
circles (of the same length) and let Υ be the space obtained by gluing the ends of a
cylinder С = S χ [0, 1] to H\X along со and c\ respectively. Let С be the image of
С in Y.
(1) If the images ofco and с \ are not the same, then π ι Υ does not contain a subgroup
isomorphic to Z2.
(2) If the images ofco and с \ are the same, then π\ Υ contains subgroups isomorphic
to Ί?, and any such subgroup is conjugate to a subgroup ofn\ С С τΐ\Υ.
Proof. Note that the local gluing theorem (II. 11.6) implies that Υ is non-positively
curved in the quotient path metric, and H\X <^-> Υ is a local isometry.
(2) If the images of Co and с ι coincide, then С is an isometrically embedded flat
torus or Klein bottle, and by (II.4.13) it\C с it\ Υ is isomorphic to the fundamental
group of the surface. Moreover, since H\X is locally CAT(—1), the image of any
local isometry from a torus to Υ must be contained in С The Flat Torus Theorem
tells us that every monomorphism φ : 1? —» it\ Υ is represented by such a map from
a torus, and therefore the image of φ is conjugate to a subgroup of it\ С С π{Υ.
(1) Suppose that the images of Co and с ι do not coincide, and consider a point ρ that
is in the image of Co but not in the image of с ι. In Y, this point ρ has a neighbourhood
that is obtained by gluing a Euclidean half-disc (a neighbourhood of a boundary

492 Chapter Ш.Г Non-Positive Curvature and Group Theory
point in C) to a CAT(— 1) space B(p, ε) С H\X. In particular, given any Euclidean
disc 5(0, г) С Ε2, there does not exist an isometric embedding/ : 5(0, r) —> Υ with
ДО) = p. Thus if со is a geodesic line in Υ that covers Co, then Co cannot be contained
in any flat plane. Hence the strip Μ χ [0, 1] in the pre-image of С that is attached to
Co is not contained in any flat plane. Since the complement of the £reimage of С is
locally CAT(—1), we have proved that there are no flat planes in Y, and therefore,
by the Flat Torus Theorem, no subgroup of πι Υ is isomorphic to Z2. G
5.23 Corollary. Let N and К be as in the modified Rips construction (II.5.45) and
fix one of the generators a £ a described there.
(1) If a subgroup of the HNN-extension
(jt\K, τ | τ"'ατ = a)
is isomorphic to Z2, then it is conjugate to a subgroup of (α, τ).
(2) Ifye tt\K, then the HNN-extension
Ny := (N, 11 t~lat = γ~{αγ)
contains a copy ofL2 if and only if γ e N.
Proof. We pointed out in (Π.5.46) that the free homotopy class of loops representing
the conjugacy class of α e π\Κ is represented by a unique isometrically embedded
circle, с : S —> К say. By the Seifert-van Kampen Theorem, the HNN extension
displayed in (1) is the fundamental group of the space obtained from К by
attaching both ends of a cylinder S χ [0, 1] along с Thus assertion (1) is an immediate
consequence of 5.22(2).
(2) In N\K, the conjugacy classes in N of a and γ~χαγ e N are each represented
by a unique isometrically embedded circle — call these circles со and c\, respectively.
These circles have the same image if and only if a and γ~ιαγ are conjugate in N.
К is a CAT(— 1) space, so we can apply the lemma with X = К and Η = N. By
the Seifert-van Kampen Theorem, the space Υ obtained by attaching the ends of a
cylinder to Co and c\ is Νγ. According to (5.22), this group contains Z2 if and only
if a is conjugate to γ~χαγ in N.
If there exists η e N such that n~lan = γ~ιαγ, then ηγ~ι e Cr(a). But, as we
noted in the proof of (5.19), Cr(a) = (a) с N. Thus a is conjugate to γ~ιαγ in N
if and only if γ e N. D
We need one more lemma in order to prove Theorem 5.21.
5.24 Lemma. Let Η be a group. For all a,heH, the following HNN extensions are
isomorphic:
(H, 11 Clat = hT[ah) = (Η, τ | τ~ιατ = a).
Proof. The desired isomorphism sends Η to itself by the identity and sends ttorh.
The inverse sends τ to th~'. G

Isomorphism Problems 493
The Proof of Theorem 5.21. Let Ν, Κ and Ρ be as in (5.17) and let Г = щК. We
shall work with the presentation of Γ = π\Κ described in the proof of (5.15); in
particular Γ has generators (a U x). We write В to denote the set of generators for
Ρ described immediately following (5.16), namely a\ = (a,·, 1), af — (1, a,·), with
a,· e a, and (χ, χ), χ e x. Let (В \ Щ be a finite presentation for P.
Given a reduced word in the letters χ±ι, we can use the relations xfaixj~e = m,j?£
in our presentation of Γ to rewrite ιυ~'αιΐυ as a word VF in the letters a*1 that
represents the same element of N С Г. We write W7, for the word in the generators
of Ρ obtained from W by replacing each a,· by a\ = (a,·, 1).
The length of W is bounded by an exponential function of the length of w and the
process of passing from w to W is entirely algorithmic. Thus we obtain a recursive
family of group presentations indexed by the set of reduced words w in the letters
Vw = (B, tw | π, t~xa\tw = WL, t~laftw = af Va, e a).
Note that Vw is a presentation for the HNN extension of Ρ by a single stable letter
tw that commutes with Ρ Π ({1} χ Γ) and conjugates a\ to WL. Let Pw denote this
HNN extension.
We also consider the following HNN extensions of Γ:
Гш = (Г, tw | t~laiiw = w~laiw = W),
and regard Гш χ Γ as an HNN extension of Γ χ Γ with stable letter tw := (tw, 1).
Let Pw be the subgroup of Гш χ Γ generated by Ρ and ?ш.
Each element of Ρ С Γ χ Г is of the form ρ = (γ, γα) where a € N. Thus
ρ e (WL, αχ) if and only if ρ e (WL) χ Γ. Similarly, /? e (a{,aR) if and only if
/? e (αϊ) χ Γ. It follows that if an element of the HNN extension Pw is in reduced
form (in the sense of (6.4)), then its image in Гш is also in reduced form. Hence, for
each word w, the natural map Pw —> Pw С Гш χ Γ is injective.
By combining these natural maps with the explicit isomorphisms Гш —» Г
described in Lemma 5.24, we get explicit embeddings into Γι χ Γ of the groups given
by the presentations Vw:
РшДР^Г„хГ4Г,хГ.
Γι is the fundamental group of the compact non-positively curved complex K\
obtained by attaching both ends of a cylinder along the closed local geodesic in К
that represents a.\. We claim that a space X as required in the statement of Theorem
5.21 can be obtained by setting X = K\ χ Κ, in which case π\Χ = Γι χ Γ. We
will be done if we can show that there does not exist an algorithm to decide if Pw is
isomorphic to Pi.
There is no algorithm to decide if a word w in the generators x±l represents an
element of N С Г (see 5.15), and therefore the following lemma completes the proof
of Theorem 5.21.
5.25 Lemma. Pw is isomorphic to Pi if and only ifPw contains a subgroup
isomorphic to Ί?, and it contains such a subgroup if and only ifw represents an element of
NCT.

494 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
Proof. Since Г is torsion-free and hyperbolic, any homomorphism Z3 —> Г must
have cyclic image. Applying this observation to the projection of Pw сГшхГ onto
the second factor, we see that Pw contains a subgroup isomorphic to Z3 if and only
if Iw := Pw П (Гш χ {1}) contains Ζ2. Note that since the action of {x, x) e Ρ by
conjugation on Гш χ {1} is the same as the action of (x, 1), in fact Iw is normal in
Гш χ {1}; the quotient is G, in the notation of (5.17), which is torsion-free.
Applying (5.23(1)) to Гш (via the isomorphism Гш = Γι in (5.24)), we see that
Iw contains Z2 if and only if Pw contains a conjugate of a subgroup of finite index in
(β,, tw(wL)~l), where wL = (w, 1) e Гш χ Γ. Since Iw is normal and contains a\, it
contains such a conjugate only if it contains a power of tw(wL)~l. Since G is torsion-
free, this happens only if tw(wL)~l € Pw. And since tw e Pw, this is equivalent to
saying that wL = (w, 1) lies in P, in other words (w, 1) e Ν χ {1} = Ρ Π (Γ χ {1}).
D
5.26/?emarfo
(1) One can subdivide the complex constructed in (II.5.45) and remetrize it as
a cubical complex. If one does so, then one gets an induced cubical structure of
non-positive curvature on the complex X constructed in the proof of (5.21).
(2) In the notation of the proof of (5.21), the group Nw of (5.23(2)) is given by
the relative presentation (N, tw \ t~latw = W). As above, for each word w in the
generatorexmereisanaturalmonomorphismA^ <^-> π\Κ\, and there is no algorithm
to decide if Nw is isomorphic to Λ^. Indeed Nw S N[ if and only if Nw contains Z2,
and by (5.23(2)) we know that this happens if and only if w e N, and membership
of N is algorithmically undecidable.
Distinguishing Among Non-Positively Curved Manifolds
Closed 2-manifolds were classified in the nineteenth century — they are determined
up to homeomorphism by orientability and Euler characteristic. If Thurston's Ge-
ometrization Conjecture [Thu82] is true then the homeomorphism problem for
compact 3-manifolds is also solvable. In other words, there is an algorithm which takes
as input pairs of compact 3-manifolds and answers YES or NO, after a finite amount
of time, according to whether or not the manifolds are homeomorphic. An easy
consequence of this is that there is an algorithm that will produce a list of compact
3-manifolds such that every compact 3-manifold is homeomorphic to exactly one
member of the list. (We are implicitly assuming that the manifolds under
consideration are described as finite objects. For the sake of argument, since all 3-manifolds
are triangulable, let us suppose that they are specified as finite simplicial complexes.)
Markov [Mark58] showed that in dimensions η > 4 there can be no algorithm
to decide homeomorphism among closed (smooth, PL or topological) manifolds,
because the existence of such an algorithm would contradict the fact that the isomor-

Distinguishing Among Non-Positively Curved Manifolds 495
phism problem for finitely presented groups is unsolvable60. However, if one restricts
one's attention to closed manifolds of negative curvature then the homeomorphism
problem is solvable in dimensions η > 5. To see this one needs the following deep
topological rigidity theorem of Farrell and Jones [FJ93].
5.27 Theorem. Let η > 5 and let Μ and N be closed non-positively curved
manifolds. lfit\M = Tt\N then Μ and N are homeomorphic.
Farrell and Jones worked in the smooth category. For an extension to the
polyhedral case see [Hu(B)93]. Fundamental groups of closed negatively curved manifolds
are hyperbolic and torsion-free so, as Sela points out in [Sel95], by combining the
solution to the isomorphism problem for torsion-free hyperbolic groups [Sel95] with
(5.27) one gets:
5.28 Theorem. Let η > 5 be an integer. There exists an algorithm which takes as
input two closed η-manifolds that support metrics of negative curvature, and which
(after afinite amount of time) will stop and answers YES or NO according to whether
or not the manifolds are homeomorphic.
There is a technical problem here with how the manifolds are given. They must be
given by a finite amount of information (from which one can read off a presentation
of the fundamental group). Cautious readers should interpret (5.28) as a statement
regarding the homeomorphism problem for any recursive class of negatively curved
manifolds.
5.29 Remarks
(1) Let η e N. At the time of writing it is unknown whether or not there exists an
algorithm which takes as input two compact non-positively curved и-manifolds, and
(after a finite amount of time) stops and answers YES or NO according to whether
or not the manifolds are homeomorphic.
(2) By a process of relative hyperbolization, one can embed any compact non-
positively curved piecewise Euclidean complex isometrically into a closed manifold
of higher dimension that has a piecewise Euclidean metric of non-positive curvature
(see [Gro87], [Hu(B)93]). Thus, in the light of (5.26(1)), the space X in Theorem
5.21 can be taken to be a closed manifold with a piecewise Euclidean metric. The
connected covering spaces of this manifold are Eilenberg-MacLane complexes and
hence thehomotopy type of each is determined by its fundamental group. Thus (5.21)
implies that there exists a closed non-positively curved manifold Μ and a sequence of
covering spaces M,· —> M, i e N, with each ττχΜι finitely presented, such that there
does not exist an algorithm to determine whether or not M,· is homotopy equivalent
60 For each η > 4, there is an algorithm that associates to any finite presentation V a closed
rc-manifold with fundamental group \V\. Markov shows that if one could decide
homeomorphism among the resulting manifolds then one could decide isomorphism among the
groups being presented.

496 Chapter III.Γ Non-Positive Curvature and Group Theory
to M0. (Again, one has to be careful about how the sequence M,· is given in this
theorem.)
In this section we have seen that the finitely presented subgroups of the
fundamental groups of compact non-positively curved spaces form a much more diverse
class than the fundamental groups themselves. We have essentially shown that if
a property of cocompact groups is not obviously inherited by all finitely presented
subgroups, then in general it will not be inherited. However we should offset this
statement by pointing out that in general it is difficult to distinguish those subgroups
which can act cocompactly by isometries on CAT(O) spaces from those that cannot.
For instance:
5.30 Theorem. Let Μ be a closed hyperbolic 3-manifold that fibres over the circle.
There exist subgroups G с nl(M χ Μ χ Μ) such that
(1) G has a compact K(G, 1),
(2) G has a quadratic Dehn function,
(3) G has a solvable conjugacy problem,
(4) the centralizer С of every element ofG has a compact K(C, 1), but
(5) G is not semihyperbolic.
The point here is that G acts properly by semi-simple isometries on a CAT(O)
space and satisfies most of the properties associated with groups that act cocompactly,
and yet G cannot be made to act properly and cocompactly by isometries on any
CAT(O) space.
We should also mention that, in contrast to the general theme of this section,
there are certain classes of non-positively curved spaces for which it is true that if X
is in the class then any finitely presented subgroup of π\Χ is also the fundamental
group of a space in the class. We saw in (II.5.27) that 2-dimensional Μ,,-complexes
have this property. Compact 3-dimensional manifolds 61 form another such class —
this is explained in [Bri98b].
6. Amalgamating Groups of Isometries
A basic way of constructing interesting new groups is to combine known examples
using amalgamated free products and HNN extensions. The Seifert-van Kampen
theorem tells us that these processes appear naturally in geometry and topology:
they describe exactly what happens when one starts gluing spaces along 7ri-injective
subspaces.
In this section we shall use our earlier results concerning gluing and isometries
of CAT(O) spaces to address the following:
61 The manifolds may have boundary, and the boundary need not be convex.

Amalgamated Free Products and HNN Extensions 497
6.1 Question. If the groups Y\ and Y2 act properly and cocompactly by isometries
on CAT(O) spaces, then under what circumstances can one deduce that various
amalgamated free products of Y\ and Г 2 also act properly and cocompactly by
isometries on a CAT(O) space? Similarly, if Υ acts properly and cocompactly by
isometries then under what circumstances can one deduce that a given HNN extension
of Υ acts in the same way?
The analogous questions for groups which act semisimply will also be considered,
but only in passing.
We shall give criteria for showing that certain amalgamated free products and
HNN extensions do act nicely on CAT(O) spaces, and we shall also present some
sobering counterexamples to illustrate what can go wrong.
For results concerning the question of when an amalgamated free product of
5-hyperbolic groups is 5-hyperbolic, see [BesF92].
Amalgamated Free Products and HNN Extensions
Amalgamated free products and HNN extensions have appeared a number of times
at earlier points in this book, but since they are central to the present discussion we
take this opportunity to recall their definitions.
6.2 Definitions. Let Я be a group and let (Γλ : λ e Λ) be a family of groups.
Associated to any family of monomorphisms (φχ : Η —> Γλ : λ е Л) one has an
amalgamated free product, which is the quotient of the free product *х<=дГ\ by the
normal subgroup generated by the conjugates of the elements {0λ(/ϊ)0λ'(/ϊ)~' | h €
Η, λ, λ' e Λ}. The natural map from each Γλ to the amalgamated free product is
injective, and we identify Γλ with its image in order to realise it as a subgroup of the
amalgamated free product.
In the case Λ = (1, 2}, it is usual to write Γι *Η Y2 to denote an amalgamated free
product (suppressing mention of the given maps Η <^> Γ,), and to refer to Γι *Η Γ2
as "an amalgamated free product of Γι and Γ2 along #."
Let Г be a group and let φ : A1 —> A2 be an isomorphism between subgroups
of Γ. Associated to this data one has an HNN extension61 of G: this is the quotient
of Γ * (t) by the smallest normal subgroup containing {а~~Чф(а)Г1 \ а е А1}. This
quotient may be described by the relative presentation
Г*0 = (Г, t Ι Γιαι = φ(α), VaeA,),
or by a phrase such as "an HNN extension of Υ by a stable letter t conjugating A1 to
A2".
62 The initials HNN are in honour of Graham Higman, Bernhard Neumann and Hanna
Neumann, who first studied these extensions [HNN49]. See [ChaM82] for historical details.

498 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
The group Г is called the base of the extension. The natural map Г —> Г*^ is
injective, and thus we regard Г as a subgroup of Г*^.
Following common practice, we shall normally use the more casual notation Г*^
to describe Г*ф, whereA is an abstract group isomorphic toA ι andA2. And we shall
refer to Г*а as "an HNN extension of Г over Л." In the special case whereA с Г is
a specific subgroup and φ is the identity map, one has the trivial HNN extension of
Γ over A, which we denote
Γ*„ =(Γ, 11 rlat = a, VaeA).
6.3 The Bass-Serre Tree. Free products with amalgamation and HNN extensions
are the basic building blocks of graphs of groups63 in the sense of Bass and Serre
[Ser77]. (The fundamental group of any graph of groups can be described by taking
iterated amalgamated free products and HNN extensions.)
An amalgamated free product of the form Γι *Η Γ2 is the fundamental group of
a 1-simplex of groups, where Γι and Γ2 are the local groups at the vertices of the
simplex ("the vertex groups") and Я is the local group at the barycentre of the 1 -cell
("the edge group"). An HNN extension Г*А is the fundamental group of a graph of
groups with a single vertex group Г and a single edge group A. The homomorphisms
between the local groups are those implicit in the notations Γι *н Г2 and Г*А.
All graphs of groups are developable in the sense of Chapter С (see C.2.17).
The universal development gives an action of the fundamental group of the graph
of groups on a tree called the Bass-Serre tree. The tree for Γι *Η Γ2 was described
explicitly prior to (II. 11.18), and the tree for Γ*Α was described explicitly in the proof
of (II. 11.21). The vertex stabilizers for these actions are the conjugates of the given
vertex groups. Thus, as a special case of (II.2.8), we see that every finite subgroup
of Γι *н Г2 is conjugate to a subgroup of Γι or Γ2, and every finite subgroup of T*A
is conjugate to a subgroup of the base group Г.
We shall need the following well-known fact (see [Ser77], 5.2, Theorem 11).
6.4 Lemma. Let G — T\*h Г2· Letxo, ...,x„eTi andy0,... ,y„ еГ2Ье elements
such that χ, φ Η ifi > 0 and y,· φ Hifi < n. Then xoyo... xny„ φ 1 in G. When an
element ofG is expressed as such a product, it is said to be in reduced form. Every
element ofG ν {1} can be written in reduced form.
In G*0 = (Γ, t I t~lat = φ(α), Va e A), consider a product zof"'zi... zmf""zn,
where the гщ are non-zero integers. Suppose ζ,- φ A ifm, < 0 and ζ,- φ φ(Α) if mi > 0.
Then zot""zi ■ ■ ■ Zmf'Zn Φ 1 in G*^. When an element ofG*,!, is expressed as such
a product, it is said to be in reduced form. Every element of G*^ can be written in
reducedform.
The above fact concerning HNN extensions is often called Britton's Lemma.
1-complexes of groups in the language of Chapter C.

Amalgamated Free Products and HNN Extensions 499
6.5 Doubling Along Subgroups and Extending Over Them. In (II. 11.7) we
discussed the construction of doubling a space X along a closed subspace Y. As a set,
the double of X along Υ is the disjoint union of two copies of X (denoted X and X)
modulo the equivalence relation generated by \y ~ y, Vy e Y]. When endowed with
the quotient topology, this space is denoted D(X; Y). More generally, one can take m
copies of X and identify them along Y; the resulting space is denoted Dm(X; Y). We
also considered the analogous construction Χ*γ, which is the disjoint union of X and
Υ χ [0, 1] modulo the equivalence relation generated by \y ~ (y, 0) ~ (y, l)Vy e Y].
There are analogous constructions for groups. As in (5.1), given a group Γ and
a subgroup H, one can consider the amalgamated free product of m copies of Γ
along H, denoted Δ„(Γ; Η). In the terminology of (6.2), this is an amalgamated free
product whose index set Λ has m elements, each Γλ is Γ, and each of the maps φχ is
the inclusion Η <^-> Γ. One can also consider the trivial HNN extension Г*н.
If X is a path-connected topological space, У is a path-connected subspace that
has an open neighbourhood which retracts onto it, and Υ <^> Χ induces an injection
of fundamental groups, then by the Seifert-van Kampen Theorem the fundamental
group ofDm(X; Y) is Δ„,(7ΓιΧ; π\Υ) and π\{Χ*γ) = {π\Χ)*πχΥ.
6.6 Proposition. Let Γ be the fundamental group of a compact non-positively
curved space X, let Ζ be a compact geodesic space and letf : Ζ —> Xbe a local isometry.
Let Η = /*(7ΓιΖ) С Г. Then Г*н and Ат(Г; Η) are the fundamental groups of
compact non-positive ly curved spaces.
Proof. These are special cases of our earlier results concerning equivariant gluing,
(11.18) and (11.21), but for clarity of exposition we repeat the details in the present
simpler setting.
(II.4.14) and (II. 11.6) show that the quotient Q of X Ц (Ζ χ [0, 1]) by the
equivalence relation generated by f(z) ~ (г, 0) is non-positively curved. Let Υ denote the
image of Ζ χ {1} in β and note that Dm(Q; Y) and Q*Y are non-positively curved
(II. 11.7). The obvious homotopy equivalence between Q andX gives an isomorphism
Tt\Q = Γ identifying it\ Υ с n^Q with H. We know from (II.4.13) that π]Υ -> Tt\Q
is an injection, so the general remarks immediately preceding the proposition apply.
α
6.7 Exercises
(1) Show that if the trivial HNN extension Γ*Α acts properly and cocompactly by
isometries on a CAT(O) space, then A is quasi-isometrically embedded in Γ. (Hint:
Argue, perhaps using (6.4), that the centralizer of the stable letter ί is A x (t). Apply
(4.14).)
(2) If Η is normal in Γ and Y/H = Z, then Δ2(Γ; Η) is isomorphic to the trivial
HNN extension Г*н.
We shall show in (6.16) that the converse to (6.7(1)) fails: there exist groups
Г that act properly and cocompactly by isometries on a CAT(O) space and quasi-
isometrically embedded subgroups Л С Г such that Г*А does not act properly and
cocompactly by isometries on any CAT(O) space.

500 Chapter Ш.Г Non-Positive Curvature and Group Theory
Amalgamating along Abelian Subgroups
In Chapter II. 11 we proved that if two groups Γι and Г2 act properly and cocompactly
by isometries on CAT(O) spaces, then so too does any amalgamated free product of
the form Γ] *ζ Γ2. In this subsection we examine the extent to which this result can
be generalized to the case of amalgamations along other abelian groups. In the next
subsection we shall consider amalgamations along finitely generated free groups.
6.8 Proposition. If Г acts properly and cocompactly by isometries on a CAT(O)
space, then so too does every amalgamated free product of the form
Ат(Г; А)
and every HNN extension of the form
(Г, 11 t~lat = a, a eA),
where A is virtually abelian.
Proof We proved in (II.7.2) that if Г acts properly and cocompactly by isometries
on a CAT(O) space, and Л С Г is a virtually abelian subgroup of Г, then there
is a proper cocompact action of Л on a Euclidean space E'" and an A-equivariant
isometry Em —> X. Thus we may apply the equivariant gluing techniques described
in (11.11.18) and (11.11.21). G
One would like to generalize the preceding proposition to cover, for example,
the case of arbitrary amalgamations of the form Γι *z» Γ2, where the Г,· are the
fundamental groups of compact non-positively curved spaces. One can do so in
certain special cases — for example (II. 11.37) where the groups Г,· are non-uniform
lattices in SO(«, 1). However, if η > 2 then in general one cannot make Γι *£,. Γ2 the
fundamental group of a compact non-positively curved space. Let us consider why.
In the case η = 1, one can take compact non-positively curved spaces Χι andX2
with Γ,- = Tt\Xi and then scale the metrics on them so that the closed geodesies in the
free homotopy class of the generators of the subgroups being amalgamated have the
same length; one then attaches the ends of a tube to these closed geodesies. Consider
what happens when we try to imitate this construction with subgroups A-, = Z",
where η > 2: the Flat Torus Theorem provides us with local isometries E"/A,- —■► X,·
that realise the inclusions A,· <^> Γ,· (these maps play the role of the closed geodesies
in the case η = 1), but in general one cannot make the flat tori E"/Ai and E"/A2
isometric simply by scaling the metric on each by a linear factor — it might be that
they are not conformally equivalent.
In some cases one can vary the metric on the spaces that are to be glued and thus
realise the desired amalgamated free product as the fundamental group of a compact
non-positively curved space (cf. II. 11.36). But in other cases the shape of the torus
associated to a particular abelian subgroup is an invariant of the group rather than

Amalgamating along Abelian Subgroups 501
simply the particular non-positively curved space at hand. We illustrate this point
with some specific examples.
6.9 The Groups T{n). Let
T{n) = (a, b, ta,tb\ [a, b] = 1, t~lata = (ab)n, t~lbtb = (ab)n).
As Dani Wise pointed out in [Wi96b], T(n) is the fundamental group of the non-
positively curved 2-complex X(n) that one constructs as follows: take the (skew)
torus Σ„ formed by identifying opposite sides of a rhombus whose sides have length
η and whose small diagonal has length 1; the loops formed by the images of the
sides of the rhombus are labelled a and b respectively; we attach to Σ„ two tubes
S χ [0, 1], where S is a circle of length n; one end of the first tube is attached to the
loop labelled a and one end of the second tube is attached to the loop labelled b; in
each case the other end of the tube wraps around the image of the small diagonal η
times.
There is some flexibility in how one attaches the tubes in this example, but
because a is conjugate to b and to {ab)n, the shape of the torus Σ,, с X(n) supporting
the abelian subgroup An = (a, b) is entirely determined by the algebra of the group.
More precisely, given any proper action of T(n) by semisimple isometries on a
CAT(O) space one considers the induced action of A„ on each of the flat planes Ε
yielded by the Flat Torus Theorem, modulo a constant scaling factor, the quotient
metric on An\E must make it isometric to the torus Σ„ described above, because the
translation lengths of a, b and (ab)" have to be the same.
Ипфт, then the tori Σ„ and Em cannot be made isometric simply by scaling
the metric on each. Thus we have:
6.10 Proposition. In the notation of (6.9), if η φ m and if Υ is an amalgamated free
product of T(n) and T(m) obtained by identifying An with Am, then Γ does not act
properly by semisimple isometries on any CAT(O) space.
The interested reader should find little difficulty in producing many variations
on this result, we note one other:
6.11 Example. Let Γ = (a, b, t \ [a, b] = 1, rlat = b2). This is the fundamental
group of the non-positively curved 2-complex obtained as follows: one takes the
torus formed by identifying opposite sides of a rectangle whose sides c\ and cj. have
length 1 and 2, respectively; to this one attaches a cylinder S χ [0, 1], where S is a
circle of length 2; the end S χ {0} wraps twice around the image of с ι and the end
S χ {1} is identified with the image of съ
We take two copies of Γ and consider the amalgamated free product G = Γ *Ζ2 Γ
obtained by making the identifications a — b and b = a. Writing ~ to denote
conjugacy in G, we have a ~ b2 = a2 ~ b = a4. We claim that G cannot
act properly by semisimple isometries on any CAT(0) space. Indeed, given any
action of G on a CAT(0) space, the translation lengths (II.6.3) of a and b satisfy:

502 Chapter ΙΠ.Γ Non-Positive Curvature and Group Theory
2\b\ = \b2\ = \a\ = \a4\ = 4|a|,thus \a\ = \b\ = 0. Thus if the action is semisimple,
a and b must be elliptic, and since they have infinite order in G, the action will not
be proper.
The same calculation with algebraic translation numbers in place of geometric
ones shows that if Q is semihyperbolic, then the image of (a, b) under any homo-
morphism G —> Q must be finite (see 4.18).
For our last example of a bad amalgamation along abelian subgroups we take
two и-torus bundles over the circle with finite holonomy and equip them with flat
metrics, then we ask when they can be glued along their fibres so as to produce a
compact non-positively curved space.
6.12 Proposition. Let φι e GL(«, Z), i = 1,2 be elements of finite order and
consider the corresponding semidirect products G, = Ζ" χ ψ, Ζ. The amalgamated
free product Τ{φι, φ2) :— G\ *%« G2 is the fundamental group of a compact non-
positively curved space if and only if the subgroup (φι, ф2) С GL(«, Ζ) is finite.
Moreover, if (φι, φι) is infinite then Γ(φι, φ2) does not act properly by semi-simple
isometries on any CAT(O) space, it is not semihyperbolic, and its Dehn function
grows at least cubically.
Proof. The quotient οίΥ(φι, φ2) by the normal subgroup Z" is free of rank two, so
r(0b02) = Z'^F2,wheremeimageof<£:F2 -> GL(n, Z)is#:= (φι,φ2).1ΐΗ
is finite then we can choose an Я-equivariant flat metric on the и-torus Σ. Regard each
φ\ as an isometry of Σ. For i = 1, 2, the mapping torus Μ, := Σ χ [0, 1]/[(jc, 0) ~
(φι(χ), 1)] is a compact non-positively curved space with fundamental group G,, i =
1, 2. By gluing Mi to M2 along Σ χ {0} we obtain a non-positively curved space with
fundamental group Τ(φι, φ2).
If Η is not finite, then the normal subgroup Z" is not virtually a direct factor
of its normalizer in Υ(φι, φ2), and therefore Υ(φι, φ2) does not act properly by
semi-simple isometries on any CAT(0) space (ΙΙ.7.17). To see that Υ(φι, φ2) is not
semihyperbolic in this case, one can argue that Η = (φι, φ2) contains an element
of infinite order and appeal to (4.17). Alternatively64, since we know that the Dehn
function of a semihyperbolic group is linear or quadratic (4.9(2)), we can appeal to
the Main Theorem of [Bri95b], which implies that if (φι, φ2) is infinite, then the
Dehn function of Υ{φι, φ2) will be either exponential or polynomial of degree d,
where 3 < d < η + 1 (it will be exponential if and only if Η contains an element
with an eigenvalue of absolute value bigger 1). G
6.13 Example. SL(2, Z) is generated by the following two matrices of finite order
^ = (_°! o) and 02=(_°! J)·
64 This argument was discovered independently by Natasa Macura and Christian Hidber
[Hid97].

Amalgamating Along Free Subgroups 503
Since SL(2, Z) contains matrices that have eigenvalues of absolute value bigger than
1, the group Τ(φι, φ2), which is the fundamental group of a space obtained by gluing
two flat 3-manifolds along tori, has an exponential Dehn function and does not act
properly by semi-simple isometries on any CAT(O) space.
Amalgamating Along Free Subgroups
Let F be a finitely generated free group. In order to realise Γ ι *fT2 as the fundamental
group of a compact non-positively curved space, one should look for compact non-
positively curved spaces Xt with Γ, = Tt\Xi where the inclusions F <^-> Γ, can be
realized by local isometries Υ —> X,, where Υ is a metric graph with fundamental
group F. A simple instance of this is:
6.14 Lemma. Let В = B{1\,..., /„) be a metric graph that has one vertex and η
edges with lengths l\,..., l„. Let φι : В —> Χι and φι : В —> Х2 be local isometries
to compact non-positively curved spaces. The double mapping cylinder
X,U(Sx [0, 1])UX2
is non-positively curved (when endowed with the quotient path metric).
Such embeddings В —> Xt arise naturally in the case of spaces built by attaching
gluing tubes as in (11.11.13).
Let X and Y\,..., Y„ be compact non-positively curved spaces with basepoints
ρ and qi,... ,q„ respectively, and let φ0·', φ1·' : Yt —> X be local isometries with
ф°-'Ш = <Pu{qi) = Ρ for all i. (II. 11.13) tells us that
Хи(У,х[0, 1])Ц-и(У»х[0, 1])
(yh 0) ~ ф°'Ш (yi, 1) ~ Ф1Ъ.) Yy/ eYi,i=l,...,n
is non-positively curved when endowed with the quotient path metric. Associated to
this description of Q we have the relative presentation
ni(Q,P)= (ni(X,p),nl(Yi,qi),si,i= l,...,n\
*7]Ф°/(8>>, = Ф1;Ш vgi e mw, q,)).
The homotopy class s, contains the loop based at ρ e Q which is the image of the
path t м> (qi, t) in У, χ [0, 1]; let σ; denote this loop.
6.15 Lemma. Let Q be as in the preceding paragraph. Let В be the metric simplicial
graph that has one vertex ν and η edges of length 1. Let ψ : Β —> X be the map that
sends υ top € Q and sends the i-th edge of В isometrically onto the loop σ\. Then ψ
is an isometry onto its image.

504 Chapter III.Γ Non-Positive Curvature and Group Theory
Proof. By construction ψ is length-preserving and injective. By composing the
projection Υ, χ [0, 1] -> [0, 1] with the arc length parameterization of the г'-th edge in
В one obtains maps Yt —> B. These maps, together with the constant map X —> {υ}
induce a map/ : Q —> 5 that does not increase lengths and which is such that
f-ψ = idg. Thus ψ is an isometry. G
An application of this lemma is given in (7.9).
Bad Amalgamations Along Free Subgroups
Recall that a subgroup Я с Г is said to be a retract if there is a homomorphism
Γ —> Я that restricts to the identity on Я. Given any finite generating set for Я, one
can extend it to a finite generating set for Г by adding generators for the kernel of
the retraction Г —> Я. With respect to the resulting word metrics, Я <^-> Г is an
isometry.
Part (2) of the following proposition illustrates the point that we made
following (6.7): in order for G*a to be semihyperbolic it is necessary for A to be quasi-
isometrically embedded in G, but this condition is not sufficient.
6.16 Proposition. Let F\ and F2 be free groups of rank two. Choose generators
a\,b\ e F\ and a2,b2 e F2 and let L denote the subgroup ofF\ χ F2 generated by
a = a\a2 and β = αφ2. {By projecting into F2 one sees that L is free.)
(1) L с F\ χ F2 is a retract. (In particular, for a suitable choice of word metrics
L <^-> F\ χ F2 is an isometry.)
(2) The trivial HNN extension Ε = (F\ χ F2, t \ t~xlt = /V/ e Z,) is not
semihyperbolic.
(3) The double Δ = A2(Fi χ Fi\ L) is not semihyperbolic.
(In fact both Ε and Δ have cubic Όehn functions.)
Proof. One obtains a retraction from F\ χ Fj, to L by sending Fi to the identity, a2
to a andbi to β.
In order to see that Ε is not semihyperbolic, we look at centralizers (4.15). We
claim that the intersection of the centralizers of t, αχ and b\ is LD Fi, which is the
kernel of the map L —> Ζ that sends each of α and β to a fixed generator. This kernel
is not finitely generated (see 5.3), so (4.15) tells us that Ε is not semihyperbolic.
To see that the intersection of centralizers is what we claim, note first that by
(6.4) the centralizer of t is L, then note that the centralizers of αϊ and b\ in Ε are
the same as their centralizers in F\ χ F2, and that the centralizer of αϊ in F\ χ F2
is (αϊ) χ F2 while that of b\ is (b\) χ F2. The second assertion is a consequence of
(6.4), but can also be seen by looking at the action of Ε on the Bass-Serre tree: each
of α and b is an elliptic element with a single fixed point, the stabilizer of which is
F\ χ F2\ and the centralizer of any elliptic isometry of a CAT(O) space must preserve
its fixed point set.

Amalgamating Along Free Subgroups 505
A similar argument shows that in Δ the intersection of the centralizers of the two
copies of a\ and the two copies of b\ is again L Π F2.
We shall sketch a proof of the parenthetical assertion concerning the Dehn
functions of Ε and Δ. The required cubic upper bound is proved in (6.20). It is enough to
establish the lower bound for E, because Δ retracts onto a copy of Ε and the Dehn
function of a group gives an upper bound on the Dehn function of any retract of that
group [Alo90]. (To get a retraction from Δ onto a subgroup isomorphic to E, leave
one of the natural copies of F\ χ F2 in Δ alone and in the other send the generator
b\ to 1 and the other generators to themselves.)
To obtain a cubic lower bound on the Dehn function of Ε we work with the
following presentation:
(αϊ, a2, bi,b2, t\ [аь а2], [ai,b2], [bu a2], [bu b2], [t, aia2], [t, αφ2}).
Let un = (a'jfrj"), let vn = {tb{)n and let w„ = v„u„v~lu~l. Note that w„ = 1 in E.
We claim that the area of any van Kampen diagram for w„ is bounded below by a
cubic function of n.
There are two important observations to be made. First, because no subword
of w„ equals 1 in E, any van Kampen diagram for w„ must be a topological disc.
Second, since the only relations involving t are of the form [t, x], if one enters a
2-cell in a van Kampen diagram for w„ by crossing an edge labelled t, then there is
a unique other edge labelled t which you can cross in order to exit the 2-cell; thus
one enters a second 2-cell, from which there is a unique way to exit crossing an edge
labelled t, and so on. Continuing in this way, one obtains a chain of 2-cells crossing
the diagram (see figure Γ.11). The chains of 2-cells that one obtains in this manner
are called t-comdors. There is one such ί-corridor incident at each edge labelled t
in the boundary of the diagram. See [BriG96] for a careful treatment of i-corridors.
In exactly the same way, one obtains a bi-corridor incident at each edge labelled
b\ in the boundary of the diagram. The orientations on the ends of the corridors force
them to cross the diagram in the manner shown in figure Γ. 11.
The key parts of the diagram on which to focus are the gaps between corridors.
The top and bottom of each ί-corridor is labelled by a word in the free group on
(αφ2) and (a\a2) that freely reduces to (α\α2)η(αφ2)~η. The side of each bi-corridor
is labelled by a word in the free group on a2 and b2 that freely reduces to <£&2ι. The
subdiagram between each pair of adjacent corridors is as drawn in figure Γ. 12. It may
be regarded as a diagram over the natural presentation of (a2, b2) χ (αϊ) S F2 χ Ζ,
and as such it has area ~ n2. Hence the area of the whole diagram in figure Γ.11 is
>пъ. О

506 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
V*W
/N
vn=V\T
Fig. Г.11 A van Kampen diagram for u„v„un 'vn '.
Fig. Г.12 The subdiagram showing that a"b2 "
(α\αι)"(αφ2)~
Att,
Subgroup Distortion and the Dehn Functions of Doubles
The following notion of distortion has been lurking in the background of several
earlier proofs (for example the preceding proposition and (5.15)).
6.17 Definition (Subgroup Distortion). Let Я с Г be a pair of finitely generated
groups, and let dr and dH be the word metrics associated to a choice of finite
generating set for each. The distortion of Я in Г is the function
а£(и) = тах{й?н(1, h) \heH with dr(l, h) < n].
One checks easily that, up to Lipschitz equivalence, this function is independent of
the choice of word metrics dr and d#.

Subgroup Distortion and the Dehn Functions of Doubles 507
6.18 Remarks
(1)Η=-> Γ is a quasi-isometric embedding if and only if there exists a constant
К such that S£(w) < Kn.
(2) The definition of distortion that we have adopted differs from that of Gromov
[Gro93] in that we have omitted a normalizing factor of \/n on the right hand side.
Authors vary in this convention.
6.19 Exercises
(1) Suppose that Я and Γ are finitely generated and that Я С Г is normal. Prove
that there is a constant к > 0 (depending on the choice of word metrics) such that
distortion of Я in Г is bounded above by η ы> к". (Hint: Take к to be the maximum
of the distances dH(l, b~~lab) where b runs over a set of generators В = B~l for Г
and a runs over the generators of Я.)
(2) Let/ : N —> Μ be a bi-Lipschitz map from one closed Riemannian manifold
into another. Suppose that/ is 7Ti-injective and consider a lifting/ : N —> Μ to the
universal cover. Consider the relationship between the length metric ^д, on N given
by its Riemannian structure and the metric d(x, y) := d^{f{x),f{y)). After taking
note of (1.8.19), bound а^(х,у) in terms of d(x, y) and the distortion οίπιΝ С π\Μ.
We have introduced subgroup distortion at this time so that we can formulate
the following theorem. We recall that Δ„(Γ; A) is the amalgamated free product
of m copies of Γ along A. As we noted prior to (5.1), if m > 3 then there is a
canonical retraction of Δ„(Γ; A) onto Δ2(Γ; A), sending the first two copies of Γ
to themselves and sending the other (m — 2) copies of Γ to the first copy by the
identification implicit in the notation. Similarly, by identifying the m natural copies
of Γ, one obtains a retraction from Δ„(Γ; A) onto (the first copy of) Γ.
We recall that modulo the equivalence relation ~ described in (I.8A.4), the Dehn
function of a finitely presented group does not depend on the choice of finite
presentation. And if Я is a retract of a finitely presented group Г, then the Dehn function
of Я is ^-bounded above by that of Г in the sense of (I.8A.4) — see [Ak>90].
6.20 Theorem. Let m > 2 be an integer. Let Г be a finitely presented group with
Dehn function fγ and let Η С Т by a finitely presented subgroup. Let 8(n) be the
distortion of Η in Γ with respect to some choice of word metrics. The Dehn function
/Δ„, ο/Δ„, = Δ„(Γ; Я) satisfies
тах{/г(и), δ(η)} ^/δ» < nfr(8(n)),
and the Dehn function of the trivial HNN extension Γ = Г*н satisfies
тах{/г(и), ηδ(η)} <fy{n) < nfr(S(n)).
Proof. First we consider the lower bound for Am. Because Δ„ retracts onto Γ, we
have/r(«) < /δ,,,(ί), so it only remains to show that δ(η) < /δ,,,(ί)· Because Δ„
retracts onto Δ2, we may assume that m — 2.

508 Chapter Ш.Г Non-Positive Curvature and Group Theory
We fix a finite presentation (Л | TZ) for Г, where A includes a generating set
В for H. Let ан be the word metric on Η associated to B. We shall work with the
following presentation of Δ2:
(А А \TZ,TZ', h~lti = lVheB).
We choose a sequence of geodesic words w„ in the generators A so that w„ e
H, \w„\ < η and dH(l, w„) = δ(η). Let w'n be the same word in the alphabet A and
consider a least-area van Kampen diagram D„ for the word w~' wj, (which represents
the identity in Δ2).
Across the interior of each 2-cell (digon) corresponding to relators of the form
h~~lh' we draw an arc connecting the vertices. These arcs form a graph Q с D„,
whose edges we label with letters h e В in the obvious way. The basepoint of the
diagram D„ and the endpoint of the arc of 3D,, labelled w„ are two vertices of Q.
We claim that these two vertices lie in the same connected component of Q. If this
claim were false, then there would be an arc through the interior of D„ that joined a
point on the arc of 3D,, labelled wn to a point on the arc labelled w'n and did not pass
through any digon labelled h~~lh'. But this is clearly impossible, because each edge
in the portion of the boundary labelled w„ lies in the closure of a 2-cell labelled by a
relator from TZ and each edge in the portion of the boundary labelled w'n lies in the
closure of a 2-cell labelled by a relator from TZ', and the only relators involving both
primed and unprimed letters are those of the form h = h'.
Thus we can connect the basepoint of D„ to the endpoint of the arc of 3D„ labelled
wn by a path in Q\ we choose a shortest such path σ. The edges of σ are labelled by
generators A,·,,..., him e B±] such that
К ■ ■ · K„ — w"
in Г. Thus m > δ(η), and since the 2-cells (digons) of D„ corresponding to the edges
of σ are all distinct, we deduce that D„ has at least δ(η) 2-cells. The integer η is
arbitrary and 3D,, has length < 2n, thus the Dehn function for the given presentation
of Δ2(Γ; Η) satisfies/д2(2и) > δ (ή) for all п.
We now consider the case of HNN extensions. Again, since Г is obviously a
retract of Г, the inequality fr(n) < /f.(n) is clear.
Г*я has finite presentation {A, t \ TZ, [t, h] = 1 V/г e B). For each positive
integer η consider a least-area van Kampen diagram D'n for t"w„t~"w~l, where the
words w„ are chosen as above. One obtains a lower bound of ηδ(η) on these diagrams
by getting lower bounds on the len gth of ί-corridors as in (6.16) — we omit the details.
We now turn our attention to the upper bounds. We give the details only in
the case of Δ2. The argument for Δ„ is entirely similar (but notationally a bit more
complicated). The HNN case is also entirely similar, except that one uses the reduced
form for elements in HNN extensions in place of the reduced form for amalgamated
free products (see (6.4)).
We continue to work with the presentation {A, A | TZ, TZ', h~~lti — 1 V/г e B)
of Δ2. Let δ{η) be the distortion of Я С Г with respect to the word metrics associated
to В and A.

Subgroup Distortion and the Dehn Functions of Doubles 509
Any word in the given generators is equal (purely as a word) to a product of the
form w = mi hi ...uivi, where each щ is a word in the generators Л and each υ,- in
the generators Ά, and all but possibly и ι and i>/ are non-empty. The integer / is called
the alternating length of w. The idea of the proof is to reduce the alternating length
of words representing the identity by applying a controlled number of relations.
Let n= \w\ and suppose that w — 1 in Δ2. Lemma 6.4 implies that one of the щ
or Vj represents an element of H. Suppose that it is щ and let t/, be the shortest word in
the generators В that represents the same element of H. By definition, |t/,| < 5(|и,|).
Let m — \Ui\+\u;\. Let U\ be the word obtained by replacing the letters of t/, by the
same letters primed.
By the definition of the Dehn function, in the free group F(A) there is a product
Ρ of at most/r(w) conjugates of relators r e ΊΖ±ι such that и, t/f1 = P. And there is
an obvious way to write UjU'^ as the product Q of |t/,| < m conjugates of relators
b~~xb'. Thus PQP'~ is a product of at most 2m + 2fr(m) conjugates of relations.
In the free group on A U A' we have: щ = (и,- J7f' )(£/;£/; ~' )(£/(-и< ~' )и< =
PQP'"1 и]. Thus
w — u\vi... Vi^.\PQP'~ u'jVj... i>/.
We move the subword PQP'~[ by conjugating it with the suffix s of w that follows
it. As a result we obtain the following equality in the free group
w — U{V\ ... [и;_1И;и,·].. ■ V[(s~~lPQP'~~ s).
The subword of the right hand side that precedes (s~' PQP' ~' s) has smaller alternatin g
length than w because the term in square brackets is a word over A'. Repeating this
argument some number N < n— \w\ times, we obtain w = WT1, where W is a word
of length at most η in either the generators A or the generators A', and where Π is
the product of at most 2N(m +fr(m)) conjugates of relations obtained by gathering
the products PQP'"1 at the right after each stage of the argument.
Finally, by the definition of the Dehn function of (A \ TZ), we may write W as
a product of at most/r(«) conjugates of relations from 1Z or TV. Thus w can be
expressed in the free group on A U A' as a product of at most/r(«) + 2N(m +/r(w))
conjugates of relators. By definition,/r is non-decreasing, η < δ(η), Ν < η, and
m = \Ui\ + \Ui\ < η + δ(η). Thus'the theorem is proved. G
In the light of 6.19(1), the following proposition shows that if Γ is hyperbolic
then the distortion in Γ of any infinite, finitely generated, subgroup of infinite index
is actually ~ k", where к > 1. (Recall that k" ~ 2" for any constant к > 1.)
6.21 Proposition. Let Г be a hyperbolic group and let N с Г be a finitely generated
normal subgroup. Suppose that N and T/N are both infinite. Then there exists a
constant к > 1 such that Sj^(n) > k". (The value of к depends on the choice of word
metric, but its existence does not.)
Proof. Let Л be a generating set for N and extend this to a generating set A U В
for Г. We also regard β as a generating set for Γ/Ν. Let С be the Cayley graph

510 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
of Г with respect to the generators A U В and let С be the Cayley graph of Γ/Ν
with respect to the generators B. There is a unique way to extend the quotient map
Г —» Γ /Ν to a map π : С —> С that sends directed edges labelled by each b e В
isometrically onto edges labelled b and collapses edges labelled a e A. Note that
d(n(x), n(y)) < d(x, y) for all x, у е С
Assume α e Л С N is an element of infinite order, η ы> a" is a quasi-geodesic
(3.10), hence there exists an integer λ > 0 such that d{\, aXr) > 4\r\ for all r e Z.
Moreover, since (a) is quasiconvex (3.6), there exists a constant if > 0 such that
for every r e Ζ and у е Г, any geodesic in С joining γ to yar must lie in the
if-neighbourhood of the edge-path labelled ar that begins at the vertex γ.
For each integer η > К we choose a geodesic edge path of length η beginning at
1 e C. Let u„ be the word in the letters B±l that labels this path. Let w„ be a word
of minimal length in the generators A that is equal to u„aknu~l in Γ. Our goal is to
show that the length of w„ is bounded below by an exponential function of n. Since
u„ax"u~' is a word of length и(2 + λ), this will prove the proposition.
We fix a geodesic segment in С joining u„ e Г to unax" e Г; let m be the midpoint
of this segment. In the light of (H.1.16), we will be done if we can show that the
edge-path in С that begins at u„ e Г and is labelled u~lw„u„ lies outside the ball of
radius (n — K) about m (see figure Г.13). For the subpaths labelled u„ and и"1 this
is clear, because they have length η and they have an endpoint a distance at least 2и
from m. And the subpath labelled w„ is sent by я to 1 e C, whereas the geodesic
containing m is send to a point a distance at most К from u„ e C. Since π does not
increase distances, the arc labelled w„ must lie entirely outside the ball of radius
(n — K) about m. D
Hyperbolic Surface Bundles. There are many hyperbolic 3-manifolds which fibre
over the circle with compact fibre (see [Ot96]), indeed Bill Thurston conjectured
that every closed hyperbolic 3-manifold has a finite-sheeted covering that fibres in
this way [Thu82]. If M3 admits such a fibration then from the long exact sequence
of the fibration we get a short exact sequence l->-S->-7riM3->-Z->- 1, where S
is the fundamental group of the surface fibre. The previous proposition shows that
the distortion of S in it\Mb is exponential, so as a consequence of (6.20) we have:
6.22 Corollary. Let M3 be a hyperbolic 3-manifold that fibres over the circle with
compact fibre Σ and let S — πχ Σ. The Dehn function ο/Δ2(7ΓιΜ3; S) is ~ 2".
Similar arguments apply to hyperbolic knot complements that fibre over the circle,
and to free-by-cyclic hyperbolic groups, examples of which are given by [BesF92].

7. Finite-Sheeted Coverings and Residual Finiteness 511
N
w
η
1
Fig. Γ.13 Normal subgroups are exponentially distorted
7. Finite-Sheeted Coverings and Residual Finiteness
In this section we shall explain how to construct compact non-positively curved
spaces that have no non-trivial, connected, finite-sheeted coverings. We shall also
construct groups which act properly and cocompactly by isometries on С AT(0) spaces
but contain no torsion-free subgroups of finite index.
Residual Finiteness
A connected, locally simply-connected space X has no non-trivial, connected, finite-
sheeted coverings if and only if π\Χ has no non-trivial finite quotients65. Thus as a
first step towards trying to build spaces with no connected, non-trivial, finite-sheeted
coverings, we look for spaces whose fundamental groups do not satisfy the following
condition.
7.1 Definition (Residual Finiteness). A group G is residually finite if for every
g eG\|l) there is a finite group Q and an epimorphism φ : G —»■ Q such that
<p(g) Φ 1. Equivalently, the intersection of all subgroups of finite index in G is {1}.
Let X be a connected complex and suppose that π\Χ is residually finite. The
topological content of residual finiteness is that given any homotopically non-trivial
If G is a group and Η С G is a subgroup of finite index, then there is a subgroup of finite
index N С Я that is normal in G.

512 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
loop in X, there is a connected finite-sheeted covering of X to which this loop does
not lift. And the reformulation of residual finiteness given in the second sentence of
the above definition implies that if π\Χ is residually finite then the universal covering
of X is the inverse limit of a sequence of finite-sheeted coverings. (If X is compact
and π\Χ is infinite, then this can be useful because the finite-sheeted coverings are
compact whereas the universal cover is not.)
The fundamental groups of the most classical examples of non-positively curved
spaces, quotients of symmetric spaces of non-compact type, are residually finite. In
the light of this, it may seem natural to ask whether the fundamental group of every
compact non-positively curved space is residually finite. However, in the classical
setting residual finiteness appears as an artifact of linearity rather than considerations
of curvature per se: the fundamental groups in question can be realised as groups
of matrices, and a theorem of Mal'cev [Mal40] shows that every finitely generated
linear group is residually finite.
Dani Wise produced the first examples of compact non-positively curved spaces
whose fundamental groups are not residually finite [Wi96a]. Some of the groups
he constructed have no non-trivial finite quotients. Subsequently, Burger and Mozes
[BuM97] constructed compact non-positively curved 2-complexes whose
fundamental groups are simple. The situation for negatively curved spaces is less clear: it is
known that hyperbolic groups are never simple [Gro87], [0195], but it is unknown
whether they are always residually finite.
In the following statement the term non-trivial is used to mean that a covering
Ζ —»■ Ζ is connected and Ζ —> Ζ is not a homeomorphism.
7.2 Embedding Theorem. Every compact, connected, non-positively curved space
X admits an isometric embedding into a compact, connected, non-positively curved
space X such that X has no non-trivial finite-sheeted coverings. IfX is a polyhedral
complex of dimension η > 2, then one can arrange for X to be a complex of the same
dimension.
We shall give a self-contained proof of this theorem, following [Bri98a]. The
main work goes into the construction of a compact non-positively curved 2-complex
К whose fundamental group has no finite quotients (7.9). With К in hand, the proof
of (7.2) becomes straightforward.
Proof. Choose a finite set of generators γ\,... ,Υν for π\Χ, where no y, = 1, and
let с ι,..., Cn be closed local geodesies in X representing the conjugacy classes of
these elements. Lemma 7.9 (alternatively [Wi96a] or [BuM97]) gives a compact non-
positively curved 2-complex К whose fundamental group has no finite quotients; fix
a closed local geodesic cq in K. Take N copies of К and scale the metric on the г-th
copy so that the length of со in the scaled metric is equal to the length 1(a) of a, then
glue the N copies of К to X using cylinders St χ [Ο, L] where Si is a circle of length
1(d); the ends of 5, χ [0, L] are attached by arc length parameterizations of cq and a
respectively (cf. II. 11.17). Call the resulting space X.

7. Finite-Sheeted Coverings and Residual Finiteness 513
(II. 11.13) assures us that X is non-positively curved. Moreover, if the length L of
the gluing tubes is sufficiently large, then the natural embedding X ^-» X will be an
isometry. The Seifert-van Kampen Theorem describes the fundamental group of X:
it is an iterated amalgamated free product; one first amalgamates π\Χ and it\K by
identifying (y\) (ΖττχΧ with an infinite cyclic subgroup С С π\ Κ represented by the
closed local geodesic cq\ the result of this first amalgamation is then amalgamated
with a further copy of я\K, identifying (уг) С π\Χ with the copy of С in this second
π\Κ, and so on. The key point to note is that π\Χ is generated by the N (obvious)
copies of τΐχ Κ that it contains, and since each of these copies of π\ Κ must have trivial
image under any homomorphism from π\Χ to a finite group, Tt\X has no non-trivial
finite quotients. D
By using equivariant gluing (II.11.19) instead of local gluing (II.11.13), onecan
extend the above result as follows:
7.3 Theorem. If a group Γ acts properly and cocompactly by isometries on а С AT(0)
space Υ then one can embed Γ in a group Γ that acts properly and cocompactly by
isometries on a CAT(O) space Υ and has no proper subgroups of finite index. If Υ is
a polyhedral complex of dimension η > 2 then so is Y.
If the group Γ in (7.3) is not torsion-free, then Γ will be a semihyperbolic group
that does not contain a torsion-free subgroup of finite index, and the complex of
groups associated to the action of Γ on У will be a finite, non-positively curved,
complex of finite groups that is not covered (in the sense of Section C.5) by any
compact polyhedron (i.e. a complex of groups whose local groups are trivial). An
explicit example of such a group is given in (7.10).
The Hopf Property
An effective way of showing that certain finitely generated groups are not residually
finite is to exhibit an isomorphism between the given group and one of its proper
quotients.
7.4 Definition (Hopfian). A group Η is said to be Hopfian if every epimorphism
Η -» Η is an isomorphism. In other words, if N С Η is normal and H/N = Η then
N={1}.
7.5 Proposition. If a finitely generated group is residually finite then it is Hopfian.
Proof. Let G be a finitely generated group and suppose that there is an epimorphism
φ :G —»■ G with non-trivial kernel. We fix go e кет φ χ {1} and for every η > 0 we
choose g„ e G such that φ"^η) =
golf there were a finite group Q and a homomorphism ρ : G —> Q such that
p(go) Φ 1, then all of the maps φ„ := ρφ" would be distinct, because 0„(g„) φ 1
whereas фт(^„) = 1 if от > п. But there are only finitely many homomorphisms from

514 Chapter ΠΙ.Γ Non-Positive Curvature and Group Theory
any finitely generated group to any finite group (because the image of the generators
determines the map). D
The Hopf property is named in honour of H. Hopf. It arose in connection with his
investigations into the problem of deciding when degree one maps between closed
manifolds give rise to homotopy equivalences. Zlil Sela [Sel99] has shown that if
a torsion-free hyperbolic group does not decompose as a free-product, then it is
Hopfian. In particular this shows that the fundamental groups of closed negatively
curved manifolds are Hopfian.
7.6 Examples. The following group was discovered by Baumslag and Solitar
[BSo62]:
BS(2,3)= (a,t | rla2t = a3).
The map а м> a2, t м> t is onto; a is in the image because a = аъа~~2 = {t~~la2t)a~2.
However this map is not an isomorphism: [a, t~lat] is a non-trivial element of the
kernel. Meier [Me82] noticed that the salient features of this example are present
in many other HNN extensions of abelian groups. Some of these groups were later
studied by Wise [Wi96b], among them
T(n) = (a, b, ta,tb\ [a, b]=\,ralata = (ab)n, t~lbtb = (abf),
which is the fundamental group of the compact non-positively curved 2-complex
X(n) described in (6.9). If и > 2 then certain commutators, for example go =
[ta(flb)t~l, b\, lie in the kernel of the epimorphism T(ri) -» T(n) given bya h>
a", b ι-* b", ta м> ta, tb ы> tb. Britton's Lemma (6.4) implies that go φ 1 in T(ri).
The proof of (7.5) shows that g0 has trivial image in every finite quotient of T(ri).
Groups Without Finite Quotients
In this subsection we describe a technique for promoting the absence of residual
finiteness to the absence of subgroups of finite index.
7.7 Proposition. Let Q be a class of groups that is closed under the formation of
HNN extensions and amalgamated free products along finitely generated free groups.
IfG e Q is finitely generated, then it can be embedded in a finitely generated group
G e Q that has no proper subgroups of finite index.
The following proof is not the most direct possible. It is chosen specifically so
that each step can be replicated in the context of compact non-positively curved
2-complexes (Lemma 7.9). In [Bri98a] this construction is used to prove other
controlled embedding theorems. A similar construction was also used in [Wi96a].
Proof. We may assume that G contains an element of infinite order go € G whose
image in every finite quotient of G is trivial, for if it is does not then we can replace it

Groups Without Finite Quotients 515
withG*BS(2, 3) e Q by forming G*(i) = G*(a) and then taking an HNN extension
by a stable letter t conjugating a2 to a3. We may also assume that G is generated by
elements of infinite order, because if {^i,..., b„} generates G then {a, t, tb\,..., tb„}
generates G * BS(2, 3), and the natural retraction G * BS(2, 3) -» BS(2, 3) sends tb-,
to t, which has infinite order in BS(2, 3).
Let Λ = (αϊ,..., a,,} be a generating set for G where each a, has infinite order.
Step 1: We take an HNN extension of G with η stable letters:
£i = (G, si,...,s„ | j'O/j,· = g%, i—l,...,n),
where the pi are any non-zero integers. Now, since each a, is conjugate to a power of
go in Ει, the only generators of Ει that can survive in any finite quotient are the j,·.
However, since there is an obvious retraction of Ει onto the free subgroup generated
by the Sj, the group Ει still has plenty of finite quotients.
Step 2: We repeat the extension process, this time introducing stable letters τ, to
make the generators s,- conjugate to go'·
E2 = (Ει, tu ..., τ„ | Tf'jiT,- = g0, i = 1,..., и).
Step 3: We add a single stable letter σ that conjugates the free subgroup of Ei
generated by the j,- to the free subgroup of £2 generated by the τ,:
Еъ = (E2, σ | σ~'ί,σ = τ,-, i = 1,..., η).
At this stage we have a group in which all of the generators except σ are conjugate
to go. In particular, every finite quotient of £3 is cyclic.
Step 4: Because no power of αϊ lies in either of the subgroups of E2 generated
by the Si or the τ-„ the normal form theorem for HNN extensions (6.4) implies that
{αϊ, σ} freely generates a free subgroup of £3.
We define G to be an amalgamated free product of two copies of £3,
G = £3 *f £3,
where F — F(x, y) is a free group of rank two; the inclusion of F into £3 is χ м> αϊ
and у \-> σ, and the inclusion into £3 is χ м> <т and у ι-* αι. All of the generators
of G are conjugate to a power of either go or g~o, and therefore cannot survive in any
finite quotient. In other words, G has no finite quotients. Π
Complexes With No Non-Trivial Finite-Sheeted Coverings
In (7.2) we reduced the Embedding Theorem to the claim that there exist compact
non-positively curved 2-complexes whose fundamental groups have no non-trivial
finite quotients. We shall construct such a complex by following the scheme of the
proof of (7.7), but first we must condition our complexes in the following manner.
Recall that a closed local geodesic in a metric space is a local isometry from a
circle to the space.

518 Chapter ΙΙΙ.Γ Non-Positive Curvature and Group Theory
and any path in K\ С ^з is я. Thus the free subgroup (a\, σ) is the π\-image in
ni(Ki,xq) of an isometry from the metric graph Ζ with one vertex (sent to xq) and
two edges of length L — λχ (cf. 6.15). In fact, we have two such isometries Ζ —» Кз,
corresponding to the free choice of which edge of Ζ to send to the image of υ χ [Ο, L].
We use these two maps to realise Step 4 of the construction on (7.7), appealing to
(6.14) to see that the resulting 2-complex is non-positively curved. D
7.10 A Semihyperbolic Group That Is Not Virtually Torsion-Free. We close this
section with an explicit example to illustrate the remark that we made following
Theorem 7.3. This example is essentially contained in Wise's thesis [Wis96a].
In the hyperbolic plane we consider a regular geodesic quadrilateral Q with vertex
angles 7r/4. Let a and β be hyperbolic isometries that identify the opposite sides of
Q. Then β is a fundamental domain for the action of Γ = (α, β). The commutator
[α, β], acts as a rotation through an angle π at one vertex of Q, and away from the
orbit of this vertex the action of Γ is free. Thus the quotient orbifold V = Γ\Η2 is a
torus with one singular point, and at that singular point the local group is Z2.
Let X(n) and Tin) be as in (7.6), let go be a non-trivial element in the kernel
of a self-surjection T(ri) -» Tin) and consider the amalgamated free product G =
T(n) *z Г in which go is identified with β. Note that since go has trivial image in
every finite quotient of T(n), the commutator [α, β] = [a, go], which has order two,
has trivial image in every finite quotient of G.
The equivariant gluing described in (II. 11.19) yields a proper cocompact action
of G by isometries on a CAT(O) space. The quotient space Υ can be described as
follows: scale the metric on X(ri) so that the closed local geodesic с representing go
has length I = \a\ = \β\, then glue X(ri) to V using a tube Si χ [0, 1] one end of
which is glued to с and the other end of which is glued to the image in V of the axis
of β. When viewed as a complex of groups, Υ has only one non-trivial local group,
namely the Z2 at the singular point of V.
In the case η — 2, if we choose go as in (7.6), then G can be presented as follows:
(a, b,s,t,a,fi Ι β = [s(ab)s~l, b], [a, b] = [α, β]2 = 1, tbt"1 = sas"1 = (abf).

Chapter III.C Complexes of Groups
Roughly speaking, complexes of groups were introduced to describe actions of groups
on simply connected simplicial complexes in terms of suitable local data on the
quotient. They are natural generalizations of the concept of graphs of groups due to
Bass and Serre. A technical problem arises from the fact the quotient of a simplicial
complex by a simplicial group action will not be a simplicial complex in general.
Indeed, even if the set-wise stabilizer of each simplex is equal to its point-wise
stabilizer, faces of a given simplex might get identified. (For example, regard the
real line as a one-dimensional complex with vertices at the integers and consider the
action of Ζ by translations.) Because of this problem, it is more natural to work with
polyhedral complexes.
To describe a polyhedral complex К combinatorially, it is natural to take its first
barycentric subdivision K' and view it as the geometric realization of the nerve of
a category whose objects are the barycentres of cells of К and whose non-trivial
morphisms correspond to the edges of K'. Such a category is a particular case of
what we call a small category without loops (scwol). The geometric realization of
a category without loops is like a simplicial complex, but the intersection of two
distinct simplices may contain more than a common face. Small categories without
loops serve as combinatorial descriptions of polyhedral complexes.
In the first section of this chapter we introduce scwols and their geometric
realizations, and we define what it means for a group to act on a scwol. The reader in a
hurry may wish to read only the definitions (1.1), (1.5) and (1.11) and then proceed
to the next section.
In the second section we give the basic definitions of a complex of groups GQ?)
over a scwol У and of morphisms of complexes of groups. Associated to each action
(in a suitable sense) of a group G on a scwol X there is a complex of groups G(y) over
the quotient scwol У = G\X. The construction of G(3O depends on some choices
but it is unique up to isomorphism. A complex of groups that arises in this way is
called developable. In contrast to the case of graphs of groups, in general a complex
of groups G(y) does not arise from an action of a group when the dimension of У is
bigger than one. We give a quick proof of the developability of complex of groups
over scwols of dimension one (a result due to Bass and Serre [Ser77]).
The fundamental group G of a complex of groups G(3O is defined in the third
section and a presentation of this group is given. In the case of a developable complex
of groups G(3O, it is possible to construct a simply connected category without loops

520 Chapter III.C Complexes of Groups
X and an action of the fundamental group G on X so that G(3O is the complex of
groups associated to this action.
Every complex of groups is locally developable; this is explained in the fourth
section where the local development is constructed explicitly; thus non-developability is
a global phenomenon. In the next chapter we shall prove that if a complex of groups
carries a metric of non-positive curvature, then it is always developable.
The theory of coverings of complexes of groups is explained in the last section.
It should be compared to the theory of coverings of graphs of groups in the sense of
Bass [Bass93].
It turns out that many concepts such as morphisms of complexes of groups,
coverings of complexes of groups, and so on, become more natural if we interpret
them in the framework of category theory. For this reason we recall in an appendix
the basic notions of small category, the fundamental group of such a category and
coverings. We hope that this will help the reader to realize that many complicated
formulas displayed elsewhere in this chapter are simply explicit descriptions of
elementary concepts.
Our treatment of complexes of groups is very similar to the combinatorial
approach of Bass-Serre to graphs of groups. A special case of complexes of groups,
triangles of groups, was first studied by Gersten and Stallings [Sta91]. They proved
the developabiliry theorem for non-positively curved triangles of groups. The general
case of complexes of groups was considered by Haefliger [Hae91] and (in dimension
2) independently by J.Corson [Cors92].
There is one important aspect of the theory of complexes of groups that we have
neglected here, namely the construction of their geometric realizations and their
relationship to complexes of spaces. This interpretation is useful for topological
and homological considerations. For this aspect of the theory we refer the reader to
[Hae91,92], [Cor92] and, in the case of graphs of groups, to [ScoW79].
1. Small Categories Without Loops (scwols)
Small categories without loops (abbreviated scwol) are algebraic objects which can
serve as combinatorial substitutes for polyhedral complexes. Canonically associated
to each polyhedral complex К there is a scwol X whose set of vertices is in bijection
with the set of cells of K. The geometric realization of Л' is a cell complex isomorphic
to the first barycentric subdivision of K. In this section we first give the basic
definitions of scwols, their geometric realizations, and morphisms between them. The
notions of fundamental group and covering for scwols are then introduced; these
correspond to the usual topological notions for their geometric realizations. We also
define what it means for a group G to act on a scwol X; when X is associated to
a polyhedral complex K, actions of G on X correspond to actions of G on К by
homeomorphisms that preserve the affine structure of the cells and are such that if
an element of G maps a cell to itself, then it fixes this cell pointwise. At the end we
analyse the local structure of scwols (this will be needed in sections 4 and 5).

1. Small Categories Without Loops (scwols) 521
Scwols and Their Geometric Realizations
1.1 Definitions. A small category without loops X (briefly a scwol)66 is a set X
which is the disjoint union of a set V(X), called the vertex set of X (the elements of
which will be denoted by Greek letters τ,σ, p,...), and a set E(X), called the set
of edges of X (the elements of which will be denoted by Latin letters a,b,c,...).
Two maps are given
i: E(X) -* V(X) and t: E(X) -* V{X);
i(a) is called the initial vertex of a e E(X) and t(a) is called the terminal vertex of a.
Let E<-2){X) denote the set of pairs (a, b) e E(X) χ Ε(Χ) such that i(a) = t(b).
A third map
Е^\Х) -> E(X)
is given that associates to each pair (a, b) an edge ab called their composition61.
These maps are required to satisfy the following conditions:
(1) for all (a, b) e E<-2)(X), we have i(ab) = i(b) and t(ab) = t(a);
(2) associativity: for all a,b,c e E(X), if i(a) = t(b) and i(b) = t(c), then (ab)c =
a{bc) (thus the composition may be denoted abc)\
(3) no loops condition: for each a e £(A0> we have i(a) φ t(a).
If we exchange the roles of i and t, we get the opposite scwol.
Consider the equivalence relation on V(X) generated by τ ~ σ if there is an
edge a such that i(a) = σ and t(a) = τ. The scwol X is connected if there is only
one equivalence class.
A subscwol X' of a scwol X is given by subsets V(X') с ν(Λ') and £(Л") с
£(Л0 such that if a e E(X'), then г'(а), г(а) е У(А"), and iia,b e E(X') are such
that i(a) = t[b), then ab e £(Л"). Every scwol X is the disjoint union of connected
subscwols, called its connected components.
We can consider X as the set of arrows of a small category 68 with set of objects
V(X): each vertex σ is identified to a unit element 1σ and we define г'( 1σ) = t( 1σ) =
σ; each edge corresponds to an element of X which is not a unit. When we view X
in this way, unspecified elements of X will be denoted by Greek letters α, β,...; an
element a e X is either an edge a or the unit 1σ identified to an object σ. A subscwol
corresponds to a subcategory.
1.2 Examples
(1) To any poset Q we can associate a scwol as follows. The set of vertices is Q
and the edges are pairs (τ, σ) e Q χ Q such that τ < σ; the initial (resp. terminal)
66 In [Hae92] a scwol (or rather its geometric realization) was called an ordered simplicial cell
complex.
67 If (a, b) € Ei2\X), one says that a and b are composable. One thinks of ab as "a following
fc".
68 See the appendix for basic definitions concerning categories.

522 Chapter III.C Complexes of Groups
vertex of the edge (τ, σ) is σ (resp. τ); the composition (τ, σ)(σ, ρ) is defined to be
(т. Р)·
(2) Product ofscwols. Given two scwols X and X', their product Χ χ Χ', is the
scwol defined as follows: V(X χ X') := V(X) χ V(X'),
E{X χ Χ') := (£(#) χ V(X')) \J(E(X) χ £(Λ")) JJcV(A') χ Ε(Χ')).
The maps z, t : Χ χ Λ" -> V(X) χ У(Л") are defined by /((a, a')) = (г'(се), ζ"(α'))
and ί((α, a')) = (i(a). *(<*')), and the composition (a, a')(/5, /J'X whenever defined,
is equal to (αβ, α'β'). This is a particular case of the product of two small categories
(see A.2(2)).
Notations. For each integer к > 0, let ESk\X) be the set of sequences (αϊ,..., ak)
of composable edges — (a,, a,+i) e £(2)(A0 for i = 1,..., к — 1. By convention
E(-°\X) = V(X). The dimension к of Л' is the supremum of the integers к > 0 such
that E^k\X) is non empty.
1.3 The Geometric Realization. For an integer к > 0, let Δ* be the standard k-
simplex, i.e. the set of points (ίο,..., г*) e ϋ*+1 such that г, > 0 and J2k=01,■ = 1.
Recall that the i, are the barycentric coordinates in Δ*.
We shall define the geometric realization |X\ of X as a polyhedral complex whose
cells of dimension к are standard £-simplices indexed by the elements of E^(X).
For к > 1 and i = 0,... ,k, let Э,- : E«\X) -> E<k~ X\X) be the maps defined by
9ο(αι, ...,ak) = (a2, ...,ak)
3,·(αι, ...,ak) = (αϊ,..., α,·α/+ι,..., ak) I <i <k
dk{au ...,ak) = (ar,...,ak^i).
For к = 1, we define 3ο(αι) = i(a\) and d\{a\) = t(ai). The following relations
follow from the associativity axiom 1.1(2):
(1.3-1) Э1^ = ^_,Э/, i<j.
We also define maps
φ : Δ*"1 -> Ak
for i = 0,..., k.
By definitiond", sends (ίο,..., in) e Δ*"1 to (ίο,.
Thus dj(Ak~~') is the face of Δ* consisting of points (to,
generally, the face of codimension r defined by г,-, = ·
the subset dh ... dir(Ak~r).
We have the obvious relations:
.,г,_1,0,г,-,...,гц) еАк.
.., tk) such that tj = 0. More
• = К =0, ζΊ>···> ir, is
(1.3-2)
djdj = dJ+ldi, i<j.

1. Small Categories Without Loops (scwols) 523
The Construction of the Geometric Realization
On the disjoint union
\jAkx{A},
k,A
where к > 0 and A e E^k\X), we consider the equivalence relation generated by
(4(*),А)~(*,Э,(А)),
where (x, A) e Δ*~' x E(-k\X). We define \X\, the geometric realization ofX, to be
the quotient by this equivalence relation; it is a piecewise Euclidean complex whose
£-cells are isometric to Δ*. The simplex Δ* χ {A} maps injectively to the quotient
(see exercise below) and its image will be called the k-simplex of\X\ labelled by A,
or simply "the k-simplex A". Thus the letters denoting elements of V{X) and E(X)
will also be used to denote the corresponding vertices (O-simplices) and 1-simplices
of \X\. If A = (ab ..., ak), we call i(ak) the initial vertex of A.
1.3.3 Remarks
(1) Note that in general the intersection of two simplices in \X\ is not a common
face, rather it is a union of faces.
(2) In general we consider \X\ as a piecewise Euclidean complex with the
topology associated to its intrinsic metric. Later we shall identify each simplex Δ& χ {Α}
to a simplex in some MnK in such a way that the inclusions (x, 3,A) м> (ά,(χ), A) are
isometries, thus metrizing \X\ as a MK-polyhedral complex. If the set of isometry
types of simplices is finite, then the topology associated to this new metric is the
same as the topology defined above.
(3) The geometric realization of the scwol associated to a poset Q is the same
as the geometric (or affine) realization of Q as defined in (II. 12.3). The geometric
realization of the product of two scwols is homeomorphic to the product of their
geometric realizations.
(4) The dimension of X is equal to the dimension of its geometric realization.
As observed above, the 1-simplices of \X\ correspond bijectively to the edges of
X; they inherit an orientation from this correspondence. If one follows an edge path
consisting of 1-simplices in the direction of the given orientations, one never gets a
closed circuit. The set of vertices of a ^-simplex is ordered.
1.3.4 Exercise. We use the notations of 1.3. Prove that the map λΑ associating to
(x, A) e Ak χ E^ its equivalence class in \X\ is injective on Δ* χ {A}.
(Hint: Suppose that χ e Δ* is contained in the interior of a face of codimension
r defined by tit = · · · = tir = 0, i\ > · · · > ir. Let χ be the unique point of Ak~~r
such that d-h ... dir(x) — χ and let A = 3,-, ... 3!r(A). Using the relations 1.3-1 and
1.3-2, show that (x,A) is the unique representative of (x, A) in Ak~~r χ E(k~~r){X).
This implies that the restriction of λΑ to {(χ, Α) | χ e interior of Δ*} is injective. The
injectivity of λΑ follows from 1.1(3).)

524 Chapter III.C Complexes of Groups
Fig.C.l The geometric realization of a scwol of dimension 2
1.4 Examples
(1) If Λ' is connected and 1-dimensional, its geometric realization is a connected
graph with two kinds of vertices: sources are initial vertices of edges and sinks are
terminal vertices of edges. This graph is bipartite and oriented. There are no edge
loops, but it is possible to have several edges connecting a given source to a given
sink.
т~ о
To О
Fig. C.2 A scwol of dimension one
(2) Naturally associated to any MK-polyhedral complex К there is scwol X whose
geometric realization is the first barycentric subdivision of K. The set of vertices V{X)
of X is the set of cells of К (equivalently the set of barycentres of the cells of K). The
edges of X are the 1 -simplices of the barycentric subdivision K' of K: each 1 -simplex
a of K' corresponds to a pair of cells Τ С S; we define i{a) to be the barycentre of S
and t(a) to be the barycentre of T.
A connected, one-dimensional scwol X is associated to a 1-dimensional
polyhedral complex (i.e. a graph) if and only if each vertex of X which is a source is the
initial vertex of exactly two edges.

1. Small Categories Without Loops (scwols) 525
V(y) = {p,a,r}
Е(У) = {ai,a2,b1,b2,bs,c1,C2,c3}
(elements with the same
label should be identified)
a2 al T2
Fig.C.3 The scwol associated to the dunce hat (cf. 1.7.41(2))
Fig. CA The scwol associated to two congruent rc-gons glued along their boundary
(3) Figure С A shows the scwol X associated to the polyhedral complex obtained
by gluing two congruent и-gons in Ml along their boundary. There are η vertices
τ*., corresponding to the vertices of the polygons, η vertices ak corresponding to the
barycentres of the sides and two vertices ρ and p' corresponding to the barycentres
of the two polygons. There are 6n edges, ak, a'k, bk, b'k, ck, c'k with к = 1, ..., и,
and i{bk) = i{ck) = p, i(b'k) = i(c'k) = p', t(bk) = t(b'k) = ak, t{ck) = t{c'k) =
4, i(ak) = i{dk) = ak and t{ak) = тк, t(a'k) = тц mod n. We have ck = akbk =

526 Chapter III.C Complexes of Groups
1.5 Morphisms of Scwols. Let X and У be two scwols. A non-degenerate morphism
f : X -» У is a map that sends V(X) to V(y% sends E(X) to Е(У) and is such that
(1) for each a e E(X), we have i(/(a)) = f(i(a)) and ί(/(α)) = /(i(a)),
(2) for each (a, fc) e £(2)(A') we have/(afc) = f(a)f(b), and
(3) for each vertex σ e ν(Λ'), the restriction of/ to the set of edges with initial
vertex σ is a bijection onto the set of edges of У with initial vertex/(a).
More generally, a morphism f : X —» У is a functor from the category X to the
category У (see A. 1). An automorphism of Л' is a morphism from X \a X that has
an inverse, i.e. a functor/-1 : X -*■ X such that/"1/ =ff~l is the identity of X.
A morphism/ : Л" —> У induces a map 1/| : |ΛΊ —> |УI of the geometric
realizations: [/| maps each vertex σ of \X\ to the vertex/(σ) of \У\ and its restriction
to each simplex of \X\ is affine. In the particular case where/ maps E(X) to Е(У)
(for instance when/ is non-degenerate), \f\ maps a point of the ^-simplex of \X\
labelled (αϊ,..., ак) to the point in the simplex of \У\ labelled (f(ai),... ,/(a*)) that
has the same barycentric coordinates. If/ is an isomorphism, then \f\ is an isometry.
If X and У are associated to polyhedral complexes К and L, then a morphism
f:X-*yis non-degenerate if and only the restriction of \f\ to the interior of each
cell of К is a homeomorphism onto the interior of a cell of L.
The Fundamental Group and Coverings
1.6 Edge Paths. Let Л' be a scwol. Let E±(X) be the set of symbols a+ and a~,
where a e E(X). The elements e of E±(X) can be considered as oriented edges of
X. For e = a+, we define i(e) = t(a), t(e) = i(a) and e~x = a"; for e = a", we
define i{e) = i(a), t(e) = t(a) and e~x = a+. An edge path in A' (or a path in X)
joining the vertex σ to the vertex τ is a sequence с = (еь ..., e*) of elements of
E±{X) such that i(ei) = σ, i(e,) = i(ei+\) for 1 < i < fc - 1, and t(ek) = τ. We call
σ (resp. τ) the initial vertex i(c) of с (resp. the terminal vertex t(c) of c). If σ = τ,
we allow к to be 0, giving the constant path at σ. Note that X is connected if and
only if for any σ, τ e V(X), there is a path joining σ to τ. If d = (e\,..., e'k.) is
a path in Λ' joining σ' = τ to τ', then one can compose с and c' to obtain the path
с * c' = (ei,..., ek, e'[,..., e'k) joining σ to τ', called the concatenation of с and
c'. Note that this concatenation operation is associative. The inverse of the path с is
defined to be the path c~l = (e\,..., enjoining τ to σ, where ej = e^'+1.
If i(c) = t(c) = σ, then с is called a Zoo/j at σ.
1.7 Homotopy of Edge Paths. Let с = (еь ..., ек) be an edge path in A'joining σ
to τ. Consider the following two operations on c:
(i) Assume that for some j < k, we have e, = e^1,. Then we get a new path d by
deleting from с the subsequence e,, e,+i.
(ii) Assume that for some7 < £, we have e, = a+, ey+i = b+ (resp. e^ = b~, e,+i =
a-), and therefore the composition ab is defined; we get a new path c' in X by
replacing the subsequence e,-, e,+ | of с by (ab)+ (resp. (ab)~).

1. Small Categories Without Loops (scwols) 527
Under the circumstances of (i) or (ii), we say that с and c' are obtained from
each other by an elementary homotopy (thus implicitly allowing the inverses of
operations (i) and (ii)). Two paths joining σ to τ are said to be homotopic if one can
pass from the first to the second by a sequence of elementary homotopies (cf. I.8A
and A. 11). The set of homotopy classes of edge paths in X joining σ to τ is denoted
Tt\{X, σ, τ). The set Tt\{X, σ, σ) will also be denoted π\(Χ, σ).
Note that the path с * с"' is homotopic to the constant path at σ and that c~' * с is
homotopic to the constant path at τ. The homotopy class of the concatenation c*c'
of two paths depends only on the homotopy classes of с and c'.
1.8 Definition of the Fundamental Group. Let Л' be a connected scwol and let
σ e V(X). The set ηι(Χ, σο) of homotopy classes of edge loops at σο, with the law
of composition induced by concatenation, is a group called Has, fundamental group
of X at σο·
If с is an edge path in Л'joining σ to τ, then the map that associates to each loop
/at σ the loop с-1 * /* с at τ induces an isomorphism of я [(Λ', σ) onto πι(Χ, г). А
scwol X is said to be simply connected if it is connected and πι(Χ, σ) is the trivial
group for some (hence all) σ e V(X).
It is a classical fact that there is a natural isomorphism from it\ (Χ, σο) to
7r,(|^|,CTo)[Mass91].
1.9 Definition of a Covering. Let Λ' be a connected scwol and let X' be a (non-empty)
scwol. A morphism ρ : X' —> Λ' is called a covering of X if, for every σ' e V(X'),
ρ sends the set {a' e E(X')\ ί(α') = σ'} bijectively onto {a e E(X)\ t(a) = ρ(σ')}
and sends the set {a' e E{X')\i{a!) = σ'} bijectively onto {a e E(X)\ i(a) = ρ(σ')}.
If ρ : X' —» X is a covering then one says that X' covers X.
Remarks
(1) Note that the geometric realization \p\ : \X'\ —> \X\ of ρ is a topological
covering. If X' covers X and Λ' is a scwol associated to a poset, then X' is also
associated to a poset. The converse is not true in general.
(2) Let ρ : X' -» X be a covering. An edge path c' = (e'p ..., e^.) in Λ" with
i(c) = σ' is called а /г/?ш# at σ' of the path с = /j(c') = {e\,..., ek) if e,- = /?(е^)
(where, by definition, p{d ) = (^(a'))* if a' e £(Л"))· It follows immediately from
the definition of a covering that if ρ(σ') = σ, then any edge path in X issuing from σ
can be lifted uniquely to an edge path in X' at σ'. In particular, as X is assumed to be
connected, ρ is surjective. Moreover if two edge paths c\, c'2 in X' issuing from σ' are
mapped by ρ to homotopic edge paths, then c\ and c'2 are also homotopic. Therefore
the map associating to the homotopy class of an edge loop d at σ' the homotopy class
of its image under ρ induces an injective homomorphism ττι(Χ', σ') —> π\{Χ, σ).
(3) If X is not connected, it is reasonable to define a morphism ρ : X' —> X to
be a covering if/j is surjective and if, for each connected component Xq of Λ\ the
restriction of ρ to ρ"' (ΛΌ) is a covering of Xq as defined above.

528 Chapter III.C Complexes of Groups
1.10 Construction of a Simply Connected Covering. Let Л' be a connected scwol
and let σο be a base vertex. We construct a simply connected scwol X and a covering
map ρ : X —> X as follows. The set of vertices V(^) of X is the set of pairs (σ, [с]),
where σ e V(#) and [c] is the homotopy class of an edge path с in Λ' that joins σο
to σ. The set E(X) consists of pairs (a, [c]) where a e E(X) and [c] is the homotopy
class of an edge path с joining σο to i(a). The initial vertex of the edge (a, [c]) is
(i(a), [c]) and its terminal vertex is (t(a), [c * a~]); the image under ρ of (a, [c]) is
a. The composition (a, [c])(a', [c']) is defined if [c] = [t/ * a'-], in which case it is
(aa', [c']). It is clear that/? is a morphism which is a covering. For instance, if [c] and
α are such that t(c) = t(a), then (a, [c * a+]) is the unique edge of X with terminal
vertex (t(a), [c]) that is sent by ρ to a. It is clear that X is connected. To see that X
is simply connected, consider σο = (σο, [cq]) e V(X), where [c0] is the homotopy
class of the trivial loop at σο; the lifting at σο of an edge loop с in X based at σο will
be an edge loop if and only if the homotopy class of [c] is trivial.
Group Actions on Scwols
1.11 Definition. An action of a group G on a scwol Λ' is a homomorphism from
G to the group of automorphisms of X such that the following two conditions are
satisfied:
(1) for all a e E(X) and g e G, we have g.i(a) φ t{a)\
(2) for all g e G and a e E(X), if g.i(a) = i(a) then g.a = a.
(Here g.a denotes the image of a e X under the automorphism of A' associated to
g e G; thus condition (2) means that the stabilizer of i(a) is contained in the stabilizer
of a.)
1.12 Remarks
(a) Condition (1) is automatically satisfied if X is finite dimensional. It is not
satisfied, for example, if X is the category associated to the poset Ζ with its natural
ordering and G is the infinite cyclic group whose generator maps η to η + 1.
(b) The induced action on \X\. The action of G on X induces an action of G
by isometries of the geometric realization \X\ in the obvious way; the isometry
induced by g e G is denoted \g\. Geometrically, condition (2) in the above definition
means that if |g| fixes a vertex σ, then it fixes (pointwise) the union of the simplices
corresponding to composable sequences (αϊ,..., α^.) with г'(а^) = σ. In particular,
if X is the scwol associated to a polyhedral complex К and |g| leaves a cell of
К invariant, then its restriction to that cell is the identity. (When К is a graph, an
action satisfying condition (2) is called an action without inversions.) This condition
is always satisfied for the action induced on the barycentric subdivision of К by a
polyhedral action on K.

Group Actions on Scwols 529
1.13 The Quotient by an Action. Condition (1) ensures that the quotient У = G\X
of X by the action of G is naturally a scwol У with VQO = G\V(X) and Е(У) =
G\E{X). The initial (resp. terminal) vertex of the G-orbitof an edge a is the G-orbit
of i(a) (resp. t{a)), and if the composition ab is defined, then the orbit of ab is the
composition of the orbit of a with the orbit of b. Let ρ : X —» У denote the natural
projection. Condition (2) implies that/? is a non-degenerate morphism. Note that if
У is connected, then ρ : X —> У is a covering provided the action of G on V(X) is
free.
A Galois covering of a scwol У with Galois group G is a covering /? : Л' —» У
together with a free action of the group G on X such that /? induces an isomorphism
from G\X to У. In particular G acts simply transitively on the fibres of p.
1.14 Examples
(0) Let Л' be a connected scwol with a base vertex σο and let ρ : X —» Λ' be
a covering such that X is simply connected. Then ρ can be considered as a Galois
covering with Galois group it\(X, σο).
To see this, let σ0 e V(X) be such that /?(σο) = σο· For each σ e V(A?), there
is an edge path с joining σο to σ; this edge path is unique up to homotopy. Given
[с] е π·ι(Λ\ σο), by definition the image [c].cr of σ under the action of [c] is the
terminal vertex of the lift at σο of the edge path с * p(c). For each a e E(X) with
/?(a) = a and ί(α) = σ, we define [c].a to be the unique edge that projects by ρ to a
and has initial vertex [c].a. This completes the definition of the action of πι(Χ, σο)
on X. The action is simply transitive on each fibre of p.
Fig. C.5 The quotient of an rc-gon by a rotation
(1) Let Ρ be a regular и-gon in E2 centred at 0 (see figure C.5) with vertices
P\,..., pn (in cyclic order). Let σ^ be the midpoint of the side joinings to pk+\ (the
indices are taken modulo n). Let X be the scwol whose set of vertices V(X) is the

530 Chapter III.C Complexes of Groups
union of the origin τ = 0, the η points ak and the η points pk, where 1 < к < п. The
set of edges E(X) contains An elements: η edges ak with i{ak) = ak and t(ak) = τ;η
edges fej. (resp. b'k) with *(&*) = p* and t(bk) = ak (resp. i(b'k) = pk and ί(^) = σ*_ι);
and и edges ck with /(с*) = p* and t(ck) = τ. Moreover, ck = akbk = ak-\b'k. The
geometric realization of X is isomorphic (as a simplicial complex) to the barycentric
subdivision of P.
Let G be the cyclic group of order n. The action of G by rotations of E2 fixing
the origin induces an action on X; the quotient is the scwol У with 3 vertices τ, σ, ρ
and 4 edges a,b,b',c, where τ = ί(α) = i(c), σ = i(a) = t(b) = t{b') and
ρ = i(b) = i(b'). The geometric realization of У is a union of two triangles whose
intersection is the union of two sides. Although the category X is associated to a
poset, У is not associated to a poset.
Note that if one were to replace X by the opposite category (interchanging i and
t), then condition (2) in the definition of an action would not be satisfied.
a\ σ, α
-A jr ^- -l
Fig. C.6 The action of the subgroup G С Isom^E2) that preserves a tesselation by regular
hexagons
(2) Suppose that we are given a tesselation of M2. by regular и-gons with vertex
angles 27Г/3 and let G be the group of orientation preserving isometries of M2 that

The Local Structure of Scwols 531
preserve this tesselation. We consider an associated polyhedral subdivision К of Μ2:
the 2-cells of К are the quadrilaterals formed by pairs of triangles from the bary centric
subdivision of the tesselation that meet along an edge joining the barycentre of an
и-gon of the tesselation to a vertex of that и-gon (figure C.6 shows the case of the
tesselation of E2 by regular hexagons).
The group G preserves the cell structure of К and if an element of G leaves a cell
invariant, then it fixes it pointwise. It follows that G acts on the scwol X associated
to К (in the sense of 1.11). The quotient У of X by this action is the scwol associated
to the polyhedral complex L obtained by identifying two pairs of adjacent sides in a
quadrilateral 2-cell OABA in figure C.6.
У has 6 vertices, three of them, τ, τ', τι, correspond to the three vertices of L,
two, σ and σι, correspond to the 1-cells of L, and one, p, corresponds to the 2-cell of
L.Andy has 12edges: eight of thembi, b\, b\, b\, c\, c\, c\, c[ have initial vertex/),
and t(bi) = t(bi) = σ, Щ) = t(b\) = out{cx) = τ, t{cx) = ть t{d{) = t{c\) = τ'.
1.15 Lifting Actions to a Simply Connected Covering. Let Λ' be a scwol with base
vertex σο and let ρ : X —> X be the simply connected covering constructed in 1.10.
Given a group G acting on X, we consider a group G whose elements are the pairs
(M, g), where g e G and [c] is the homotopy class of an edge path in Λ'joining σ0 to
g. σο; the composition of elements in G is defined by ([c], g)([c'], g) = ([c*g.c'], gg').
The group G acts on X as follows: given (a', [c']) e X, the action of ([c], g) e G is
(M, g)-(oc\ [c']) = (g-a\ [c*g.c']).~LetΦ : G -»■ G be the homomorphism mapping
(g, M) to g- We have an obvious exact sequence:
1 -» m(X, σ0) -> G ->■ G -+ 1.
The projection ρ : X ->■ X is Φ-equivariant and induces an isomorphism of G\X to
G\X. The subgroup π\{Χ, σ0) of G acts freely on X with quotient Λ'.
The Local Structure of Scwols
1.16 Definition of the Join. Let X and X' be two scwols. The join69 X * X' of X
and A" is the following scwol, which contains the disjoint union of X and X' as a
subcategory:
V(X * Λ") := V(X) ]_[ V(X'), E(X * X') := E(X) \J(V(X) χ V{X')) ]_[ E(X')
(the edge corresponding to (σ, σ') e V(X) χ У(Л") is denoted σ * σ'); the maps
i, ί : £(Λ" * Χ') -> V(X * Λ") restrict to E(X) and £(Λ") in the obvious way and
ί(σ * σ') := σ' and ί(σ * σ') := σ. The composition of the edge a e E(X) with
the edge i(a) * σ' e V(A^) χ V(X') is given by a(i(a) * σ') := ί(α) * σ' and the
composition of the edge σ * ί(α') e V(A') χ У(Л") with the edge a' e E(X') is
defined by (σ * t(a'))d := σ * i(a').
In general X * Λ' is not isomorphic to A?' * X

532 Chapter III.C Complexes of Groups
Given two morphisms of scwols/ : X -> У and/' : X' -> ЗЛ their join is the
unique morphism/ */' : X * X' -»■ У * У which extends/ and/' and which maps
the edge σ * σ' to the edge/(a) */'(σ').
The geometric realization of X * X' is affinely isomorphic to the simplicial join
of the geometric realizations of X and X'.
1.17 Definitions of Lk,, (X), U?(X), Χσ, Χσ, Χ(σ).
For σ € V(X), we define a scwol Ькст(Л') = Ll^, called the upper link of σ, by
setting
У(Цс„(ДГ)) = {α e £(#): ί(α) = σ}
£(1Л„(ДГ)) = {(α, b) e ^(ЛГ): ί(α) = σ},
denning maps i, ί: E(Lka (X)) -> V(Lko- (АО) by i((a, &)) = ab, t((a, b)) = a, and
defining composition when ab = d by
(a, b)(a\ b') = (a, bb1).
The natural morphism Ll^ -»■ Λ' mapping (a, fr) e £(Lko-) to b e E(X) is in general
not injective on the set of vertices.
The lower link \}ца(Х) is a scwol denned similarly:
V(Lka(X)) ={ae E{X): i(a) = σ}
E(Lka(X)) = {(a, b) e EF>(X): i(b) = σ],
with i((a, b)) = b, t((a, b)) = ab and composition is defined by
(a, b)(a', b') = (aa', b').
There is a natural morphism LkCT —»■ Λ" mapping (a, b) e £(Οίσ) to a e £(A0-
Regarding σ as a scwol with a single element σ, we define
Xa = a*LK(X), Χσ =1±σ(Χ)*σ
Χ(σ) = U? {Χ) * σ * LkCT(A0 = Λ"7 * LkCT(A0 = Lk^AO * Χσ.
A non-degenerate morphism/ : X ^- У induces, for each σ e V(A^), an
isomorphism from Xе onto yf-a\ Note that if У is connected and X non-empty, then/
is a covering if and only if/ induces an isomorphism from Χ(σ) to y(f(a)) for all
σ e V(A').
Note that if we replace X by the opposite scwol, then the roles of Χσ and Xе are
exchanged.
We have a natural morphism
Χσ = σ # LkCT —»■ X,
mapping a to σ, whose restriction to Lka is as above and which maps the edge σ * a
to a. Similarly we have a morphism

The Local Structure of Scwols 533
1*1 = \X(r)\
Fig. C.7 Examples
\X°
σ
\Χσ
\Χ(σ)
We also have a morphism
X" ->X.
Χ(σ) -* X
that extends the morphisms defined above and maps the edge a * b e E(Lka * LkCT) to
the edge ab e E(X), where i{a) = t{b) = σ. This morphism is injective if and only
if any two distinct edges with initial vertex σ (resp. terminal vertex σ) have distinct
terminal vertices (resp. initial vertices).
1.18 Examples
(1) If Λ' is associated to aposet V (as in 1.2(1)), then 1^{Χ) (resp. X" or Χ{σ))
is the category associated to the subposet LkCTCP) (resp. V or V{a)) as defined in
II. 12.3.
(flvb3) (a2,b3)
Fig.C.8 hkT(X) in the scwol associated to the dunce hat (see figure C.3)
(2) Let X be the scwol associated to an MK -polyhedral complex К (see example
1.4(2)). Let σ e V(X) be the barycentre of a cell of К which is the image under ρλ
of a convex cell Cx (in the notations of 1.7.37). Note that the geometric realization
of Xе is the barycentric subdivision of Cx. Therefore, the morphism X" -> X is
injective if and only if the map ρλ is injective.

534 Chapter III.С Complexes of Groups
If σ is a 0-cell of K, then there is a natural bijection from the geometric realization
of Ll^ to the geometric link of σ in the barycentric subdivision of К as defined in
1.7.38. (See figure C.8.)
(3) Let 1 be the scwol whose geometric realization is the barycentric subdivision
of the unit interval. Thus 1 has three vertices τ0, τι, σ and two edges ao,a\ with
ί(α0) = το, t{a\) = τ\, and г'(ао) = i{a\) = σ. The geometric realization of the
rc-fold product I" is the barycentric subdivision of the rc-cube.
A scwol X is associated to a cubed complex if and only if, for each σ e V(X),
the scwol X" is isomorphic to lk for some к.
1.19 Exercise. Show that the geometric realization of the morphism Χ(σ) —> X
induces an affine isomorphism of the open star of σ e ΙΑ'(σ)! onto the open star of
ae\X\.
2. Complexes of Groups
Let G be a group acting on an MK -polyhedral complex К by isometries that preserve
the cell structure. Assume that if an element of G leaves a cell of К invariant, then it
fixes the cell pointwise. The quotient of К by this action is again an MK -polyhedral
complex, which we denote L. For each cell σ of L, we select a cell σ of К in the orbit
σ. Let Ga be the isotropy subgroup of σ. If τ is a face of σ, the isotropy subgroup
of the corresponding face of σ is a subgroup of GCT> therefore the isotropy subgroup
GT of the chosen representative Tof τ is conjugate to a subgroup of Ga. The family
of abstract groups (GCT) together with the injective homomorphisms Ga —> Gx and
some additional local data define on L what we call a complex of groups G(L) over
L. The Ga are called the local groups of G(L). When L is one dimensional, G{L) is
a graph of groups. Remarkably, if К is simply connected then one can reconstruct
from G(L) the complex K, the group G, and its action on K. On the other hand,
when L is of dimension greater than one, it is in general not the case that an abstract
complex of groups G(L) over L is developable, i.e. that it arises from an action of a
group on a polyhedral complex K.
In order to describe complexes of groups over polyhedral complexes combina-
torially, it is convenient to pass to the associated scwol and to develop the theory in
this framework. We can then come back to geometry using geometric realizations of
scwols.
The essential points of this section are the definition (2.1) of a complex of groups
G(y) over a scwol У, the definition (2.4) of morphisms of complexes of groups, and
the construction in (2.9) of the complex of groups G(3O associated to an action of a
group G on a scwol X (where У is the quotient of X by G), and finally the notion
of developability. If G{y) is a complex of groups associated to an action of a group
G on a scwol X, then there is a morphism φ : G(y) —> G which is injective on the
local groups. Conversely, one can associate to any such morphism an action of G on

Basic Definitions 535
a scwol D(y, φ) called the development of У with respect to φ. (This construction,
described in (2.13), generalizes the Basic Construction in 11.12.18.)
Basic Definitions
The next definition will be justified by the considerations in (2.9).
2.1 Definition of a Complex of Groups
Let У be a scwol. A complex of groups G(3O = (GCT, ψα, gab) over У is given
by the following data:
(1) for each σ e У(У), a group Ga called the local group at σ,
(2) for each a e Е(У), an injective70 homomorphism ψα : G,(a) —> G,(a),
(3) for each pair of composable edges (a, b) e £(2)(3O, a twisting element gaib e
Gt(a),
with the following compatibility conditions:
(i) ^{ga.bHab = -фа-фь,
where Ad(gab) is the conjugation by gab in Gi(a);
and for each triple (a, b, с) е ЕР\У) of composable elements we have the cocycle
condition
(U) ta{gb.c)ga.bc = ga.bgab.c-
A simple complex of groups over У is a complex of groups over У such that all the
twisting elements ga^b are trivial. A complex of groups over a poset is by definition
a complex of groups over the associated scwol. A simple complex of groups over a
poset as defined in (II. 12.11) is a simple complex of groups over the scwol associated
to this poset.
Note that condition (i) (resp. condition (ii)) is empty if У is of dimension 1
(resp. < 2) because in that case Е(2)(У) (resp. Е<3)(У)) is empty.
Suppose that we are given a map that associates to each edge a e Е(У) an element
ga € Gt(fl). The complex of groups 0{У) over У given by G'a = Ga,f'a = Ad(ga)\lra
and g'ab = ga^a{gb)ga,bgab ls sa^ to ^ deduced from G(3O by the coboundary of
(ga). Note that G(y) is deduced from 0{У) by the coboundary of (g"1).
2.2 Notation. We wish to use the description of a scwol as (the arrows of) a category,
identifying vertices σ to units 1σ. We define^ ■" G,(„) —> G,(„) as before if α e £(3Ό
and as the identity of Ga if a = 1 σ. For composable morphisms α, β ey,we define
gaj as above if α, β e £(У) and as the unit element of G,(a) otherwise. Conditions (i)
70 One could develop the theory of complexes of groups without this injectivity assumption,
the only serious modification would be that local developability (in the sense of section 4)
would then be equivalent to the injectivity of the ψα.

536 Chapter Ш.С Complexes of Groups
GQ>)
У
^J* G^ Gr
a' a
ο -< · >- о
τ' σ τ
(1)
Φα
Φα'
α
ο ·
τ V. ^ σ
α'
(2)
GT1 Gri
Φ^ G^a>
Фаз
Υ
GTi
Τ"2 Τχ
ο ο
*σ
α3
Υ
ο
Τ"3
(3)
Fig. £9 Complexes of groups over scwols of dimension one
and (ii) of 2.1 are still satisfied if we replace a, b, с in those formulae by composable
elements α, β, γ € Χ.
2.3 Remark. Every complex of groups over the poset associated to the faces of an
и-simplex can be deduced from a simple complex of groups over this poset by a
suitable coboundary.
More generally, let У be a scwol which has a vertex σο such that for each vertex
σ φ σο, there is a unique edge ασ with Ί{ασ) = σο, ί(ασ) = σ. Then any complex
of groups G(y) over У can be deduced from a simple complex of groups over У
by a coboundary. Indeed it is straightforward to check that the complex of groups
deduced from G(y) by the coboundary of (g~^ ) is simple.
2.4 Morphisms of Complexes of Groups. Let G(yr) = (Ga>, -ψα>, ga\b') be a
complex of groups over a scwol У'. Let/ : У —> У be a (possibly degenerate) morphism
of scwols. A morphism φ = (φσ, φ(α)) from G(y) to G(y') over/ consists of:
(1) a homomorphism φσ : Ga -* G/(CT) of groups, for each σ e V(y), and
(2) an element φ(α) e G,(f(a)) for each α e £(3Ό, such that
(i) Ad(0(aM/(a)0i(a) = 0ί(α)ΐΑα>
andforall(a»e£(2)Cy),
(ϋ) Фк,а)(8а,ьЖаЬ) = Ф(а)-фПа){ф{Ь)^Ш(Ь).
If/ is an isomorphism of scwols and φσ is an isomorphism for every σ e V(3O,
then φ is called an isomorphism.
With the point of view of (2.2) (which we shall often take implicitly in what
follows), given a e У, we define φ(α) as above if a e £(3Ό and as the identity

Basic Definitions 537
С-з Gr
ψ
GTl-<— Ga >■ GT2 GT^— Ga Ga, »- Gr
T"3
T"3
/V
/
аг -s a3/ \аз
о -«-
Π αι σ a2 T2 η αι σ σ' α2 τ2
д., /(σ)=/(σ')=σ ^
Fig. С. 10 A morphism of one-dimensional complexes of groups (homomorphisms between
local groups with the same label are isomorphisms)
element of G/((J) if α = 1σ. Conditions (i) and (ii) are still satisfied if a, b are
replaced by composable elements α, β еУ.
Homotopy. Let φ and φ' be two morphisms from G{y) to GQ7') over/ given
respectively by (φσ, φ{α)) and (φ'σ, φ'(a)). A homotopy from φ to φ' is given by a family
of elements sa e G/(CT) indexed by σ e V(y) such that
φ'σ = Аа(ка)фа and ф'(а) = к,(а)ф(а)фт(к^) Va e VQT), a e ВД.
If such a homotopy exists, we say that φ and φ' are homotopic.
Composition. If φ = (φσ,φ(α)) : G(y) -* GO") and 0' = (φσ',φ'(α')) :
(G(y!) —> G(y') are morphisms of complexes of groups over morphisms/ : У —>
У' and/' : У' -* У", then the composition 0' ο φ : в(У) -* в(У") is the morphism
over/'/ defined by the homomorphisms (φ' ο φ)σ = 0' ο 0σ and the elements
(0' о ф){а) = ф;ш)(ф(а))ф'(/(а)).
Let us repeat the definition of morphism in the important special case where У
has only one vertex, i.e. G(y') is simply a group G.
2.5 Definition. A morphism φ = (φσ, φ{α)) from a complex of groups G(3O to a
group G consists of a homomorphism φσ : Ga —» G for each σ e V(3O and an
element φ(α) e G for each α e £(3Ό such that
«fe)1^ = Ad(0(a))0/(a) and <f>na)(ga,b)<t>(ab) = ф(а)ф(Ь).
We say that 0 is injective on the local groups if each homomorphism $σ is injective.
2.6 Remark. Let G(3O be a simple complex of groups. A morphism φ : G(3O —> G
is called iimp/e if 0(a) = 1 for all a e Е(У). We shall see later (3.10(3)) that if У
is simply connected, then every morphism φ : G(y) —■► G is homotopic to a simple
one.

538 Chapter Ш.С Complexes of Groups
2.7 Induced Complex of Groups. If/ : У -» У is a morphism of scwols and if
G(y') = (GCT-, ψα>, ga',b') is a complex of groups over 3·7', then the induced complex
of groups f*(G(y')) is the complex of groups (GCT, ψα, gab) over y, where
Ga = G/((j), ψα = ψηα), gaib = gf(a)J(b)-
(Note that there is an obvious morphism φ : f*G(y') -> G(y) over/, where 0σ is
the identity and 0(a) is the unit element of G,(f^.)
If У is a subscwol of У and/is the inclusion, thenf*G(y) is called the restriction
ofG{y') to У.
2.8 The Category CGCV) Associated to a Complex of Groups GQ0
The definitions of complexes of groups and morphisms between them appear
more natural if we interpret them in the framework of category theory (see A. 1).
We associate to the complex of groups G(y) the small category CG(y) whose
set of objects is У(У) and whose set of elements (arrows) are the pairs (g, a), where
a e У and g e Gi(a). We define maps i, t: CG(3O -* У{У) by i((g, a)) — i(a) and
t((g, a)) = t(a). The composition (g, a)(h, β) is defined if i(a) = ί(β) and then it is
equal to
(8,α)φ,β) = (8ψα(.Κ)8α,ρ,αβ).
Conditions (i) and (ii) in (2.1), when taken together, are equivalent to the associativity
of this law of composition. The map (g, а) м> a is a functor CG(y) —» y.
A morphism φ : G(y) —» G(y) of complexes of groups over/ : У ^ У gives
a functor CG(y) —» CG(y') of the associated categories, namely the functor (also
denoted φ) mapping (g, a) e CG(y) to (фца)(я)ф(а), f(a)) e Св(У). Conversely,
any functor φ : CG(y) -^ CG(y') projects to a functor/ : У -> У and defines a
morphism (also denoted φ) from G(3O to G(y') over/ — the homomorphisms 0σ
and the elements φ(α) are determined by the relations: </>(g, 1σ) = $a(g), 1/(σ)) and
0(1,(β),α) = (0(α),/(α)).
If 0' : G(^') -^ G(y") is a morphism over/' : У -► У, then the composition
0Ό 0 is the morphism G(3^) -> G(y) over/' of ■ У -> У" corresponding to the
composition of the associated functors.
Developability
2.9 The Complex of Groups Associated to an Action
(1) Definition. Let G be a group acting on a scwol X (in the sense of definition 1.11).
Let У = G\X be the quotient scwol and let ρ : X —> У be the natural projection.
For each vertex σ e У(У) choose a vertex σ e V(A') such that ρ(σ) = σ. For
each edge a e ДЛ') with i(a) = σ, condition (2) of (1.11) implies that there is a
unique edge a e X such that p(a) = a and г(а) = σ. If τ = t(a), then in general
τ φ t{a); choose ha e G such that ha.t(a) = τ. For σ e У(У), let GCT be the isotropy

Developability 539
Fig. £11 Associating a complex of groups to an action
subgroup of σ, and for each a e Е(У), let ψα : G,(a) —> G,(0) be the homomorphism
defined by
Ya(g) = hagh~l.
Condition (2) of (1.11) implies that the image of ψα lies in G,(ay For composable
edges (a, b) e ЕР-\У), define
ga.b = hahbh~bl,
an element of G,(0) (see figure С11). The complex of groups over У associated to
the action ofGonX (and the choices above) is
G(y) = (Ga,i,a,ga,b).
It is easy to check that the two conditions (i) and (ii) of (2.1) are satisfied.
There is a natural morphism associated to the action:
φ : G(y) -> G,
φ = {φσ, φ{α)), where φσ : Ga -> G is the natural inclusion and φ{α) = ha. This
morphism is injective on the local groups. (In (2.13) we shall see that all morphisms
with this local injectivity property arise from the construction described above.)
(2) Different Choices. A different choice of elements ha in the above construction
would lead to a complex of groups deduced from G(y) by a coboundary. A different
choice of representatives σ' in the G-orbits of σ and associated choices h'a would lead
to a complex of groups G'(У) = {Οσ, ψ'α, g'a b) over У that is isomorphic to G(y). An
isomorphism λ = (λσ, λ(α)) from G(3O to о'{У) is obtained by choosing elements
ka e G such that ka .σ = σ' and defining λσ = Ad(^CT)|G„ andX(a) = k,^hak^h'a~l.
(Different choices of the elements ka give homotopic isomorphisms.) Note that the

540 Chapter Ш.С Complexes of Groups
ka determine a homotopy from φ to φ'λ, where ф' : С(У) —> G is the natural
morphism described above.
Note also that if one can find a strict fundamental domain for the action of G,
i.e. if one can choose the liftings such that for each a e Е(У) with τ = t{a) we have
t(a) = 7, then by choosing the ha to be trivial, we get a simple complex of groups.
(3) When У is connected, one way of choosing the liftings σ is to first construct a
maximal tree Τ in the 1-skeleton |У|(1) of the geometric realization of У and to lift it
to a tree Τ in |ΛΊ(1) so thatp{T) = Τ (this is possible because ρ is non-degenerate).
Then choose σ to be a vertex of Τ and ha to be the identity if i(a) = σ and t(a)
is a vertex of Τ (where as above a is the unique edge in X projecting to a with
i(a) — σ). If Xo is the connected component of X containing Τ and if Go is the
subgroup of G leaving X0 invariant, then the inclusion Xq -> X induces a bijection
G\Xq —> G\^ = У and the complex of groups associated to the action of G0 on X0
is canonically isomorphic to G(y).
(4) Equivariant Actions. Let О be a group acting on a scwol X' and let p' be the
natural projection from X' to У := G'\X'. Let в(У) = (GCTS V^, ga',6') be the
complex of groups associated to this action with respect to choices σ' e V(X') and
ha, e G' as above. Let φ': G(y) —> G' be the associated morphism. Suppose that we
have a morphism L : X —» Λ" that is equivariant with respect to a homomorphism
A : G ^ G' (i.e. L(g.a) = A(g).L(a) for all α e X and g e G). This induces a
morphism /: У —> У, for which there is a commutative diagram
X —^-» Л"
Ί 4
У —U. y.
For each σ e V(y) we choose an element ka e G' such that ka.L(W) = 1{σ). For
each σ e V(3^) let λσ : Ga -> G;(CT) be the homomorphism g н> kaA(g)ka~l, and
for each α e Е(У) let λ(α) = ^Л^)^1, е GJ(;(a)).
It is straightforward to check that λ = (λσ, λ(α)) defines a morphism G(3O ->
G(y') over /: У -^- У and that we get a diagram that commutes up to homotopy
GOO —^ G(y)
G —^-» G'
The family (ka) gives the homotopy from Αφ to 0'λ.
2.10 Examples
(1) We describe the complex of groups associated to the action considered in
example 1.14 (1) (maintaining the notation established there). As representatives for

Developability 541
О := τ, σ, ρ, choose respectively Ο, σ\, ρ\. Then b = Ъ\, Ь = b\, and hy is the
rotation R about О that sends σο = σ„ to σι. All the local groups are trivial except
Gx — (R), and the only non-trivial twisting element is ga,v = R-
(2) Let G(y) be the complex of groups associated to an action of a group G on
a connected scwol X with choices σ and ha as in (2.9(1)). For convenience, as in
(2.9(3)), we assume that the vertices σ belong to a tree Τ that projects to a maximal
tree Τ in the 1-skeleton of the geometric realization of У = G\X. We pass to the
universal covering/? : X —> A'ofA^,andasin(1.15)weconstructanactionofagroup
Gon X and a homomorphism Φ : G —> G so that /j is Φ-equivariant and the kernel
of Φ acts freely on X with quotient X. There is an isomorphism G\X —> G\# = У
induced by /j, and we identify G\X with G\X by means of this map. To construct a
complex of groups associated to this action of G on X, we can lift Г to a tree Τ in
the 1-skeleton of the geometric realization of X and for each σ e VO) define the
vertex σ e V(X) by: /?(σ) = σ and σ e f. For each α e £0) with i(a) = σ, let
α be the edge in X with i(a) = σ and p(a) — a. Then choose an element ha e G
such that Φ(Αα) = Αα and ha.t{a) — τ where τ = ί(α). Note that Φ maps the isotropy
subgroup Ga isomorphically onto the isotropy subgroup G^ — Ga.
If we identify these subgroups by this isomorphism, then the complex of groups
over У associated to this action of G on X, with respect to the choices that we have
made, is exactly G(y).
(3) We describe the complex of groups associated to the action considered in
example 1.14(2). Let С be a 2-cell of К which is the union of two adjacent triangles
in the barycentric subdivision of the tesselation with vertices O, A, B, A, where О is
the barycentre of an и-gon of the tesselation and В a vertex of this и-gon (see figure
С6). Let s (resp. r~~x) be the element of G fixing О (resp. B) and mapping A to A,
and let t be the half turn fixing A ■ As representatives for the vertices of У we choose
the barycentres of those faces of С that are contained in the triangle OAB. Then the
non-trivial local groups are Gx —Ъп, GTl = Z3, Gx< = Z2, generated respectively
by s, r, t. The non-trivial twisting elements are ga_bl = s, gd ν = t, gaub\ — r~l.
(See figure С12.)
2.11 Developability. A complex of groups G(3O is called developable if it is
isomorphic to a complex of groups that is associated, in the sense of 2.9(1), to an action
of a group G on a scwol X with У = G\X.
In general, complexes of groups are not developable if the dimension of У is
greater than one. Conditions for developability will be discussed in (2.15) and Chapter
т.д.
2.12 Remark. If a complex of groups GOO over a connected scwol У is developable,
then it is isomorphic to a complex of groups associated to an action of a group G on
a simply connected scwol.
Indeed, given an action whose associated complex of groups is isomorphic to
GO)· we can fifSt restrict this action to a connected component as indicated in

542 Chapter Ш.С Complexes of Groups
Z6
Fig. C.12 The complex of groups associated to the action of figure £6
(2.9(3)) and then pass to the universal covering of this component as in example
2.10(2).
The Basic Construction
2.13 Theorem (The Basic Construction). Let G(y) = (Ga, ψα, ga.b) be a complex
of groups over a scwol У.
(1) Let G be a group. Canonically associated to each morphism φ : G(y) —> G
there is an action ofG on a scwol D(y, φ) with quotient У. (D(y, φ) is called
the development of У with respect to φ.) If φ is injective on the local groups, then
G(y) is the complex of groups (with respect to canonical choices) associated
to this action and G(y) —> G is the associated morphism.
(2) If G(y) is the complex of groups associated to an action of a group G on a
scwol X {with respect to some choices) and if φ : G(y) —> G is the associated
morphism, then there is a G-equivariant isomorphism D(y, φ) —> X that
projects to the identity ofy.

The Basic Construction 543
Proof. For (1) we define the scwol D(y, φ) as follows:
V(D(y, φ)) = {(g0ff(Gff), σ)| σ e VQ>), g0ff(Gff) e G/0ff(Gff)}
E(D(y, ф)) = {(яфца)(.вт), α)\ α e EQ>), g e g<l>m{Gm) e G/0i(a)(G/(a))};
the maps i, t: E(D(y, φ)) -> V(D(y, φ)) are
i((g0i(a)(Gi(a))), й) = (g0i(a)(Gi(a)), ί(α)),
i((gfe)(Gi(a)), a)) = {ёФ{аТхф,(а){С,(а)), t(a));
and the composition is
(g<f>i(a)(Gna), a){ht>nb){Gnb)), b) = (hf>m(Gm), ab),
where (a, ft) e £(2)СУ), «ДеС and g0i(a)(G,(a)) = кф{ЬТ{фКа){Ст).
The map r is well-defined because if χ e Gi(a), then by (2.5)
Ф«а)(х)Ф(аТ1 - Φ(αΤιφ,(α)(ψα(χ)) = φ(α)~ι modulo 0,(a)(G,(a)).
It is straightforward to check that Λ' is a scwol. For instance to see that
K(Hnb)(GKb), ab)) = r((g0/(a)(G,-(a)), α)),
we need to check that Иф(аЬ)~1фца)(С,(а)) = вф(а)~1ф,(а)(С,(а)), provided that g =
hxp{b)~x фц„)(х) for some χ e G,-(a). Using 2.5 we get:
§ф(аТ1 =ЬффГ1фтОсЖа)~1 = кф(ЬГ1ф(аТ1ф(а)фКа)(х)ф(аГ1
= Ъф(аЬ)~хΦκα)(8ά}·ψα(χ)) = Ьф{аЬу1 mod 0,(a)(G,(a)).
The group G acts naturally on X: given g,h e G and α e У. define
A.(g0i(a)(Gi(a)),a!) := Ощфца)(Сца)), a). It is clear that У = G\^. Consider
the complex of groups G(3O associated to this action and the (natural) choices
σ = (σ, Ga)andha = 0(a)is(GCT, ifa,g~ab), where Ga = φσ{ΰσ), the injection i/ra is
the restriction of Αά(φ(α)) to 0,(a)(G,(a)), and ga>6 = ф(а)ф(Ь)ф(аЬу1 е 0,(a)(G,(a)).
If each 0σ is injective and we identify GCT with its image φσ{ΰσ), then G(3O = G(y).
(2)lf(gGKa),a) e £>СУ, 0), thence еУ = G\X. Using the notation of 2.9(1), let
a be the unique element (edge or vertex) of X such that p(a) = aandi'(c7) = i(a).Itis
straightforward to check that (gG,(a), а) м> g.a defines a G-equivariant isomorphism
D(y, φ) -> X. D
2.14 Remark. Given a complex of groups G(3O = (GCT, ψα, gab) and a morphism φ :
G(y) —> G to a group (where 0 is not necessarily injective on the local groups), one
can consider the complex of groups G(3O = (GCT, ψα, g~ab) (where Ga = φσ{ϋσ),
etc.) and the induced morphism φ = (φσ, φ(α)): G(3O —» G, where φσ : Ga —» G
is simply the inclusion and φ(α) = φ(α). It is clear that D(y, φ) = D(y, φ), and by
2.13(1) the complex of groups canonically associated to the action of G on D(y, φ)
is G(y).

544 Chapter III.C Complexes of Groups
2.15 Corollary. A complex of groups G(3O is developable if and only if there exists
a morphism φ from G(3O to some group G which is injective on the local groups.
2.16 Remark. Let У be the scwol associated toaposet Q (see 1.2(1)). LetG(3Obea
simple complex of groups and let φ : G(3O —> G be a morphism that is injective on
local groups and simple in the sense that φ(α) = 1 for all a e Е(У). Then D(y, φ)
is the scwol associated to the poset yielded by the basic construction of Π. 12.18.
As mentioned in remark 2.6, if У is simply connected then any morphism from
G(y) to a group G is homotopic to a simple one (see 3.10(3)). Therefore a simple
complex of groups over Q is strictly developable if and only if the corresponding
complex of groups over the associated scwol is developable. In particular, for sim-
plices or и-gons of groups the notions of developability and strict developability are
equivalent. See parts (5) and (6) of II. 12.17 for examples of complexes of groups
which are not developable.
2.17 Corollary (Bass-Serre). Any complex of groups G(y) over a scwol У of
dimension one is developable.
Moreover, if У is finite and all of the local groups are finite, there is an action of
a finite group on a finite scwol X (which is connected if У is connected) such that
the associated complex of groups is isomorphic to G(y).
Proof. Before beginning the proof properly, we remark that whenever one has two
free actions of a group Я on a set Ζ such that the sets of orbits have the same
cardinality, one can find a permutation of Ζ that conjugates one of the actions to the
other. To see this, we write G to denote the group of permutations of Ζ and write the
actions as homomorphisms φ, φ' : Η —> G. In each orbit of both actions we choose
a representative: let Zo с Ζ and Z0 с Ζ be the sets of chosen representatives. By
hypothesis, there is a bijection^ : Zo —> Z0. As the actions are free, for each ζ e Ζ
there is a unique го e Zo and a unique ho e Η such that ζ = 0(йо)(го)· Thus we may
define a map/ : Ζ -> Ζ by sending ζ to 0'(^o)(/b(zo))· This map is a bijection, and
by construction/ф(Н) = ф'{Щ for each h e H.
We now turn to the proof of the corollary. Consider first the case where all the local
groups Ga have a bounded order. Let Ζ be a finite set whose cardinality is divisible
by the cardinality of each GCT, and let G be the group of permutations of Z. For each
σ e У(У) we can construct an injective homomorphism $σ : Ga —> G whose image
acts freely on Ζ by choosing a partition of Ζ into subsets whose cardinality is the
order of Ga and taking an arbitrary free action of Ga on each subset.
For each a e Е(У), the images of G,-(a) under the injective homomorphisms фщ
and φ,(α)ψα act freely on Ζ and in each case the action has |Z|/|G,(a)| orbits. Therefore
our opening remark yields an element φ(α) e G such that φ,^ψα = Ad($(a))$i(a).In
this way we obtain a morphism G(3O —> G, denoted φ = (φσ, φ(α)), that is injective
on the local groups. If У is finite, then the development D(y, φ) is a finite scwol
on which G acts and the associated complex of groups is G(3O· If У is connected,
we can consider a connected component of D(y, φ) as in 2.9(3) to prove the last
assertion of the corollary.

The Basic Construction 545
In the general case, we let
Z= J] Ga,
and we again take G to be the group of permutations of Z. We let Ga act by left
translations on the factor Ga and trivially on the other factors. For each a e Е(У) the
homomorphisms фщ and φ^ψα are conjugate via a permutation φ(α) of Ζ because
the cardinality of Gi(a) is equal to the cardinality of G,(a) χ Gi(a)/^(Сад)· Thus we
may conclude as above. D
2.18 The Functorial Properties of the Basic Construction. Let G(y) = (Ga, ψα,
ga:b) and G(y') = (Ga<, ψα', ёа'м) be complexes of groups over scwols У and y.
Let φ : G(y) -> G and φ' : GQ/) -> G' be morphisms and let A : G -> G' be a
group homomorphism.
(1) Let λ : G(y) -> G(y') be a morphism over I : У -> У'. Functorially
associated to each homotopy from Αφ to φ'λ there is a A-equivariant morphism
L : D(y, φ) -> D(y\ φ') that projects tol:y~> У'.
(2) If λ is an isomorphism and φ, φ' are injective on the local groups, then кегЛ
acts simply transitively on the fibres ofL, and L is surjective if and only if A is
surjective.
(3) If φ and φ' are injective on the local groups then every A-equivariant morphism
L : D(y, φ) -> D(y\ φ') arises as in (1).
Proof. (1) A homotopy from Λ0 to 0'λ is determined by a family of elements^ e G'
(notation of 2.4). It is straightforward to check that the map
is well-defined and that it is a Л-equivariant morphism L : D(y, φ) -> D(y', φ')
which projects to /.
(2) If λ is an isomorphism and if φ and φ' are injective on local groups, then ker Λ
acts freely on D(y, φ), because for each σ e VOO the equality φ'λσ = (Аака)Афа
implies that ker Λ Π φσ (Ga) = 1. Let us check that ker Λ acts transitively on the fibres
of L. If two vertices of D(y, φ) are mapped by L to the same vertex, then they are
of the form (gi<pa(Ga), σ) and ^a{Ga), σ) with A{g{)k~{ = A(g2)k~l modulo
Φ\,σ·Χβΐ(σ)). As λσ is surjective, there exists χ e Ga such that KAig^gi)^1 =
Φ'Κσ)(Κ(χ))- This implies that Aig^gi) = Α(φσ(χ)), hence h := g2φσ(x)gϊl €
кегЛ, and therefore h.(g^a(Ga), σ) = (g20CT(GCT), σ).
(3) This was already observed in 2.9(4). D
The case where Λ is the identity is already interesting:
2.19 Corollary. Let φ,φ' : G(y) —> G be two morphisms which are injective on
the local groups, φ and φ' are homotopic if and only if there is a G-equivariant
isomorphism D(y, φ) —> D(y, φ') projecting to the identity on У.

546 Chapter Ш.С Complexes of Groups
2.20 Exercise (The Mapping Cylinder). Let φ = (φσ,φ{α)) : G(y0) -> G(yO be a
morphism of complexes of groups over a non-degenerate morphism / : Уо —> ЗА
which is injective on the local groups.
(1) Define the mapping cylinder of/. This should be a scwol M/ that contains
the disjoint union of Уо and У χ as subscwols and has extra edges ua indexed by
the elements a e Уо, with i(ua) = i(a) and t(ua) — f(t{a)). For each pair of
composable elements α, β e Уо, one should have compositions defined by the
formulas 4αβ = ua$ =/(α)ιΐβ. Show thatA^ is indeed a scwol.
(2) Construct the mapping cylinder G(Mf) of φ. This should be a complex of
groups over Mj whose restriction to У ι is G(y{), for i = 0, 1. The homomorphisms
associated to the new edges should satisfy \j/Ua = φ,(α)ψα. Complete these data to
obtain the desired complex of groups G(Mf).
(3) Given σ0 e У(Уо), show that the homomorphism
*ι(σθ>ι),/(σ0)) -> 7r1(G(M/),/(a0))
induced by the natural inclusion G(yO —> G(M/) (see 3.6) is an isomorphism.
3. The Fundamental Group of a Complex of Groups
In this section we define the fundamental group of a complex of groups G(y) and we
give a presentation of this group in terms of presentations for the local groups and a
choice of a maximal tree in У. For a developable complex of groups, we construct its
universal covering explicitly. Our treatment is a natural generalization of the theory
of graph of groups due to Bass and Serre [Ser77]. Before reading this section, the
reader may wish to look at paragraphs A. 10 to A. 13 in the Appendix, where the
corresponding notions for small categories are described.
The Universal Group FGQO
3.1 Definition. Let G(3O = (GCT, ψα, gafi) be a complex of groups over the scwol y.
The group FG(y) is the group given by the following presentation. It is generated
by the set
Ц Ga U &<У)
a<=V(,y)
subject to the relations
the relations in the groups GCT,
(α+Γι =a~ and (α-)-1 =a+,
a+b+ = ga,b(ab)+, V (a, b) e £(2)СУ)
. fa(g) = a+ga~, VgeGi(a)

3. The Fundamental Group of a Complex of Groups 547
There is a natural morphism
t = (ίσ, ι(α)) : С(У) -> FGQ>),
where ίσ : Ga —> FG(y) is the natural homomorphism mapping g € G„ to the
corresponding generator g of FG(y) and ί(α) = α+. (The relations defining FG(y)
are the least set that one must impose in order to make i a morphism.)
If one regards a group G as a complex of groups over a single vertex, then FG is
canonically isomorphic to G.
3.2 The Universal Property and Functoriality of F
Given a morphism φ : G(y) —> G, there is a unique group homomorphism
Ρφ : FG(y) -> G such that (Ρφ) οΐ=φ.
More generally, if φ : G(y) —> G(y') is a morphism of complexes of groups
over a morphism f : У —> У, then associated to φ there is a homomorphism
Ρφ : FG(y) —> FG(y') such that the following diagram commutes:
G(y) —ί-s- FG(y)
*l рф[
С(У) —^ FGQT)
Proof. Let φ = (φσ,φ(α)) : GQO -* GCV')- Then F0 maps the generator g e GCT
to $CT(g) and the generator a+ to 0(α)/(α)+. It is straightforward to verify that F$ is
well defined. D
3.3 G(3O-Paths. Let У be a scwol and let G(y) = (GCT, ψα, gab) be a complex of
groups over У as defined in (2.1). Given σ0 e V(3O, a G(y)-path issuing from σο is
a sequence с = (go, ei. gi, ■ ■ ■, e*. g*) where (ei,..., e*) is an edge path in У that
issues from σο (in the sense of 1.6), go e Gao, and g, e Gi((?() for 1 < i < k. The
vertex σο is called the initial vertex i(c) of с and σ = i(e^) is called the terminal
vertex t(c) of c; we say that с joins σο to σ. A G(3O-path joining σο to σο is called a
G(yyioop at σο.
If с' = (g0, e',, g',,..., e'k,, g'k.) is a G(y)-path issuing from the terminal vertex
of c, then the composition or concatenation of с and d is the G(3O-path
с * с' = (go, ei, gi,..., ek, gkg'0, e\, g[,..., e'k, g'k,).
This partially defined law of composition is associative: if с, с', с" are G(y)-paths
such that c*c' and с' * с" are defined, then (с * с') * с" = с * (с' * с"); this G(3O-path
is denoted с * c' * c". The inverse c"1 of с is the G(3O-path (g0, e\, g\, ..., e'k, g'k)
where g,' = g-J, and ei = β;,1 ,.

548 Chapter Ш.С Complexes of Groups
3.4 Homotopy of G(3O-Paths. To each G(3O-path с = (go, e\, g\,..., eb gk) we
associate the element n(c) of FG(y) represented by the word goe\gi .. .e^gu. By
definition, two G(3O-paths с and c' joining σ to τ are homotopic if n(c) = η (с').
The homotopy class of с is denoted [c] and the set of homotopy classes of G(3O-paths
joining σο to σ is denoted n\(G(y), σ0, σ).
Note that the homotopy class of the composition of two G(3O-paths depends only
on the homotopy class of those paths.
The Fundamental Group ni(G(y), <xo)
3.5 Definition. The operation [c] [c'] : = [с * с'] defines a group structure on the set of
homotopy classes of G(3O~loops at σο. The resulting group is denoted π\ (G(y), σο)
and is called the, fundamental group of G(3O at σο.
Note that, by definition, the map π identifies the group π\ (G(y), σο) to π\ (G(3O,
σ0, σο), which is a subgroup of FGQ?). If с is a G(3O-path joining σ0 to σ, then the
map associating to each G(3O-loop I at σ0 the G(3O-loop (c~~l * I * c) at σ induces
an isomorphism 7Ti(G(3O, σο) —> Tt\(G(y), σ). If we identify these groups to the
corresponding subgroups of FG(y), then this isomorphism is the restriction of the
inner automorphism Ad(7r(c)).
If G(y) is the trivial complex of groups over У (i.e. all of the Ga are trivial),
then 7Ti(G(3O, σ0) = п\(У, σ0). More generally, the map associating to each G(3O-
loop с = (go, e\, g\,..., еь gk) at σ0 the edge loop (e\,... ,ek) induces a surjective
homomorphism Tt\(G(y), σ0) -> Я\(У, σ0).
Any morphism φ = (φσ, φ(α)) from G(y) to a group G induces a homomorphism
πι(φ,σ0) : τγι (G(3O, σ0) -»■ G mapping the homotopy class of a G(3O-loop с to the
element (F0)(7r(c)) e G.
More generally, we have:
3.6 Proposition. Every morphism φ = (φσ, φ(α)) of complexes of groups G(y) —>
G(y') over a morphism f : У —> У' induces a natural homomorphism
πχ(φ,σ0) : *ι«ΪΟ>)>σο) -> *ι(<ΪΟ>'),/(σο)),
namely the restriction ofFφ : FG(y) —» FG(y).
Proof. We only need to check that F$ maps the subset Tt\(G(y), σο, σ) с FG(y) to
the subset jri(GQ>'),/(σ0),/(σ)) с FGCV').
For each e e ДЗ·7)*, we shall define a G(y')-Patn denoted ф(е) such that
0(e-1) = ф(е)~1. Assume e = a+. If/(a) e £(У), then ф(е) is the GCV')-path
(ф(а),Да)+, 1/(/(α))); if/(α) is a unit 1σ/, then 0(e) is the G(y')-path (1σ/). For each
element g e G„, we define </>(,?) to be the G(y)-path (0σ(#)). For each G(3O-path
с = (go, e\, ..., gk), we define ф(с) to be the concatenation of the G(y)-paths
<P(go) * Ф(е\) * ·■ ■ * Ф^к)- It is clear that F$ maps the homotopy class of с to the
homotopy class of F0. Π

3. The Fundamental Group of a Complex of Groups 549
A Presentation of πι (GQ>)5 σο)
Consider the graph which is the 1-skeleton |ЗЛ(1) of the geometric realization of У:
its set of vertices is V(y) and its set of 1-cells is Е(У). We assume that У is connected
(equivalently, that this graph is connected). Let Г be a maximal tree in this graph,
i.e. a subgraph which is a tree containing all the vertices. Such a maximal tree is
not unique in general: any subgraph Τ of \У |(1) which is a tree can be extended to a
maximal tree.
3.7 Theorem. The fundamental group ni(G(y),ao) is isomorphic to the abstract
group Tt\{G{y), T) generated by the set
Ц Ga Ц Е*(У)
subject to the relations
the relations in the groups Ga,
(a+)~~l = a" and (a~~)~~l = a+,
a+b+ = ga.b{ab)+, V (a, b) e £(2)СУ)
faig) = a+ga~, Wg e Gm,
a+ = 1, Vae T.
Proof. Identify щ(аУ),а0) with ni(G(y),a0,a0) and let Φ : ni(G(y),a0) -±
n\(G(y), T) be the homomorphism obtained by restricting the natural projection
FG(y) -> ni(G(y), T). For each vertex σ e VQO, let cc = (eb ..., ek) be the
unique edge path in У joining σο to σ such that no two consecutive edges are inverse
to each other and each e, is contained in Τ (i.e. e, = af, with a, e T). Let 7rCT be
the corresponding element e\... ek £π\ (G(y), σ0,σ) с FGQ?). Note that if a e Г,
then either тг1(а) = 7Γί(α)α+ or 7Γί(α) = Яца)й~. Note also that the image of πσ in
ttiCGQO, Γ) is trivial. Let Θ : п^СКУ), Т) -> п^СКУ), σ0) be the homomorphism
mapping the generator g e GCT to itag-K~x and the generator a+ to the element
π,(α)α+πΖ\. This homomorphism is well-defined because the relations are satisfied;
in particular Θ(α+) = 1 if a e T.
The elements of the form nagn~l, where g e Ga, together with those of the
form 7Г,(а)а+7Г(до, where a e £(^), generate 7Ti(G(y), σο). Their images under Φ
are, respectively, g and a+. This shows that Φ and Θ are inverse to each other. D
The fundamental groups of the complexes of groups of dimension one shown in
figure C.9 are, respectively, the amalgamated product GT| *g„ Gr,, an fflVTV-extension
Gr*G„, and the amalgamation of the groups GT. along their isomorphic subgroup
TJr(Ga). The morphism of complexes of groups described in figure CIO induces an
isomorphism on the fundamental groups.

550 Chapter III.C Complexes of Groups
3.8 Corollary. If all of the groups Ga are finitely generated (resp. finitely presented)
and if У is finite, then it\{G(y), σ) is finitely generated (resp. finitely presented).
The criterion of developability given in 2.15 can be reformulated as follows.
3.9 Proposition. A complex of groups G(y) over a connected scwol У is developable
if and only if each of the natural homomorphisms Ga —> it ι (G(y), T) (equivalently
Ga —» FG(y)) is injective.
Proof. According to 2.15, G(y) is developable if and only if there is a morphism φ
from G(y) to a group G which is injective on the local groups Ga. The universality of
FG(y) (3.2) shows that such a morphism exists if and only if the canonical morphism
i : G(y) -> FG(y) is injective on the local groups. Moreover the quotient map
FG(y) -> iti(G(y), T) is injective on the subgroups iG(Ga). Indeed, by choosing
σο = σ in the proof of 3.7, we see that the map θ ο φ restricted to ia{Ga) is the
identity. D
3.10 Proposition. Let G(y) be a complex of groups over a connected scwol У, let
Τ be a maximal tree in \У |(1) and let G be a group.
(1) Given a morphism φ = (φσ, φ(α)) : G(y) —» G, define
ni(4>,T):ni(G(y),T)^G
by composing the natural isomorphism θ : iTi(G(y), T) —> it\{G(y), σο) with
the homomorphism π\(φ,σο) '. iT\(G(y), σο) —*■ G defined in (3.6). If φ is such
that φ(α) — 1 for each a e T, then π\(φ,Τ) maps each generator g e Ga to
0CT(g) and each generator a+ to ф(а).
(2) Every morphism φ' : G(y) —> G is homotopic to a morphism φ : G(Y) —> G
such that φ(α) = 1 for each a e T. Moreover φ is unique up to conjugation by
an element ofG. Therefore the map associating to a morphism G(y) —» G the
induced homomorphism it\{G(y), σο) —> G gives a bijection
H\G(y\ G) -> H\itx{G(y\ σο), G),
where Hl (G(y), G) is the set ofhomotopy classes ofmorphisms ofG(y) to G,
and Hl(it\(G(y), σ0), G) is the set of homomorphisms from it\{G(y), σο) to G
up to conjugation by an element ofG.
(3) IfG(Y) is a simple complex of groups over a simply connected scwol y, then
every morphism G(y) —> G is homotopic to a simple morphism.
Proof. We use the notations established in the proof of 3.7. Part (1) follows
immediately from the observation that πι(φ,σ0) maps each πσ to 1 e G.
To prove (2), consider the image ka of πσ under π\{φ', σ0). Let φ = (φσ, φ(α)) be
the morphism, homotopic to φ', defined by φσ ~ ка(ка)ф'а and ф(а) = кца)ф'(фкщ.
For each a e T, we have кца) = кца)Ф'(а) or k,^ = кца)Ф'(а)~~1, therefore ф(а) = 1.

3. The Fundamental Group of a Complex of Groups 551
It remains to show that φ is unique. Let φ = (φσ, φ(ά)) : G(y) —> G be another
morphism such that φ(α) = 1 for a e T. If (ka)aSv(y) is a homotopy between φ and
φ, the relation ф(а) = кца)ф(а)к^\ applied to the edges a e Τ implies that ka is an
element g e G independent of σ, therefore φ is the conjugate of φ by g.
To prove (3), we first observe that if φ' : G(y) —> G is a morphism, then
α ь» $'(α) gives a morphism from the trivial complex of groups over У to G. By
(2), since У is simply connected, there is a family (ka) of elements of G such that
кг(.а)Ф'(а)к^1 = 1 for each α e Е(У). Therefore (ka) defines a homotopy between φ'
and a simple morphism φ : G(y) —> G. □
3.11 Examples
(1) Assume that G(y) is a simple complex of groups (so all the elements gab
are trivial). Then the subgroup of iti(G(y), T) generated by {a+ ] a e Е(У)} is
isomorphic to the fundamental group of y. It follows that if У is simply connected,
then nx(G(y), σ0) = Tt\{G{y), T) is the direct limit (in the category of groups) of
the family of subgroups Ga and inclusions ψα. Indeed all the generators of the form
a+, a e Е(У), represent the trivial element of n\(G(y), T).
In particular if G(Q) is a simple complex of groups over a poset Q, and the
geometric realization of Q is simply connected, then the group Нтст Ga (denoted
G( Q) in II. 12.12) is the fundamental group of the corresponding complex of groups.
Itfollows that G(Q) is strictly developable (in the sense of II.12.15) if and only if it
is developable.
(2) Let φ = (φσ, φ(α)) : G(3O -> G{y') be a morphism of complexes of groups
over/ : У -> У and suppose that one can find maximal trees Τ in ]У](1) and Τ in
\y\w such that f(T) с Г and ф(а) = 1 for each a e T. Then the homomorphism
induced by φ on the fundamental groups ni(G(y), T) -» 7ii(G(y'), V) (via the
isomorphisms of 3.7) maps the generators g e Ga (resp. a+) to the generators $CT(g)
(resp. φ(α)+).
(3) A presentation of the fundamental group of the complex of groups G(y) in
example 2.10(3).
If we choose the maximal tree Τ in ]У|(1) to be the union of the edges
a, a', a\, a\, b\, then we have 1 = a = d = αϊ = a\ = b\. As ga у = gay —
8a' У = Sa У = 1' we see successively that 1 = c\ = C\ = b\ =c\.
Besides the relations s" = r3 = t1 = 1, we are left with the four relations
gaMct = a+b\, gdMc\+ = d+b\, g^b,A = a\+b\ + , ga,,,;c+ = a\b\ +.
As ga,b, = ■*. go, = L ^',.&', = ?' &»,.&', = r~' and 1 = ci = α = α' = αϊ =
a', = ci, these relations are equivalent to
s = b+, c\+ = b+, tc\+ = b[ + , r~x = b\ + .
After eliminating b\', c'j+ and ί»Ί+, we get only one relation rst = 1. Eventually we
get the presentation
Ki(G(y), T) = (t, s, r\f = r3 =s" = rst = 1).

552 Chapter Ш.С Complexes of Groups
This is a presentation that we would have obtained directly by applying Poincare's
theorem (cf.[Rh71]) to the quadrilateral С = OABA (see figure C.6).
More generally, consider the scwol У associated to the polyhedral complex
obtained by identifying the sides OA, OA and the sides BA, BA of the quadrilateral С
(in the notation of figure C.6). Consider a complex of groups G(3O such that the
local groups associated to the vertices ρ,σ,σ\ are trivial. After modification by a
coboundary, we can assume that the only twisting elements that might be non-trivial
are gafc, e Gr, gj ib' e GT> and gaub> e Gri. The same calculation as above shows
that n\{G(y), T) is isomorphic to the quotient of the free product Gr * Gr/ * Gr, by
the normal subgroup generated by gaM ga, _&, g~|&,.
(4) On the scwol У obtained by gluing two congruent и-gons along their boundary
(see 1.4(3)), we consider a complex of groups G(3O such that the only non-trivial
local groups are the groups G& = Gn. After modification by a coboundary we may
assume that the only non-trivial twisting elements are the η elements gatbt = Sk e G*.
We leave the reader to check that, as above, the fundamental group of G(3O is the
quotient of the free product of the groups G* by the normal subgroup generated by
the element i! .. .sn.
(5) Seifert-van Kampen Theorem. Let G(3O be a complex of groups over a scwol
У. Assume that У is the union of two connected subscwols У\ and y2 such that the
intersection У о = У ι Π У2 is connected. Let σο e У(Уо) be a base vertex. Let G(yL)
be the complex of groups which is the restriction of G(y) to the subscwol yit for
i — 0, 1, 2. Then Tt\(G(y), σο) is the quotient of the free product Tt\(G{y{), σ0) *
jti (G{y2), σο) by the normal subgroup generated by the elements j\ (y )J2(y)~' for all
γ e 7ti(G(y0), σ0), where j,- : Tt\{G{y0), σ0) -> ^(GO^o) is the homomorphism
induced by the inclusion of G(3^o) into G{y{), for i = 1,2.
To see this, choose a maximal tree T0 in |Уо1(1) and extend it to maximal trees T\
and T2 in |^i |(1) and \y2\w. Then Γ = Τχ U Γ2 is a maximal tree in |У|(1). It is clear
from the presentation in (3.7) that π\ (G(y), T) is the quotient of the free product of
K\(G(y{), T\) * Tt\{G(y2), Тг) by the normal subgroup generated by the elements
МуШуТ1 fora11 У e ti(GCVo), To), wherej; : jt,(GCM>), 7b) -> 7r,(G(y,), T{) is
the homomorphism induced by the inclusion of G(yo) to G{yi), for г = 1,2. And
we can identify n^G^i), Tt) with iti{G{y{), σ0) (hence ft toy';) as in (3.7).
3.12 Exercise (HNN-Extensions). Let G(3^ι) be a complex of groups over a connected
scwol У\ and let G(3^o) be its restriction to a connected subscwol Уо. We assume that
if a e Е(У{) has its terminal vertex in y0 then a e Е(У0). Let φ : G(y0) -+ GCVi)
be a morphism over a non-degenerate morphism/ : y0 —» 3^1 and assume that φ is
injective on the local groups.
(1) Following (2.20), construct a new scwol У containing Ух as a subscwol,
with the same set of vertices and new edges ua indexed by the elements a e Уо.
(Construct У from the mapping cylinder M/ of/ by identifying in M/ the subscwol
Уо Я: У\ Я: Mf to the subscwol У о Я Mf.)
(2) Construct a complex of groups G(3O whose restriction to У ι is G(3^i), the
homomorphisms associated to the new edges ua being defined by ψ^ = φ,(α)Ψα-

The Universal Covering of a Developable Complex of Groups 553
(3) Choose σ0 e V(y0) and an edge path с in У\ joining σ0 ιο/(σ0). Let Φ0 :
Jii(G(yo),cro) —> Tt\(G(yi), σο) be the homomorphism induced by the inclusion
Уо -> 3Ί and let Φι : ^(GO^-ob) -> π\(βΟ>\),σώ be the homomorphism
sending the homotopy class of each G(yo)-loop I based at σο to the homotopy class
of the GQO-loop (c * 0(0 * c~~l). Show that 7Ti(G(3O, σ0) is isomorphic to the
quotient of the free product iti(G(yi),a0) * Ζ by the relations Фо(у) = ίΦ\(γ)ί~ι,
for all γ e 7Ti(G(3O. σο)> where r is the generator of Z. (If Φο is an injection, this
group is an HNN-extension.)
The Universal Covering of a Developable Complex of Groups
Let G(y) be a developable complex of groups over a connected scwol У. Choose a
maximal tree Τ in the 1-skeleton of the geometric realization of У. Let
iT-G(y)^nl(G(y\T)
be the morphism mapping each element of the local group Ga to the corresponding
generator of Tt\{G(y), T) and each edge a to the generator a+. The developments
associated to morphisms to groups were defined in (2.13).
3.13 Theorem. The development D(y, ΐτ) is connected and simply connected.
Proof. We identify the groups Ga to their images in Tt\{G{y), Τ). Thus the elements
of D(y, it) are pairs (gGi(a), a) where a e У and g e n\{G{y), T). We prove first
that X := D(y, ij) is connected.
Foreaclm e V(y)leta"bethevertex(G(J^)ofD(K, ΐτ), and for each a e Е(У)
let a be the edge (G,(a), a). Let W be the subset of the geometric realization of D(y, φ)
consisting of the images of the edges {a \ a e Е(У)}. The images of the a with a e Τ
form a tree Fcff that projects to T. Since i(a) = i(a) for every a e Е(У), the tree
Г contains all of the vertices σ and hence W is connected. To prove that D(y, It)
is connected, it is sufficient to prove that, for each generator s of 7Ti(G(3O, T), the
intersection W Π s. Wis non-empty. But this is clear for both s = g e Ga ands = a*,
because this intersection contains σ and either t(a) or i(a).
If D{y, It) were not simply connected, we could consider as in 2.10 (2)) its
universal covering ρ : X —» X = D(y, It), which is equivariant with respect to the
homomorphism Φ : G -► G := nx(G(y), T) (in the notation of 2.10 (2)). Using the
choices made in 2.10(2), the complex of groups associated to the action of G on X
can be identified to G(y); the corresponding homomorphism φ : G(y) —> G is such
that φ(α) = 1 when a eT. Therefore the map associating to each generator g e Ga
οίτΐχ(G(y), T) the element 4>(g), and to each generator a+ the element φ(α), extends
to a homomorphism Φ : G —> G. Moreover Φ ο Φ is the identity of G. The map

554 Chapter Ш.С Complexes of Groups
ρ : X -> X sending (gC^a) to Φ(#).(α) is a morphism and ρ ο ρ is the identity of
X. AsX is connected,pis a bijection andXis simply connected. D
Note that if 0(У) is a graph of groups, then D(y, iT) is called the associated
Bass-Serre tree (see 11.18 and 11.21).
The following is a generalization of II. 12.20.
3.14 Properties of the Development D{y, φ). Let У be a connected scwol and
choose a base vertex σο € У (У)- Let Φ '■ G(y) —> G be a morphism which is
injective on the local groups and let D(y, φ) be the corresponding development on
which the group G acts. Let τΐ\{φ, σο) : n\{G{y), σ0) —> G be the homomorphism
induced by φ.
(1) The connected components ofD(y, φ) correspond bijectively to the elements
of the cosets in G of the image ofn^, σο)).
(2) IfD(y, φ) is connected (equivalently, π\ (φ, σο) is surjective), the fundamental
group of D(y, φ) is the kernel of π\{φ, σο). In particular D(Y, φ) is simply
connected if and only ifл\{ф, σο) is an isomorphism.
Proof. Let Γ be a maximal tree in the 1-skeleton of the geometric realization of
У. As D(y, φ) depends up to isomorphism only on the homotopy class of φ (see
2.19), we may assume that φ(α) = 1 for every a e T, and hence replace π\(φ, σο)
by the homomorphism Λ = ηι(φ, Τ) : ni(g(y), Τ) —» G that sends the generator
g e Ga to $σ(,?) and the generator a+ to φ(α). Let Go с G be the image of this
homomorphism. We have Air = φ, and φ is equal to the composition of a morphism
0o ·' G(y) -> G0 with the inclusion G0 -> G.
According to 2.18, the map L : D(y, tT) -+ D(y, fo) defined by
(yGjia), а) ь» (A(y)0,w(G,(a)), α),
where γ e K\{G(y),T), is a Galois covering in the sense of 1.13 with Galois
group the kernel of Л. In particular L is surjective, D(y, фо) is connected and the
fundamental group of D(y, φ) is isomorphic to the kernel of Λ. It is easy to see that
D(y, фо) is a connected component of D(y, φ) and the connected components of
D(y, φ) are in bijection with the elements of G/G0. D
3.15 Corollary. Let G be a group acting on a simply connected scwol X with
quotient У = G\X and let G(y) be the complex of groups associated to this action (with
respect to some choices). Let Τ be a maximal tree in the \-skeleton of the
geometric realization of У. Then G is isomorphic to πλ (G(y), T) and X is equivariantly
isomorphic to D(y, iT).
3.16 Example. This corollary can be used to give a presentation of such a group
G (compare with II. 12.22). For instance, one sees that the subgroup of the group of
orientation preserving isometries of S2, E2 or И2 leaving invariant a tesselation by
regular и-gons with vertex angle 2π·/3 is generated by three elements r, s, t subject
to the relations s" = r3 = t1 = rst = 1 (cf. 3.11(3)).

The Local Structure of the Geometric Realization 555
4. Local Developments of a Complex of Groups
In this section we show that complexes of groups are always developable locally.
(Thus non-developability is a global phenomenon.) We construct local developments
at the vertices (4.10, 4.21) and edges (4.13) of the underlying scwol and show that
in the developable case these are faithful models for the local structure of the
development (4.11). We shall also explain how the different developments are related to
each other (4.14 and 4.22).
For the duration of this section we fix a scwol У.
The Local Structure of the Geometric Realization
Recall that the geometric realization \У | of a scwol У is the complex obtained as the
quotient of the disjoint union of simplices
U Δ"χ ш
/i>0, А<=Е<"ЧУ)
by the equivalence relation generated by the identifications
(ф(х), А) ~ (x, д:(А))
where (x, A) e Δ""1 χ Е(п)(У\ and the maps d, : Δ""1 -> Δ" and 9, : Е^"\У) ->
Е("~1)(У) are defined as in (1.3).
4.1 Definition of st(o-). Given σ e V(3O, we define the open star st(o-) of σ in \У\
to be the union of the interiors of the simplices containing σ. (This is an open subset
ofp>|.)
Given σ € У(У) and ρ, q > 0, let Ε^·4\σ) с Е^^СУ) be the subset consisting
of sequences of composable edges
(zi, · ·. ,Zp, cu ■ ■ ■ ,ся)
such that i{zp) = σ if ρ > 0, t{c\) = σ if q > 0; if ρ = q = 0, then by definition
£(0,0)^) _ |σ| т^е eiements 0f £<Ρ·9)(σ) label the simplices of \У\ containing σ.
For 0 < i < ρ (resp. ρ <i <p+q), the map 9, : Е^+ч){а) -+ £<Ρ+«_,)(σ) restricts
to a map 9,- : &<ι\σ) -+ Ε<Ρ-]^(σ) (resp. Ε^\σ) -> £<Ρ·«_,)(σ)).
4.2 Definition of St(a). Given σ e У(У) we define St(a) to be the complex obtained
as the quotient of the disjoint union
У Ap+" χ {A},
p,q.A
with ρ > 0, q > 0, A e E^-q\a), by the equivalence relation generated by the
identifications
(di(x),A)^(x,d,(A)),

556 Chapter Ш.С Complexes of Groups
where (x,A) e Ар+ч~] χ Ε^ι\σ) and i φ p. The vertex Δ° χ {σ}) will also be
denoted σ. There is a natural affine map St(a) —> |У I which induces a bijection from
the set of simplices containing σ e St(a) to the set of simplices containing σ e 1УI
and which restricts to an affine isomorphism from the open star of σ e St(a) onto
the open star of σ e \У\. We identify these two open sets st(a) by this isomorphism.
4.3 Remark. The projection St(o-) -» \У\ is not injective if there are at least two
distinct edges with initial vertex σ (resp. with terminal vertex σ) and the same
terminal vertex (resp. the same initial vertex).
We leave the reader the exercise of proving that St(a) is the geometric realization
of these woiy(^) defined in (1.17).
4.4 Definition of st(a). Given a e Е(У), we define the open star st(a) of a to be the
union of the interiors of those simplices in \У\ that contain the 1-simplex labelled a
(which we also denote a).
Let £<Ρ·*·«>(α) С £(р+*+<й(;у) be the subset consisting of sequences
{z\,...,Zp,ab ... ,ak,c\,.. .,cq)
such that a] ... ak = a. The elements of E&'k-i\a) label the simplices of \У\
containing α.
4.5 Definition of St(a). Given a e Е(У), to construct the complex St(a) we start
with the disjoint union of Euclidean simplices
У AP+k+i χ {A},
p,k,q,A
where ρ > 0, к > \,q > 0 and Л е E^-^ia), and form the quotient by the
equivalence relation generated by the identifications (rf,-(jc), A) ~ (jc, 3,(A)), where
(x, A) e Al'+k+i"' χ E^-^ia) &ηάίφρ,ρ + к. (We write a to denote the 1-simplex
of St(a) labelled (a) e £{0J-0)(a).)
There is a natural map St(a) —> |У | whose restriction to the open star in St(a) of
the 1-simplex a is an affine isomorphism onto st(a). As above, we identify these two
sets by means of this isomorphism.
4.6 Exercise. Show that for all σ, τ e V(y), the intersection st(a) Π st(r) is the
disjoint union of the connected open sets st(a), where a is any edge such that its
initial vertex is σ and its terminal vertex is τ or vice versa.
Let a, a' be two distinct edges of У such that t(a) = t(a'). Show that st(a)Dst(a') φ
0 if and only if there exists b e Е(У) such that either a = a'b or a' = ab. In the
latter case st(a) Π st(a') is contained in the disjoint union \\b st(b), where b e Е(У)
is such that a' = ab.
Consider the case where i(a) = i(a') and st(a) Π st(a') φ 0.

The Geometric Realization of the Local Development 557
The Geometric Realization of the Local Development
Let G(y) = (GCT, ψα, gatb) be a complex of groups over У. For each σ € У(У) we
shall construct a complex St(<?) with an action of the group Ga such that St(a) is the
quotient of St(a) by the action of Ga. (The construction is motivated by Proposition
4.11.) First we need another definition.
4.7 Definition of E^-^Ca). The elements of £^,9)(σ) are sequences
{girc{Gi(a)), Z\,...,Zp,C\,..., Cq),
where (zi,..., zp, c\, ■.., cq) = A e Ε^·^{σ), с = с\,... cq and g^j/c{Gi(c)) e
Ga/^c{Gi(C)). If q = 0, by convention VcCGfc)) = GCT, so £^'0)(σ) is in bijection
with £^0)(a)·
We define
9,- : EfP'4\a) -^E^'u\a) for 0 < i < ρ and
9, : ΕΜ{σ) ^E^q~x\a) ίοτρ < i < ρ + q
bysending(gVfc(G,{c)),A)to(gVfc(G;(C)), 3,-A)ifi < p+qori =p+qand0 < q < 1,
and if i = ρ + q, q > 2, then by definition
dp+q(gYc(G<(c)), Zl,---,Zp,C],...,Cq) =
(g8t!c,Yc(Gi(c')), Z\,-..,Zp,C\,..., Cfy_i),
where c' = C\ .. -cq~.\.
The group Ga acts naturally on EP'q)(S) by left translation on the first component
and the maps Э, are GCT-equivariant. The quotient by this action is Eij,'<!\a).
4.8 Lemma. Ifi and] satisfy 0 < i < j < ρ or ρ < i < j < ρ + q, then
Щ = dj+]di.
Proof. Only the case j = p + q and / = ρ + q — 1 is not obvious. In that case, one
must apply condition 2.1 (ii). D
4.9 The Construction of St(<r) and st(tr). We start from the disjoint union of
Euclidean simplices
У Δ'+* χ {Α},
ρ,η,Α
where ρ >0,q>0 and A e ^^(σ). On this we consider the equivalence relation
generated by
(</;(*), А) ~(*,Э;(А)),
where (x, A) e Δ^"1 χ Ε^·4\σ) and 0 < г < ρ от ρ < ί < ρ + q.

558 Chapter III.C Complexes of Groups
The group Ga acts naturally on this disjoint union by the identity on the first factor
and as in (4.7) on the second factor. This action is compatible with the equivalence
relation.
In the light of Lemma 4.8, we know that the quotient by this equivalence relation
is a complex (with an action of GCT); it is denoted St(a). The (p+^-simplex {Ap+q, A)
is mapped bijectively onto a (p + q)-cell of St(a) which we regard as being labelled
A e E^'q)(S). For ρ = q = 0, there is only one such 0-simplex and it will be denoted
σ. The set of simplices of St(a) containing σ are in bijection with the set of their
labels.
If У is the scwol associated to a poset Q and if G(3O is a simple complex of
groups, then St(a) is isomorphic to the complex constructed in 11.12.24.
4.10 Definition. The open star of σ in St(a) is denoted st(a); it is invariant by the
action of Ga. We call st(a), equipped with this action of Ga, the geometric realization
of the local development ofG(y) at σ.
The natural projection £^·9)(σ) -> £^·9)(σ) induces a projection St(a) -> St(a)
and an isomorphism Ga\St(a) —> St(a). The restriction of the projection to st(a) is
denoted
ρσ : st(a) -> st(<r) с \у\.
4.11 Proposition. If the complex of groups G(y) is associated to an action of a
group G on a scwol X (with respect to some choices, as in 2.9(1)), then there is a
canonical Ga-equivariant isomorphism
fa : st(a) -► st(CT),
where ~σ is the vertex of \X\ chosen as the representative of σ.
Proof. We use the notations established in (2.9). As the projection ρ : X —» У
is non-degenerate, given a simplex of \У\ that contains σ and is labelled A =
(z\, ■. ■ ,zp,c\,.. .,cq) e Ep-q(a), there is a unique simplex of \X\ labelled A
projecting to A with initial vertex the chosen representative of the initial vertex i(cq)
of A. The simplices of \X\ containing σ and projecting by \p\ to the simplex
labelled A = (z\, ■ ■., zp, c\, ■ ■., cq) e E{p'q){a) are of the form (ghc).A, where
g e Ga,c = c\.. .cq and hc e G is the chosen element of G mapping t(c) to
σ. They are in bijection with the elements of Ga/^c{Gi(C))- Therefore the map
(gfc(Gi(c)\ Л) ь» ghc.A
gives a bijection denoted λρ·ι : £^·9)(σ) —> Ε^^ζσ), which is GCT-equivariant.
We claim that these maps \p-q commute with the maps 3,. It will then follow that
the maps (x, А) ь» (χ, λρ·«(Α)) from Ap+<1 χ Ε^·ι\σ) to Ар+ч х £<?■«>(σ) induce a
GCT-equivariant isomorphism from St(a) to St(CT).

The Geometric Realization of the Local Development 559
To see this, first observe that, for A = (zu ..., zp, c\, ■ ■., cq) e E^^io), we
have 3;(A) = 3,-(A) for i φ p,p + q and dp+q(A) = hCq.dp+q(A). We check commu-
tativity only in the non-trivial case i = ρ + q and q > 2. In this case, λ^"1 maps
dp+q(gfc(Gi(c)),A) = (gg~l4fd(Gi(d)), dp+qA) to
{gg^chc')-dp+q{A) = (ggc-,cqhdhCii).dp+q(A) = (ghc).dp+q(A),
which is equal to dp+q(XM(gil/c(Gj(c))), A). (Here we appealed to (2.9(1)) for the last
equality.) D
The construction of St(a) is functorial with respect to morphisms of complexes
of groups.
4.12 Proposition. If φ = (φσ, φ(α)) : G(3O —> G(Z) is a morphism of complexes
of groups over a non-degenerate morphism f : У —> Ζ, then it induces, for each
vertex σ e У(У), φα-equivariant maps St(a) —> St(/(a)) and st(a) —> st(/(a)).
Proof. For ρ > 0, g > 0, let фР-Ца) : Ε^'4){σ) -> £{p^)(/(a))befhe0(j-equivariant
map
(girc(Gi(c)), А) ь» {фП^ЖсЩ(с){Стс)))ЛА)),
where A = (z\, ■■ ■ ,Zp, c\,..., cq) e EP'^ia), с = сь ...cq and
/(A) = (^(zi),...,/(^),/(c1),...,/(c,)).
We claim that the maps фр-ч{а) commute with the maps 3f defined in (4.7). Again
we check only the non trivial case where i = p+ q and q > 2. We have
(φ™-\σ) ο dp+q)((gi,c(Gl(c)), A)) =
Waigg^Wc'WwiGw»), f(dp+qA)),
where c' = c\, ... cq-\. Using condition (ii) of (2.4), we see that this is equal to
(dP+q о фт)@){^ЫСКс)\ А)) =
WaigWitigf^lf^fic-^Gi^lfidp^A)).
Therefore the maps
(id, φΡ'ΐ{σ)) : /±P+q χ E^-q\a) -> Ap+<1 χ £Μ(/(σ))
induce a 0CT-equivariant map St(a) —» St(/(a)), and by restriction a map st(a) —>
st(/(a)). D

560 Chapter III.C Complexes of Groups
4.13 The Construction of St(a) and st(a). Let a e Е(У) with i(a) = σ and t(a) = τ.
Let Е^-к^(а) С £<Ρ+*·«>(σ) be the set of sequences
A = (g^c(Gi(a)), Z],..-,Zp,a],...,ak,c],..., cq),
where a = a] ... ak, с = c\,... cq and girc(G-,(c)) e G/Vfc(Gi(c)). We have maps
9; : Ε^^ζα) -^"'-^(a) 0 < i < ρ
EJJ><k^(a) -н> £<?·*-' ·<?)(£) p<i<p + k
Е^'к^(а) -^Е^'к-ч-])(а) p + k <i<p + q + k
which are the restriction to E^-k-^(a) of the maps 3; defined above in (4.7).
To cons tract the complex St(2z) we start from the disjoint union
У /У+к+ч χ {Λ},
p,k,q,A
where p > 0,k > \,q > 0 and A e E&^'^fa), and then we form the quotient
by the equivalence relation generated by (d,{x), Λ) ~ (x, 3,(A)), where (x, A) e
ΔΡ+k+q χ Ε(ρΧΦ(α) and 0 < i < ρ + q + k, ίφρ,ρ + k.
The group Ga acts naturally on this disjoint union and the action is compatible
with the equivalence relation.
The quotient of this equivalence relation, equipped with the induced action of
GCT, is denoted St(2z). The (p + k + g)-simplex (/SP+k+q, A) is mapped injectively to
the quotient; its image is a simplex which we label A. For/? = q = 0 and к = 1,
there is only one such 1-simplex and it will be denoted a.
Finally, we define st(2z) to be the union of the interiors of those simplices of
St(2z) that contain a. Note that the natural projection St(a) -» St(a) induces an affine
isomorphism GCT\St(2z) -» St(a).
Note also that there is a natural GCT-equivariant map St(2z) -> St(a) induced by
the inclusion
£<P.M)(2) _» £(Ρ+*·«)(σ).
This map sends st(2z) by an affine isomorphism onto the union of the interiors of
the simplices that contain the 1-simplex labelled (GCT, a). This simplex will be also
denoted a. We identify st(2z) to its image in st(a).
4.14 Proposition. Let a e Е(У) with i(a) = σ and t{a) = τ. There is α ψα-
equivariant map
fa : st(a) -* st(?)
sending st(2z) isomorphically onto an open set, and the induced map that one gets
st(a) = GCT\st(2z) —> st(r) = GT\st(r) is the natural inclusion. Moreover, for all
(a, b) e Е<-2)(У), we have
fa{fb{x)) = ga,bfab(x)
for each χ e st(2z) nf^\st(b)).

The Geometric Realization of the Local Development 561
st(c '
Fig.C.13 Local developments of the complex of groups G(y) of figure С12
Proof. There are Va-equivariant maps
£(p.*-9)(2) -* £<ρ·*+9)(γ)
defined by
(g^c(G;(C)), А) ь» (ggfl,c^ac(G/(aC)), A),
where Л = (zi,..., z,„ au ..., ak, c,,..., cq) e Е^+^СУ) with a = a]...ak.
As above, one checks that these maps commute with the maps 3,- and hence
induce a Va-equivariant map St(2z) -> St(r). The restriction of this map
fa : st(o) -* st(r)
is an affine isomorphism onto the open star of the image of a.
For (a, b) e £(2)(3Ό, the open set st(2z) n/b~'(st(b)) is the union of the interiors
of those simplices of St(a) labelled by
{g^c{Gi(c)),z\, ...,Zp,au...,ak,bu...,bi,C], ...,cq),
where a = a.\,... a^, b = b\ ...bi, and с = c\,... cq.
The composition/дД maps the interior of the simplex labelled (gVcCG,^)), A)
to the interior of the simplex labelled {fa{^b{g))fa{gb.c)ga,bcfabc{Gi(c)),A). The
map fab maps the interior of this simplex to the interior of the simplex labelled
(^ab(g)gab^abc(Gi(c)), A). One checks that the conditions (i) and (ii) of (2.4) imply
that
^a{4>b{gWa{gb,c)ga,bc = ga,b^ab(gab,c)-
О

562 Chapter III.C Complexes of Groups
4.15 Remark. One can show that St(a) and St(a) are the geometric realizations of
scwols and that the maps defined above St(2z) -» St(a) and St(a) -> St(r) are
geometric realizations of morphisms of scwols (see 4.21).
Local Development and Curvature
Assume that the geometric realization \У\ of the scwol У is endowed with the
structure of an MK-complex with a finite set of shapes in the sense of Chapter 1.7
(cf. 1.3.3(2)). For each σ e У(У), we can use the affine maps discussed above to
endow St(a) and St(a) with an induced MK-complex structure such that St(a) —»
St(a) restricted to each simplex is a local isometry. From this st(a) inherits an
induced length metric and the maps/a : st(2z) —» st(a) constructed in (4.14) are local
isometries.
4.16 Definition. Let G(3O be a complex of groups over a scwol У such that \У\ is
metrized as an M^-complex. We say that G(3O has curvature < к if st(a), in the
induced metric described above, has curvature < к for each σ e У(У). If к < 0 then
we also say that G(y) is non-positively curved.
In the next chapter, we shall prove the following theorem which is a generalization
of Gersten-Stallings theorem [St91] fortri angles of groups. See also [Gro87], [Hae90,
91], [Spi92] and [Cors92], in the case dim}7 = 2.
4.17 Theorem. If the complex of groups G(y) is non-positively curved then it is
developable.
If У is connected and G(3O is of curvature < к, where к < 0, then the geometric
realization of the simply connected development of У as constructed in Theorem
3.13 is an MK-polyhedral complex which is CAT(/c). It is equipped with an action of
the fundamental group of G(3O by isometries, and G(y) is isomorphic to the complex
of groups associated to this action.
4.18 Remark. Assume that У is the scwol associated to an M^-polyhedral complex
K. Then G(3O is of curvature < к in the sense of (4.16) if and only if, for each
vertex τ of K, the geometric link of 'τ in st(r), with the induced spherical structure,
is CAT(l). To see this one simply applies the argument given in (II.5.2).
4.19 Examples
(1) In the notation of 1.4(3) and 3.11(4), if we assume that the two congruent
и-gons in M2K have an angle > я/и^ at the vertex ц, where nk is the order of Sk e G*,
then the complex of groups G(3O is of curvature < к. If к < 0, then G(3O is
developable.
(2) The following is an alternative construction of an example due to Ballmann
and Swiatkowski (see Theorem 2 and section 4 of [BaSw97]). We take m copies

Local Development and Curvature 563
У], ■ ■ ■ Ут of the scwol У described in figure C.6 (see 1.14(2)), and glue them along
the subscwol with edges a, a' and vertices σ, τ, τ'; the remaining vertices and edges
in У к will be indexed by the subscript к (see figure C.14). Let Ζ be the scwol obtained
in this way; the set of vertices V(Z) is σ, τ, τ', тк, а'к, рк, к= 1,..., m. We define a
complex of groups G(Z) over Ζ where the local groups associated to pk, ak, o'k are all
trivial, GT> — Z2 is generated by t, Gn = Z3 is generated by sk and Gr is a group H.
We assume that the only non-trivial twisting elements are ga<bt = sk e H, get..v. = t
and gaty = rk. We compute the fundamental group using 3.11(3); in 34 we take the
maximal tree considered in 3.11(3) and we take the union of these trees to be our
maximal tree Τ in Z. This yields the following presentation of G := Tt]{G{Z), T):
the generators are the elements of Η and an extra element t, and these generators
are subject to the relations in Η and the extra relations t2 = 1, (skt)3 = 1 for
к = 1,..., m.
Fig. C.14 Example 4.19(2)
We metrize Ζ as a piecewise Euclidean complex so that each 34 is the union of
two Euclidean triangles with angles π/6 at τ, π/2 at τ' and я/З at тк. At the vertex

564 Chapter III.C Complexes of Groups
Tk the local development is isometric to the interior of a hexagon in the Euclidean
plane, and at the vertex τ' the local development is isometric to the interior of m
Euclidean rectangles glued along a geodesic; in particular these local developments
are of non-positive curvature. In the local development st(r") at τ, the link of ΐ is the
following graph L: the vertices of L are the elements of H; the edges of L are labelled
by the integers 1,..., m and each edge has length π/З; two vertices h, h! are joined
by an edge labelled к if h' = hs^.
The complex of groups G(Z) has non-positive curvature if and only if the girth
of L is at least 6. If this is the case, then G(3O is developable and the geometric
realization of the simply connected development of У as constructed in (3.13) is a
piecewise Euclidean complex К whose 2-cells are equilateral Euclidean triangles;
the link of each vertex will be isomorphic to L. The fundamental group G acts on K;
the action is transitive on the set of vertices and the isotropy subgroup of a vertex
is isomorphic to H. The isotropy subgroup of each 2-cell is the dihedral group with
six elements. (The triangulation of К corresponding to the triangulation of \У\ is
obtained by dividing each 2-cell of К into three congruent quadrilaterals and passing
to the barycentric subdivision of this cell decomposition of K.)
The Local Development as a Scwol
Here we construct the local development purely in the framework of scwols.
Specifically, given a complex of groups G(y) = (Ga, ψα, ga,b) over a scwol У, for each
vertex σ e У(У) we construct a scwol У(а) with an action of Ga such that the
geometric realization is equivariantly isomorphic to St(a).
4.20 The Construction of Lkj. The upper link LkCT of σ e У(У) was defined
in (1.17). The natural morphism LkCT —> У defined in (1.17) induces a complex
of groups G(LkCT) over Lka from G(y) by the construction of (2.7). There is a
natural morphism ψσ : GiLka) —> Ga mapping the local group G,(a) associated to
a e V(LkCT) с Е(У) to the group Ga by ψα and associating to each edge (a, b) of
LkCT the element ga<b e Ga. The development 0(1^, ψσ) of G(LkCT) associated to
this homomorphism shall be denoted Lk^.
More explicitly, the scwol Lkj is defined as follows:
V(Lks) = {(gMGna)), a)\ae Е(У), t(a) = σ; gfa(Gm) e G„/^a(G;(a))}
£(Lk5) =
{(g\MG/№>). a, b) | (a, b) e £(2)(У), Κα) = σ; gMGm) e GJifab(Gm)}
The maps i, t: E(Lk^) —> V(Lk^) are defined by
i((g^ab(Gm), a, b)) = (gi/rab(Gm), ab)
KigMGiu,)), a, b)) = {gg~lifa{GKa)\ a).
Composition is defined when a' = ab and g' = gga\b' m°d т/гачу^ць·)) by the
formula

The Local Development as a Scwol 565
(g^ab(Gi(b)), a, b)(g'i/fa>b>(Gm), а', У) = (g'faw(Gm), a> bb')-
The group Ga acts naturally on Lk^, namely h e Ga acts on (g^abiG^b)), a, b) e
Е(Ц%) by
h-(giJrab(Gnb)), a, b) = (hgifab(Gm), a, b).
There is also a natural morphism Lkj —> Lka mapping (gifabiGw), a, b) to (a, b);
this induces an isomorphism GCT\Lk„ —» Ll^.
4.21 Definition of the Local Development У(а). The local development of the
complex of groups G(3O at the vertex σ e V(3O is the scwol У(а) := (Уа *
Lkj) = (Ькст * σ * Lk^) together with an action of Ga on it: Ga acts trivially on the
first factor and as above on the second factor. We write y„ to denote the subscwol
(σ*υ^)ς5;(σ).
We have a natural morphism
У&) -> У(а)
inducing an isomorphism Ga\y(a) —» У(а). This is simply the join of the identity
on the first factor У and the morphism Lk„ —» Lka on the second factor.
4.22 Proposition. St(a) is isomorphic to the geometric realization of У (5).
Proof. We leave the proof as an exercise for the reader. We only point out the natural
bijections
V(Lki) <* &0J)(5), σ <н> £(0·0)(σ), V{Uaa) <н> £(,·0)(σ).
D
4.23 Proposition. Lei φ = (φσ, φ(ά)): G(y) —> G(2) fee α morphism of complexes
of groups over a non-degenerate morphism of scwols f : У —> Ζ, and let σ e
У(У). Then φ induces a φα-equivariant morphism У(5) —> Z(f(a)) of the local
developments.
Proof. The non-degenerate morphism/ : У —*■ Ζ induces a morphism LkCT —>
Lk/(tj) sending the vertex α e У(Ькст) to the vertex/(a) e Lk/(tj) and the edge
(a, fc) e Ε(1±σ) to the edge (f(a),f(b)) e 1±Λσ). The desired morphism У(а) =
(Ькст * σ * Lk^) -> Z(f(a)) = (Ькст */(σ) * Lk,~) is the join of the following
morphisms of the three factors. On the first it is the morphism just described; on the
second it maps σ to/(a); and the morphism on the third factor is described on the
set of vertices (resp. edges) of Lk^ by the following formulae (where a e Е(У) has
Ha) = σ, (a, b) e ЕР\У) and g e Ga):
igfa(Gm), a) ь» ($a{.g№(a)ilfm(Gma))),f{a))
(gfabifinab)), a, b) ь» (ф^)ф(аЬ)фдаЬ)(Ста,ы)),f(a),f(b)).
It is clear that this morphism is $CT-equivariant. (Compare with 2.18(1).) Π

566 Chapter Ш.С Complexes of Groups
5. Coverings of Complexes of Groups
In this section we define coverings of complexes of groups and reformulate the
definitions in several ways. We also define the fibres and monodromy of a covering.
We show that a covering is determined up to isomorphism by its monodromy. The
theory of coverings in the framework of small category theory is somewhat simpler
and its relation with the theory of coverings of complexes of groups is indicated at
the end of the appendix.
In the particular case of graphs of groups, our theory of coverings should be
equivalent to the theory of Bass [Bass93], although it is not clear if his notion of
morphism is the same as ours.
Definitions
5.1 Definition. Let φ : С(У) —» G(3O be a morphism of complexes of groups over
a non-degenerate morphism of scwols/ : У —*■ У, where У is connected. We say
that φ is a covering if for each vertex σ' e У(У):
(i) the homomorphism φσ· : Ga· —» G/v) is injective, and
(ii) the$CT--equivariantmapst(a') —» st(/(a'))mducedby$(see4.12)isabijection.
Z2
-э-о D,
st(f)
st(f)
Fig. £15 A covering of a 1-dimensional complex of groups
5.2 Lemma. Condition (ii) is equivalent to the condition:
(ii)' For each a e Е(У) and σ' e V0>') with t(a) = σ =/(σ'), the map
] I Ga'/ira>(GKa:)) -> Ga/yjfa{GKa))
α'<=Γ'(α)
Г(а')=ст'
induced by g t-» φσ'^)φ(α') is bijective.

5. Coverings of Complexes of Groups 567
Proof. The proof is in terms of the maps ф^д(а') defined in the proof of (4.12). If (ii)
is satisfied, then the map φ°·](σ') : Ε^°·ι\σ') —> ^'^(σ) is bijective. The bijectivity
of this map is clearly equivalent to (ii)'.
If0°^a') is bijective, then <^9(σ') is bijectivefor all/J, q. This is obvious if q = 0
because/ is assumed to be non-degenerate. Let us prove surjectivity for q > 1. Let
A = (girc(GKc)),A),wheTeA = (z\,..., Zp, cu ..., cq) e ΕΡ·4)(σ) and с = с\ ...%,.
By hypothesis (ii)', there exists a unique element (gVc'^c1))), c') € ^"'^(σ')
mapped by φ°·](σ') to (g^c(Gi(C)), c). As/ is non-degenerate, there is a unique
A' = (z\,..., z'p, c\,...,c'q)e E^-q\a') with initial vertex i(c') = i(c'q) projecting
by/ to A. Therefore {g'^diG;^)), A') is mapped to A. The proof of the injectivity is
similar. D
5.3 Remark. Conditions (ii) or (ii)' are equivalent also to the condition that the map
Lk^- —» Lk^ induced by/, as defined in 4.23, is bijective. This in turn is equivalent
to the condition that the map У'(а') -> У(а) is bijective.
5.4 Examples
(1) In the case where G(y') and G(3O are trivial complexes of groups (i.e. all
the local groups are trivial), definition (5.1) reduces to the definition of a covering
of scwols (1.9). However if the local groups are not trivial, then the underlying
morphism of scwols will not in general be a covering. (This is the case in figure
CAS.)
(2) In the notation of 2.9(1), consider an action of a group G on a scwol X and
an associated complex of groups G(3O over the quotient y. We shall describe how
to associate to this situation a covering λ : X —» G(y) (where we consider X as the
trivial complex of groups over X) over the morphism ρ : X —> У = G\X. For each
σ e V(X) we choose an element ga e G such that g^.W = σ, where σ is the chosen
representative in the orbit of σ. We define λ to be the morphism that maps each edge
a e E(X) to the element
It is straightforward to check that λ is a morphism, namely
λ(αΕ) = ^&)irpia)(4^))gp^,PCb)
for all (a, b) e E^2)(X). Moreover λ is a covering, because condition (ii)' of 5.2
is satisfied. In this natural sense, the scwol D(y, it) constructed in 3.13 can be
considered as a simply connected covering of G(y).
We generalize the preceding construction. With the notation of 2.9(4), consider
actions of groups G and G' on scwols X and X' respectively. Let L : X —» X' be a
morphism which is equivariant with respect to a homo morphism Λ : G —» G'. Let
λ : G(y) —> G'(y') be an associated morphism for the corresponding complex of
groups as defined in 2.9(4). Assume that X is connected. Then λ is a covering if and
only if both of the following conditions hold: L is a covering and the restriction of Λ
to the isotropy subgroup of each vertex of X is injective.

568 Chapter Ш.С Complexes of Groups
The Fibres of a Covering
5.5 Definition. Let φ = (φσ·, φ(α')): G(y') —» G(3O be a morphism of complexes
of groups over a non-degenerate morphism/ : У —» У. The fibre Fa over a vertex
σ e v(y) is the set
Fa= Ц Οσ/φσ,{Οσ·)
σ'£/-Ι(σ)
with the natural action of Ga by left multiplication.
For a e Е(У1 let
Fa '■ Fi(a) —> F,(fl)
be the i/ra-equivariant map which is the union of the maps defined by
Fa(g0/(O')(Gi(a'))) = Ψα(8)Φ(α'Τ] <P«a')(G,(a'))),
where a' e/~'(a) and g e G,(a).
One calculates (using 2.1(i) and 2.4(ii)) that for composable edges (a, b) e
Е(1\У) we have
Fa°Fb = ga,b-Fab-
5.6 Proposition. Condition (ii)' of 5.2 holds for every σ' e/_1 (σ) if and only if:
(ii)" Va e ДУ) ννζ'ίΑ t(a) = σ, the map Fa : F,(a) -> Fi(a) ii α bijection.
In the proof of this proposition we shall need the following general observation.
5.7 Lemma. Let Η and К be subgroups of a group G, and let (g/);ej be a family
of elements of G. We consider the action of Η on G/K by left translations and
similarly the action of К on G/H. For eachj e 7, let Щ be a subgroup of Η that fixes
gjK e G/K. Note that the subgroup Kj := gj~]Kgj fixes gj~] Η e G/H. The following
conditions are equivalent:
(1) The H-equivariant map
\}h/Hj^g/k
j<=J
given by hHj (->· hgjK is a bijection.
(2)
Ц.= Η П gjKgr* and G = U HgjK.
jsJ
(This last condition means that {gj}jSj is a set of representatives for the double
cosets H\G/K, i.e. the set of subsets ofG of the form HgK.)
(3) The K-equivariant map
Цк/Kj^G/H
j<=j
given by kKj (->· kgj~]H is a bijection.

The Fibres of a Covering 569
Proof. The injectivity of Н/Ц —> G/K is equivalent to the condition that Ц is the
isotropy subgroup Я Π gjKg~' of g/ίΓ, and also to the condition that K, is the isotropy
subgroup of gj~]H e G/H. The map associating to the Я-orbit of gK e G/K the
double coset HgK induces a bijection from the set of Я-orbits in G/K to the set of
orbits in G under the action (h, k) : g t-» /igfc-1 of Я χ if on G. Therefore {gjK}jSj
is a set of representatives for the Я-orbits in G/K, if and only if {gj}jSj is a set of
representatives for the double cosets H\G/K in G. Interchanging the roles of Я and
K, we see that it is also equivalent to say that (g~' Я);<=у is a set of representatives for
the /f-orbits in G/H. Thus (1), (2) and (3) are equivalent. D
Proof of the proposition 5.6. Condition (ii)" is equivalent to the statement that for
every σ' e/_1 (σ) and every a e Е(У) with t(a) = σ, the map
] I Gna)/4>Ha')(Gna>)) ~> GCT/0CT'(GCT'),
α'<=/-'(α)
ί(α')=σ'
which is the restriction of Fa, is bijective.
We apply the lemma with the following choices:
G = Ga, К = φσ'(ΰσ,), H = tya(Gna))
J={a' ef-\a)\t(a') = a'},
ga> = φ(α')~] Va' e 7 and Ha = ^α0,·(α')(σί(α')·
Condition (ii)" corresponds to condition (1) of the lemma, while condition (ii)'
corresponds to condition (3). D
Our next goal is to establish a converse to 5.6: given an appropriate system of
bijections Fa one can reconstruct an associated covering.
5.8 Proposition. Let G(y) be a complex of groups over a connected scwol У. Suppose
that for each σ e V(y) a set Fa is given with an action of Ga, and for each
edge a e Е(У) a ψα-equivariant bijection Fa : F,-(a) —> Ft(a) is given such that
Fa о Fb = ga,b-Fab far all (a, b) e £(2)(3O- Then one can construct a covering
φ : G(y') —> G(y) over a non-degenerate morphismf : У —> У such that the Fa
are naturally the fibres of φ. Moreover G(y') is unique up to isomorphism and φ is
unique up to homotopy.
Proof. For each σ the subset f~] (σ) С Vty) will be a set of representatives for the
GCT-orbits in Fa. For each a' e/~'(a) there will be an edge a! = (a, a') in У that
projects by/ to a; the initial vertex i(a') is σ' and the terminal vertex ί(α') will be
the chosen representative in the G^-orbit of Fa(a'). When the composition of two
edges (a, a')(b, p') is defined, it is equal to (ab, p'). This information specifies the
scwol У and the morphism/ : У —> У.
Define Ga> to be the subgroup of Ga fixing σ' e Fa and let φσ, : Ga> —> Ga be the
natural inclusion. For each a' e V(y')w'ithf(a') = a, choose an element ф(а') е G,(0)

570 Chapter III.C Complexes of Groups
G,(a)-orbit
Fm FKa)
t(a) о ■< . i(a)
Fig. C-16 Construction of a covering from the monodromy
such that (p(a').Fa(i(a')) = t(a') and define a homomorphism ψα> : G,(a-) —» G,^ by
the formula
ΨΑ8) = Φ(α')Μ8)Φ(α'Γ].
For composable edges (a', b') e Е(1\У), define
ga,,b, = ф(а')ЫФ(Ь'))8а,ьФ(а'Ь'Т] е G((e0.
It is straightforward to check that the above data define a complex of groups G(y')
over У and a morphism 0 : G(y) —> G(3O over/ : 3·7' —> У defined by the $σ-
and the elements φ(α'). The fibre of 0 over σ is the disjoint union of the Ga/Ga> for
σ' e /~' (σ); it can be identified to Fa by the map sending the coset gGa> to the point
g.a'. Under this identification, the map from the fibre over i(a) to the fibre over t(a),
associated to a as in (5.5), is precisely Fa. Moreover, as these maps are bijections,
(5.6) shows that the morphism φ is a covering.
It is clear that any other choice for the representatives σ' of the orbits, or for the
elements φ(α'), would lead to a homotopic morphism. D
5.9 Galois Covering. An important special case of the preceding construction arises
naturally when one has a morphism Φ = (Φσ, Φ(α)) from a complex of groups G(3O
to a group G.
For each σ e У(У) we define Fa = G with the action of Ga given by right
translations via Φσ (i.e. for h € GCT, g e G, define h.g = £Фст(/г-1)). If Fa : G ->
G is defined to be the right translation by Φ(α)-1, then Fa is ^fa-equivariant and
{Fa о Fb){g) = Fab{g)®t(a){g~l) = ga,b-Fab{g)- Following the construction of (5.8),
we give an explicit description of the covering φ : G(y) —» G(3O over the morphism
/ : У —> У in this special case.
G,(a)-orbit

The Fibres of a Covering 571
У will be the development D(y, Ф) as constructed in (2.13) and/ : D(y, Ф) ->
У will be the natural projection. Thus/~'(a) is the set of pairs ((g^a(Ga), σ) €
G/<S>a(Ga) χ {σ} and/-'(α), а е Е(У), is the set of pairs a' = (g%(a)((GKa)), a)
with i(a') = (g<f>iia)(Giia)), i(a)) and ί(α') = (§Φ(α)Φί(α)(σί(α)), ί(α)). The group G
acts on У = ЩУ, Φ) and/ induces an isomorphism G\y —> 3·7·
For each σ' e /_,(σ) <Ξ V0O. the group GCT- is the kernel of Φσ and φσ· :
Gat —> GCT is the inclusion. In order to define ^а> and ga> j,> we need to choose for
each Фст(£ст)-со8е1 σ' a representative in G; we again denote this σ', thus identifying
σ' to the coset σ'Φσ(Οσ). We also choose for each edge a' e f~x(a) с £(У) an
element $(a') e Gf(0) such that
/(α')Φ(α)-,Φί(α)(0(α'Γ,) = ί(α').
We then define ψα> : G,-(a-) -^- G,^) by ^ = Αάφ(α') ο ^τα.
For composable edges (a', U) e еР-\У) with α =f(a'), b =f(b'), we define
g0<,&< = ф(а')МФ(Ь'))8а,ьФ(а'ЬТ] е кегФг(а).
The complex of groups G(3O is defined to be (GCT-, ^a-, ga',i>0 and the morphism
φ : G(y') —» G(y) is given by the homomorphisms φσ> and the elements φ(α').
5.10 Examples
(1) Even if G(3O is a simple complex of groups, in general a Galois covering
G(y') will not be isomorphic to a simple complex of groups, as the following example
illustrates.
Let У be the scwol associated to an и-gon P. The set of vertices of G(3O has η
vertices τ^ corresponding to the vertices of P, has η vertices σ* corresponding to the
barycentres of the sides of P, where к = 1,..., η, and has a vertex ρ corresponding
to the barycentre of P. The set of edges has 2n elements a*, a'k with i(a,0 = i(a'k) =
Ok, t(cik) = Tk, t(a'k) — -Гц (indices mod и), and a further 2n elements bt, Ck with
i(bk) = i(cu = P, t{bk) = ak, t(ck) = τ*. Moreover akbk = ck = a'k+lbk+]. We
consider on У the complex of groups where Gp is trivial, Gat is cyclic of order two
generated by Sk, and GXk is the dihedral group of order 2rik generated by the images
of Sk andifc+i.
Let Φ : G(3O —> Z2 be the morphism associating to each element Sk the generator
t of Z2 and to each edge the trivial element. We construct the associated Galois
covering φ : G{y) —> G(3O- The scwol У is associated to the complex obtained
by gluing two copies of Ρ along their boundary — in the notation of figure C.4,
the vertex ρ corresponds to the coset {t} and p' corresponds to the coset {1}. The
local groups associated to the vertices τ* are the cyclic groups of order щ generated
by SkSk+]; the other local groups are trivial. To determine the twisting elements we
choose coset representatives, these are trivial except for p, where we have the coset
{t}. The elements ф(а') associated to the edges a' of У are trivial except ффк) = Sk
and ф(Ск) = Sk+\. With these choices, all the twisting elements are trivial except
gat,bt = skSk+l for k = 1, . . . , П.

572 Chapter III.C Complexes of Groups
(2) We reconsider the triangle of groups Gk(y) studied in Π. 12.34(5) and maintain
the notation established there. Thus У is the scwol associated to a Euclidean triangle
Δ with angles (k — 2)n/(2k, π/2, π /к at the vertices το, τι тг, respectively. Consider
the dihedral group D3 generated by the elements t\ and ti of order 2. For к odd, there
is a simple morphism Φ : G(y) —» D3 sending i2 to tj., both s\ and s$ to t\, and ίο
to 1.
The corresponding Galois covering gives a simple complex of groups G(y'),
with respect to suitable choices (cf. [Hag93]), where У is the scwol associated to
the polyhedral complex К obtained by gluing three isometric copies of Δ along the
edge [το, τι ]. Let τ2, i = 1, 2, 3, denote the remaining three vertices of K. The local
group at τ{ is the dihedral group Dt generated by two involutions, t and x,; the local
group at το is the cyclic of order two generated by V, and the local group at το is
isomorphic to Z2 χ Ζ2, the three non trivial elements beingx\,xi,хъ (the elements
labelled by the same letter are identified by the edge monomorphisms). The groups
associated to the barycentres of the 2-simplices and to the edge [το, τι] are all trivial,
and the groups associated to the edges other than [το, τι] are cyclic of order two. We
leave the reader to check that GtW) is simply connected if к = 3.
The Monodromy
5.11 Construction of the Monodromy. Let G(y) be a complex of groups over a
connected scwol У and let the system of maps (Fa, Fa) be as in (5.8). For each
g e Ga, let Fg : Fa —> Fa be the bijection ii-> g.x given by the action of g on
Fa. For each oriented edge e e E{X)± define Fe : Ft(e) —> Рце) to be Fa if e = a+
and F~' if e = a~. Given a G(3O-path с = (g0, e\, g\,..., et, gk) joining σ to τ,
let Fc = Fg0FeiFgl ... FetFgt : Fx -> Fa. Note that if с and d are homotopic, then
Fc = Fc>. Also Fc-i = F~\ and if the concatenation с * d of two G(3O-paths с and
d is defined, then Fc%ci = Fc о Fc-.
Fix a vertex σο. The monodromy associated to (Fa, Fa) is the action οι π \(GQ?),
σο) on FCT0 where the homotopy class of a G(3O~loop с at σο acts as the bijection
Fc ■ /v„ -> Fao. When (Fa, Fa) is the fibre system of a covering φ : G(y') -+ G(y%
this action is called the monodromy of the covering.
We end this section by stating the classification of coverings in terms of
monodromy.
5.12 Theorem. Let G(y) be a complex of groups over a connected scwol У. Fix a
vertex σο e У(У). Suppose that we are given an action ofn\{G{Y), σο) on a set Fq.
Then there is a covering φ : G(y') —> G(y) whose fibre over σο is Fo and whose
monodromy is the given action ofn\(G{y), σο). This covering is connected if and
only if the monodromy action is transitive.
Let φ' : G(y) -> СНУ) over f : У -> У, and φ" : СНУ") -> СНУ) over
f" : У" —> У, be two coverings such that there is an equivariant bijection /0 from
the fibre F'a of φ over σο to the fibre F" of φ" over σο (equivariance is with respect

Appendix: Fundamental Groups and Coverings of Small Categories 573
to the monodromy actions). Then there is an isomorphism φ : G(y') —>- GQ?") over
f : У —> У" such that φ" οφ = φ' and φ induces the mapfo on the fibres over σο-
Proof. The proof of this result follows from the corresponding result proved in the
more general framework of coverings of small categories in the Appendix. (The
relationship between the general framework and coverings of complexes of groups
is explained in A.24.) D
Appendix: Fundamental Groups and Coverings
of Small Categories
As we indicated in the introduction to this chapter, the notational complexity that
necessarily accompanies definitions concerning complexes of groups (morphisms,
homotopy, fundamental groups, etc.) can obscure what are actually very natural and
general constructions. In this appendix we strip away the specific notation associated
to complexes of groups and explain these definitions in the general context of small
categories. Thus the notions of morphisms of complexes of groups and homotopies
between them are shown to correspond to functors and homotopies for the associated
categories.
We give the definition of the fundamental group of a small category and give
a presentation of this group using a maximal tree in the graph naturally associated
to the category. The fundamental group of a complex of groups is the fundamental
group of the associated category.
The theory of coverings for small categories is parallel to the theory of
coverings for topological spaces. If φ : G(y') —» G(y) is a morphism of complexes
of groups which is a covering as defined in 5.1, then the corresponding functor
CG(y') —» CG(y) for the associated categories is not a covering in general, but it
is the composition of an equivalence CGty) —> С and a covering С —> CGQ}).
(The fact that the equivalence depends on some choices explains why the theory of
coverings for complexes of groups is somewhat more complicated than the theory
of coverings of categories.)
Basic Definitions
A.l Category. A (small) category11 С consists of a set C, an auxiliary set 0(C),
called the set of objects of C, and two maps i : С —■► 0{C) and t : С —> 0{C).
The maps i and t associate to each elementΊ2 γ e С its initial object i(y) and
its terminal object t(y). Given σ, τ e 0{C), we let CT(J denote the set of γ & С
71 In this chapter we deal only with small categories, so for us "category" will always mean a
small category.
72 In the literature, elements of С are often called morphisms, but we shall not use this
terminology. We interpret the elements of С as arrows 7 with source 1(7) and target ((7).

574 Chapter III.C Complexes of Groups
such that i(y) = σ, t(y) = τ. A law of composition (partially defined) is given
in C: two elements γ, γ' е С can be composed if and only if i(y) = t(y'); their
composition γ γ' belongs to C((y) ,(y'). To each object σ e 0(C) is associated an
element Ισ e 0σ<σ which is a unit: if γ & CT?(J, then γ Ισ = γ = Ιτγ. The law of
composition is associative: ifyy' and γ'γ" are defined, then (γγ')γ" = y(y'y"),
and this composition is denoted γ γ'γ".
We shall often identify 0(C) to a subset of С by the inclusion σ м> 1σ. An
element γ e Cr?(J is called invertible if there is an element у-1 е CffiT such that
γγ~χ = 1τ and γ~χγ = 1σ.
A subcategory С of a category С consists of a subset С с С and a subset
0(C) с 6>(C) such that if у е С then i(y), tiy) e 0(C); if у, у' е С are
composable in С then yy' e C; and if σ e C(C'), then 1σ e С
On C(C) there is an equivalence relation generated by [σ ~ τ if CaT φ 0]. The
category С is connected if there is only one equivalence class in 0(C). In general,
given an equivalence class, the union of the Caa with σ and τ in that class is a
subcategory of C; it is called a connected component. С is the disjoint union of its
connected components.
We write C(2) to denote the set of pairs of composable elements, i.e. pairs (yi, уг) е
С χ С with i(yi) = ?(уг). More generally, we write Cw to denote the set of fc-tuples
(γι. · · ·, Yk) for which the composition γχ ... у* is defined. In particular C(1) = C.
By convention, C(0) is the set of units of C, often identified to 0{C).
A.2 Examples
(0) A group can be considered as a category with one object where all elements
of the category are invertible.
(1) Let G be a group acting on a set X. Let С = G χ X, 0(C) — X and let
г, ί : С -> C(C) be the maps defined for у = (g, x) by г'(у) = jc, i(y) = g.jc. The
composition (g, x)(g', x1) is defined iff χ = g'.x' and is equal to (gg', x'). For each
object χ e X, we define 1л = (1, χ). These data define a category denoted GxX. The
connected components of G κ Χ are the subcategories associated with the restriction
of the action of G to the orbits.
There are two extreme cases. If X is a single point, then G κ Χ can be identified
to the group G. If G is the trivial group, then G κ Χ is reduced to its set of units X.
(2) Product of two categories. Given two categories С and С we can consider
their product С х С as a category: the set of objects is 0(C) χ C(C'); the maps
i,t:CxC'^ C(C)xC(C')send(y, y')to(/(y), i(y'))and(t(y), t(y')) respectively;
and the composition (yi, γ()(γ2, γ{), whenever defined, is equal to (У1У2, у/Уг')·
(3) A small category С is a scwol (i.e a small category without loops) if and only
if every invertible element of С is a unit and every element of С with the same initial
and terminal vertex is a unit.
A.3 Functors and Equivalence, к functor φ from a category С to a category С is
a map φ : С -» С together with a map OiC) -> C(C') (also denoted φ) such that

Appendix: Fundamental Groups and Coverings of Small Categories 575
(1) for yeC, 0(i(y)) = ί{φ{γ)) and 0(i(y)) = i(0(y)),
(2) for composable γ, у' e С, φ(γγ') = φ(γ)φ(γ'),
(3) fora e 6>(C), 0(1σ) = 10(σ).
A functor φ : С —> С is said to be an isomorphism if there exists a functor
ф~1 : С -> Csuch that 0_10 is the identity of С and 00" 4s the identity of С'.
Two functors 0o, 0i : С —> С are homotopic if for each object σ € С there is
an invertible element ка e C^i((j) Λ(σ) such that for each у e С we have 0i(y) =
/С((у)0о(у)/с^Д. If for some σο e (9(C) we have 0ο(σο) = 0ι(σο), and if κσο can be
chosen to be 1ф(а„), we say 0o and 0i are homotopic relative to σο.
A functor 0 : С —» С is an equivalence if there is functor -ψ : С —*■ С such that
i/f 0 is homotopic to the identity of С and φψ is homotopic to the identity of С In
that case we say that С is equivalent to С
A.4 Proposition. For each category C, there is a subcategory Co £Ξ С ^ис/г ί/ϊαί ί/ге
inclusion l : Co —> С й аи equivalence and Co « minimal in the sense that if γ e Co
й invertible, then i(y) — i(y). Аиу /wo ^ис/г subcategories of С are isomorphic.
Proof. In C(C) we consider the following equivalence relation: τ ~ σ if there is an
invertible element γ such that г(у) = σ and i(y) = τ. For each object σ € С, choose
a representative σο in its equivalence class and an invertible element γσ e CCTo σ;
choose yCTo = 1 σο. The set of objects of the subcategory Co is the set of representatives
of the equivalence classes, and for σο, το e C(Co) we have (Co)r0,CTo = ^τ0,σ0· Let
ψ : С -» Co be the functor mapping у е С to У(У)У У,{"у') е Co. Then ψ t is the identity
of Co and ιψ is homotopic to the identity of C, the homotopy being determined by
the choices γσ. We leave to the reader to check that other choices would lead to
isomorphic subcategories. D
A.5 Example. Let G be a group acting on a scwol X as in (1.11). Associated to such
an action is a category denoted Gx X whose set of objects is V(X) and whose set of
morphisms consists of pairs (g, a) e G χ X with i((g, a)) = i(a), t((g, a)) = g.t(a);
the composition (g, a)(h, β), whenever defined, is equal to (gh, (h~l .α)β). Let G(y)
be the complex of groups over the scwol У = G\X associated to this action (as in
2.9(1)) with respect to some choices σ and ha. Let CG(y) be the category associated
to G(y) as in (2.8). Consider the map that sends (g, a) e CG(y) to the element
(g, a) e G к X, where a is the unique element of X such that /(a) = i(a) and
p(a) = a. This map gives an inclusion of CG{y) into G κ Χ, and in this way we
identify CG(y) to a minimal equivalent subcategory of G κ Χ.
We now characterize categories associated to complexes of groups.
A.6 Proposition. A small category С is associated to a complex of groups by the
construction of (2.8) if and only if it satisfies the following conditions:
(1) For each object σ of С, the subcategory Ca<a is a group (which we denote Ga).

576 Chapter III.C Complexes of Groups
(2) For each pair of objects (σ, τ), if γ e CT(J, h e Ga and g e Gx, then the
equality gy = γ implies g = \τ, and the equality γ = yh implies h = Ισ.
(3) For each γ e CT?(J and h e Ga, there is an element ψγ(Κ) e GT such that
ψγ(Κ)γ — yh. (By (1) and (2), ψΥ(Κ) is unique and the map ψγ : Ga —> Gx is
an injective homomorphism of groups.)
(4) If У € С is invertible, then γ e G-,(Yy
Proof. These four properties are clearly satisfied if С is the category associated to
a complex of groups. Conversely, assume that С satisfies properties (1) to (4). Such
a category has a natural quotient У which is a scwol. The set of objects У(У) of
У is equal to the set of objects of C. The elements of У are equivalence classes of
elements of C: two elements γ, γ' e С are equivalent if i(y) = i(y'), t(y) = t(y')
and y' = gy for some g e Gi(y). Let ρ : С —» У be the map associating to an
element γ its equivalence class. If we define the composition of two classes to be
the class of the composition of elements in these classes, whenever defined, then ρ
is a functor. Condition (4) implies that У is a scwol.
For each edge a e Е(У), choose a representative a in the class a. Let G(3O be the
complex of groups over У given by the groups Ga, the homomorphisms ψα = ψα
and the elements ga<b e Gi(a) that are uniquely defined by the equality ga,bCib = ab in
C. Then С is clearly isomorphic to the category associated to G(y): the isomorphism
sends (g, a) to ga. Another choice of representatives would give a complex of groups
deduced from G(3O by a coboundary (in the sense defined in (2.1)). D
A.7 Definition of a Pre-Complex of Groups. A category С satisfying conditions
(1), (2) and (3) of (A.6) is called a pre-complex of groups.
A.8 Example. Let G be a group acting on a category without loops X. The category
G κ Χ defined in (A.5) is a pre-complex of groups.
A.9 Proposition. If a category С is a pre-complex of groups, then it is equivalent to
the category associated to a complex of groups, and this complex of groups is unique
up to isomorphism.
Proof. The minimal subcategory Co of С constructed in (A.4) satisfies the conditions
of (A.6), and is therefore the category associated to a complex of groups. D
The Fundamental Group
A.10 C-paths
We wish to define a notion of combinatorial path in a category C. To this end,
we associate two symbols y+ and y~ to each element γ e C. The set of symbols
y+, y~ with у е С is denoted C*. Given e e C*, we define its initial object /(e) and
its terminal object t(e) by the formulae:

Appendix: Fundamental Groups and Coverings of Small Categories 577
i(Y+) = t(y), i(y+) = i(y) and i(y ) = i(y), t(y ) = i(y).
For e = γ+ (resp. y~), we define e_1 = y~ (resp. y+).
A path in С joining an object σ to an object τ is a sequence с = (e\,..., e{),
where each ey e C±, t(ef) = i{ej+\) for 7 = 1,... ,k — 1, and i(ei) = σ, i(e^) = τ.
The initial object i(c) of с is σ, and its terminal object t(c) is τ. If σ = τ, then
we also admit the constant path at σ (with к = 0). Note that С is connected if and
only if for all σ, τ e 0(C), there is a path joining σ to τ. If c' = (e\,..., e'K) is a
path in С joining σ' = τ to τ', then one can compose с and c' to obtain the path
с * с' = (β[,..., вк, е\,..., е'к) joining σ to τ', called the concatenation of с and
c'. Note that this composition is associative. The inverse of the path с is the path
c~l = (e'[,..., e^) joining τ to σ, where ej = e^J+l.
If i(c) = t(c) = σ, then с is called a loop at σ.
A.ll Homotopy of Paths
Let с = (ei,..., e^·) be a path in С joining σ to τ. Consider the following three
operations on c:
(1) Assume that for some j < к, we have ey = >j+ and ey+i = v.^ (resp. ey = y~
and ey+i = X^i). Then the composition YjY]+\ (resp. Yj+iYj) is defined and we
get a new path c' in С by replacing the subsequence ey, e,+1 of с by (γ/γ]+ι)+
(resp. Οί+ι)})").
(2) Assume that for some j < к, we have ey = еГ1,. Then we get a new path c' by
deleting from с the subsequence ey, ey+i.
(3) Assume that for some j, the edge ey is associated to a unit (i.e. ey = 1* for some
object p). Then we get a new path c' by deleting ey.
If с and c' are related as in (1),(2) or (3), then we say that they are obtained
from each other by an elementary homotopy (thus we implicitly allow the inverses
of operations (1) to (3)). Two paths joining σ to τ are defined to be homotopic if one
can pass from the first to the second by a sequence of elementary homotopies. The
set of homotopy classes of paths in С joining σ to τ is denoted η ι (С, σ, τ). The set
7Г](С, σ, σ) will also be denoted it\{C, σ).
Note that if с joins σ to τ, then the path с * c~' is homotopic to the constant path
at σ and that c_1 * с is homotopic to the constant path at τ. The homotopy class of
the concatenation с * с' of two paths depends only on the homotopy classes of с and
d.
Another equivalent way of defining homotopy is to introduce the group FC with
presentation (С* | 71) where the relations 1Z are
(γγ')+ = γ+γ'+, V(y,y')eC®
(y+)-'=y-, Wye С
Two paths с = (ei,..., e*) and c' = (e\,..., e'^) joining σ to τ are homotopic
if and only if the elements of FC represented by the words e\ ... e* and e\ ... e'k
are equal. In this way we identify ?Г|(С, σ, τ) to a subset of FC, in such a way that

578 Chapter Ш.С Complexes of Groups
the homotopy class of the concatenation of two paths is the product in FC of the
homotopy classes of those paths.
A.12 The Fundamental Group. The set Я\{С, σο) of homotopy classes of loops at
σ0, with the law of composition induced by concatenation of loops, is a group. It is
called the fundamental group of С at σ0.
If с is a path in С joining σ to τ, then the map associating to each loop / at σ the
loop c_1 * / * с induces an isomorphism of Tt\{C, σ) onto Tt\{C, τ). A category С is
simply connected if it is connected and π\ (С, σ) is the trivial group for some (hence
all) σ e 0(C).
Let φ : С -> С be a functor. For each у e C, we define φ(γ+) = φ(γ)+ and
Φ(Υ~) = (Ф(У)Т'· And if с = (et,..., ek) is a path in С joining σ to τ, its image
under φ is the path ф(с) = (ф{е\),..., $(efc)) in С joining Φ(σ) to Φ(τ)- The maP
ci-> 0(c) induces a homomorphism
πχ(φ, σ0) : 7ri(C, σ0) -> щ{С, φ(σ0)).
In fact this homomorphism is the restriction of the homomorphism
Έφ : FC -► FC
mapping each generator γ± of FC to the generator ^(у)* of FC.
If 0o, 0i : С —> С are functors such that 0ο(σο) = 0ι(σο) and if 0o and φι
are homotopic with respect to σο (see A.3), then the homomorphisms induced by
them on 7Ti(C, σ0) are equal. Indeed if the homotopy is defined by the family of
elements (/сст)стео(С) and we write ea = к + e C/=t, then for every e e C*, the paths
(e,-(e), фо(е), e,(e)) and (0i (e)) are homotopic, so in particular the images of each loop
at σο are homotopic. This shows in particular that the inclusion into С of a minimal
equivalent subcategory Cq of С induces an isomorphism on fundamental groups.
If С is the category associated to a complex of groups G(y), then πχ (G(3^), σ) as
defined in (3.5) is canonically isomorphic to it\ (C, σ). Indeed there is a natural
isomorphism FG(y) -> FC that restricts to an isomorphism K\{G(y), σ) -> it\{C, σ);
this is obtained by sending each g e Ga to (g, 1σ)+ e C± and a± e ^(У)* to
A.13 A Presentation of the Fundamental Group. Assume that the category С is
connected. Consider the graph \C\^ whose set of vertices is 0{C) and whose set of
1-cells is C; an element γ e Cr>(J is considered as an edge joining the vertices σ and
τ. Let Τ be a maximal tree in |C|(1).
Let tti (С, Т) be the quotient of the free group on C± by the relations:
(γγ'Ϋ = y+y'\ (y+)"' = Y~ Vy, γ' eC and y+ = 1 Vy e T.
Then тт\{С, Т) is canonically isomorphic to Tt\{C, σο).
Proof. The proof is the same as the proof of 3.7.
D

Appendix: Fundamental Groups and Coverings of Small Categories 579
A. 14 Remark. For any category С one can construct a classifying space ВС which
is the geometric realization of the nerve of С (see [Qui73] and [Se68]). ВС is a cell
complex whose set of 0-cells is 0{C) and whose l-skeleton is the graph |C|(1) defined
above. The fundamental group it\{BC, σο) is naturally isomorphic to Л\(С, σο). The
covering spaces of ВС are the geometric realizations of the nerves of the coverings
of С as defined below.
Covering of a Category
A.15 Definition. Let С be a connected category. A functor φ : С —» С from a
category С to a category С is a covering if for each object σ' of С the restriction of
φ to the subset of morphisms that have σ' as their terminal (resp. initial) object is a
bijection onto the set of morphisms of С with terminal (resp. initial) object φ{σ').
It follows from A. 17 that φ is surjective. As in remark 1.9(3), if when С is not
connected, one could define a covering φ : С —> С as a functor which is surjective
and satisfies the above condition.
A.16 Examples
(1) For a connected category without loops X, the notions of coverings defined
in (1.9) and (A.15) are the same.
(2) A covering of the category CG(y) associated to a complex of groups G(y)
is in general not associated to a complex of groups, but it is a pre-complex of groups
(A.7).
(3) Let G be a group acting on a connected scwol X and let G be the extension
of G by the fundamental group of X acting on the universal covering Л' of Л' as in
(1.15). Then the natural functor GxX^-GkX is a covering.
A.17 Proposition (Path Lifting). Let φ : С -> С be a covering of the connected
category С Let σ e 0(C) and σ' e 0(C) be such that φ(σ') = σ. Any path in С
starting at σ can be lifted uniquely to a path in С starting at σ'. Moreover, if two
paths issuing from σ' project by φ to homotopic paths in C, then the paths are also
homotopic in C. Thus φ induces an injection of7ti(C, σ') into 7i\(C, σ).
Let Co be a connected category and fix σο e 0(Cq). Let φχ,φι : Co —» С be two
functors such that φ ο φι = φ ο φι and φι(σο) = $2(σο). Then φι = φ%.
Proof. The first part follows directly from the definition of a covering and of homo-
topy, and the second part follows from the first. D
A.18 Definition. Let С be a connected category. A covering ф' : С —» С, where С
is connected, is a universal covering if it satisfies the following condition: for every
covering ф" : С" -> С and every σ' e C(C') and σ" e C(C') there is a functor
ψ : С -> С" such that φ" ο ψ = φ' and ψ{σ') = ψ(σ").

580 Chapter Ш.С Complexes of Groups
It follows from the above proposition that ψ is unique, and therefore the universal
covering ф' : С —» С is unique up to isomorphism.
A.19 Proposition. A universal covering ф' : С —*■ С of a connected category
С always exists and it is necessarily simply connected. Conversely, any covering
φ" : C" —> С such that C" is simply connected is universal.
Proof Choose σ0 e 0{C). We define a category С whose set of objects 0{C) is the
set of homotopy classes [c] of paths issuing fromao- The functor φ' will map [c] to the
terminal object of c. The set of elements of С with initial object [c] are pairs (у, [с]),
where у e С with ί{γ) — ф'([с]). The terminal object of (у, [с]) is the homotopy
class of the path с * γ~. The object [с] is identified to the pair (1ф'([С)), [с]). The
composition (y', [c'])(y, [с]) is defined if [d] = i((y, [c])) and is equal to {γ'γ, [с]).
The functor φ' maps (y, [с]) to у. It is straightforward to check that С is a connected
category and that ф' : С —> С is a covering. Let σό denote the object of С which is
the homotopy class of the constant loop at σο.
We shall prove that φ' is a universal covering. Let ф" : С" —» С be a covering.
Let σ£ e 0(C") be such that 0"(σο') = σ0. Let т/г : С(С') -* C(C") be the map
associating to each object [c] the terminal object of the path in C" issuing from σό'
whose projection by φ" is c. We extend this to a functor ψ : С —> С" by mapping
(у, [с]) to the unique element y" e C" such that i(y") = ф"([с]) and φ"(γ") = у.
This functor maps the homotopy class of the constant path at σο to σ^. (If we want
ψ to map a certain object τ' to a preferred object τ" with φ'(τ') = φ"{τ") = τ, then
it is sufficient to choose σ^ appropriately: choose a path d in С joining τ' to σ^, then
define σό' to be the terminal object of the lifting of ф'{с') with initial object τ".)
To see that С is simply connected, note that the map associating to the homotopy
class of a loop с at σο the terminal point of the lifting of с with initial object σό gives
a bijection from π\{€, σ0) to φ~ι(σο), from which it follows (A.17) that every loop
at σό is homotopic to a constant loop.
If C" is simply connected, then it is easy to see that ψ is a bijection. Since
ψ~ιψ(σό) = σ'ύ and τ//-~'(τ//-(σό')) = σό, it follows from (A. 18) that ψ is an
isomorphism. D
A.20 Examples
(1) Suppose that the category С is just a group G with a single object identified
to the unit element of G. Then its universal covering is the category С whose set of
objects is G and whose set of elements is G χ G, where for each (g, x) e G χ G
we define i(g, χ) = χ and t(g, x) = gx; the composition (g, x)(h, y), when defined,
is equal to (gh, y).
(2) Let G be a group acting on a connected scwol X. One can check that the
fundamental group π\(β κ Χ, σο) of the associated category is isomorphic to the
extension G of G by ττι (Χ, σο) described in (1.15).

Appendix: Fundamental Groups and Coverings of Small Categories 581
A.21 The Monodromy of a Covering
Let φ : С —> С be a covering. The fibre of φ over σ e 0{C) is the set Fa =
Φ~\σ) с C(C'). For у е С with i(y) = σ, let FY : F,(y) -* Fi(y) be the map
associating to each σ' e FCT the terminal object of the unique element γ' of С such
that φ(γ') = γ and г(у') = σ'. This map is a bijection because φ is a covering.
Moreover, for composable morphisms (yi, уг) € C(2) we have Fyiy, = FYlFn, and
Fio is the identity of Fa. In other words F can be considered as a functor from the
category С to the category whose elements are bijections of sets.
We can reconstruct С and φ : С —> С from the functor у м> Fy, as follows.
The set 0{C) is the disjoint union of the Fa, σ e 0(C), and φ : 0{C) -> 0{C) is
the projection to the index set of the disjoint union. To recover the elements of С we
consider the set С xo(C) 0(C) of pairs (y, σ') e С χ C(C') such that г(у) = </>(σ')·
This becomes a category with set of objects C(C') if we define г((у, σ')) = σ'
and i((y, σ')) = FY(o'); the composition is (yi, σ[){γ2, σ'ι) = (γ\Υι, σ'ι), when
defined. The functor С х<э(С) О (С) —> С sending (у, σ') to у is a covering which
is equal to φ if we identify С with С Xo(q 0{C) by the map sending y' to the pair
(0(У'), »(У'))·
The monodromy of the covering 0 is defined as follows. For each e e C*,
define Fe : Fi(e) -> Fi(e) to be FY if e = y+ and F~' if e = y~. For each path
с = (ei,..., e*) joining σ to τ, define Fc = Fei · · · Fet : Ft -^- Fa. (Note that the
unique path in С that issues from σ' e Fa and projects to с joins σ' to F~'(a')·)
If the concatenation с * с' is defined, then F^ = FcFd and Fc-i = F~'. If с and
c' are homotopic paths, then Fc = Fc>. In particular the map associating to each loop
с at σο the map Fc : Fao —» FCTo givesahomomorphismMfrom7ri(C, ao)to thegroup
of bijections of Fao. This homomorphism is denoted Μ and is called the monodromy
representation of φ at σο. The image M{ji\ (C, σο)) is called the monodromy group of
φ at σο.
A.22 Proposition. Let С be a connected category and let φ : С —> С be a covering.
Let σ0 e С α«ύ? let σο = $(σό).
(1) The homomorphism π \(C, σ'0) —» π \(C, σο) described in (Α. 17) is infective and
its image is precisely the subgroup ofn\(C, σο) that fixes σ0 in the monodromy
action Μ described above.
(2) С is connected if and only if the fundamental group πι (С, σο) acts transitively
on the fibre Fao via the monodromy of φ.
Proof. The first part of the proposition follows directly from the definitions. Let us
prove the second part. Suppose that С is connected. Given x, у е Fao, there is a
path c' in С joining χ to у whose projection by φ is a loop с at σο. The bijection
Fc of Fao associated to the homotopy class of с by the monodromy maps у to x.
Conversely, if the monodromy group acts transitively on Fao, then С is connected:
given two objects χ e Fao and ζ e Fa, let с be a path in С joining σο to σ; the map
Fc : Fa —» Fao maps ζ to some у e FCa\ by hypothesis there is a loop c' such that /y
maps у to jc; therefore the path in С that issues from χ and has projection с' * с joins
χ to z. □

582 Chapter Ш.С Complexes of Groups
Given two actions of a group G on sets Xi and X2, one says that a bijection
f :X\ —> X2 intertwines the actions if g.f(x) = f(g-x) Vg e G, Ух e Xi.
A.23 Proposition. Let С be a connected category and let ф' : С —» С and φ" : С" —»
С fee /wo coverings of С. Assume that there is a bijection /0 from the fibre F'ao of φ'
over σο to the fibre F'^ of φ" over σο intertwining the monodromy representations
M' and M" of φ' and φ". Then there is a unique isomorphism φ : С -> С" with
φ" φ = φ' such that the restriction of φ to Fao is the given bijection /0.
Given an action ofn\(C, σο) on a set Fq, there is a covering ф' :С'^-С (unique
up to isomorphism) whose fibre above σο is equal to Fq and whose monodromy at σο
is the given action.
Proof. For each object σ of С, choose a path ca in С joining σο to σ. Let F'a (resp.
FJ) be the fibre of φ' (resp. φ") above σ. Let F'Ca : F'a -> F'ao (resp. F£ : F'^ F^)
be the bijection determined by ca. As in (A.21), we identify С with С xo(C) 0{C)
(resp. C" with С х0(С) 0{C")). Let
Given y' = (y, σ') e C, where уеС, г'(у) = 0'(σ') = σ and i(y) = τ, we define
Ф(У) = (γ,Μσ)).
We claim that the map φ : С -> С" is a functor. Clearly ϊ(φ(γ')) = φ(ϊ(γ')). Let
us check that ί{φ{γ')) = φ{ί(γ')). By hypothesis,/0 conjugates the monodromies
around the loop cT * y+ * c~':
/oF7 F' F'~' = F" F"F"~lfo.
Hence
ΚΦ{Υ')) = Κ(/σ(σ')) = (F;F"llfoF'cJ(a') =
(F"c!foF'c,Fy)(v') = (/τ^Χσ') = Φ(ί(γ')).
It follows that if the composition {γι,σ'ι)(γ2, a'i) = {У1У2, σ'2) is defined in C,
then the composition φ{{γ\, σ'ιΓίΦϋΥι, σ'ι)) is also defined in C" and is equal to
(Y\Yi, f(a'2)) = Φ((ΥιΥι, σΊ))· It is clear that φ is bijective, therefore it is an
isomorphism.
Given an action Μ of 7Ti(C, σο) on a set Fo, we construct an associated covering
ф' : С -> С as follows. The set C(C') will be the product O(C)xF0 and φ : 0{C) ->
0{C) will be the natural projection. The set С is the set С χ Fo. We define г((у, χ)) =
(г'(у), χ) and r((y, x)) = (i(y), M([c]).je), where с is the loop ci(y) * y+ * c,^. Given
(yi, jci ) and (yi,xi) in С such that i((yi, x\)) = i((y2, xi)), the composition is defined
to be equal to (γχΥι,χι)· Note that i((yiy2,*2)) = i((yi.*i)), because the loop
■'(/ι η) *s nomotopic to the loop c((yi) * /|+ * c,(y| ^ * c((n) * y2+ * c,·^.
This shows that С with this partially defined law of composition is a category and
that φ is a covering with the prescribed monodromy (if we identify the fibre above
σ0 to Fo by the map (σο, χ) π-* χ). □

Appendix: Fundamental Groups and Coverings of Small Categories 583
The Relationship with Coverings of Complexes of Groups
A.24 Proposition. Let G(3O be a complex of groups over a connected scwol У and
let С = CG(y) be the associated category.
(1) Let φ : G(y') —» G(3O be a covering ofG(y) in the sense of (5.1). There is a
canonical covering ф' : С —» С = CG(y) and an inclusion λ : CGiy') —» С
which is an equivalence, such that φ = φ'λ.
(2) Conversely, if φ' : С —> С = CG(y) is a covering, then С is a pre-complex of
groups, and for any inclusion λ : CG(y') —> С which is an equivalence, the
composition φ'λ : CG(y') —» CG(y) is associated to a covering φ : G(y') —>
GOO·
Proof. (1) Let φ = {φσ,,φ{α')) : G(y') -> G(3O be a covering over a morphism
/ : У -> У.Let φ : CG(y') -> Св{У) be the associated functor. For each σ e У(У),
let Fa be the fibre over σ with the action of Ga as in 5.5. Every element of CG(3O is of
the form у = (g, a), where a e У and g e G^ay, define Fy = FgFa : Fi(o,) -» Ft(ft)
using the notations of 5.11 (if α = 1 σ, then Fa is the identity of Fa). For composable
elements y, y' of CG(3O> we have Fyy> = FyFY'. Let ф' : С —> С be the associated
covering as in A.23: the set of objects C(C') is the disjoint union of the fibres Fa and
С is the set of pairs (y, x) e CG(3O xo(C) C(C) with χ e F;(y); the functor φ' maps
(y, jc) to y.
We now define the inclusion λ : CG(y') -* C. Let y' = (g', a') e Св(У),
where α' e У with ι(α') = σ', ί(α') = τ' and g' e Ga.. ThenX(y') = (φ(γ'), λ(σ')),
where λ(σ') = фа>{Са,) e ΰσ1φσ,(βσι) с /γ(σ-). То show that λ is a functor, the
main point to check is that i(X(y')) = X(i(y')). We have
ί(λ(γ')) =/>(/)(λ(σ')) = ΦΑ8)Φ{θί')ΡΑΦο'{Οσ,)
= φτ,(8')φτ,(Οτ,) = φτ.(βτ.) = ΚΚυ'))·
Thus if у,', у2' е CG(y') are composable, then λ{γ[) and X(y2') are composable and
ΗΥΪΥΪ) = (Φ(Υ{ΥΪΙ λ«/2» = (ΦίΥ'ΜΥίΧ λ(ΐ(^))) = λ(Χι')λ(^).
Part (2) essentially follows from the proof of (5.8). Indeed let ф' : С -> С =
CG{y) be a covering; for σ e У(У) let Fa = φ~' (σ), and for у е С let Fy : F!(y) ->
F,(Y) be as defined in A.21. The elements у such that г(у) = i(y) = σ form a group
isomorphic to Ga, and in this way we get an action of Ga on Fa. If у = (1 ί(0), α), we
put Fy = Fa; for (a, fr) e £(2)СУ), we have F0Ffe = ga,bFab. Let 0 : G(y') -> G(^)
be the covering over/ : У —> У constructed in (5.8). The functor λ : CGty) —» С
maps y' to the pair (φ(γ'), l„-)e CG(3O хор) O(C') = C, where σ' is the chosen
representative of г(у') in the G/(CT')-orbit. D

Chapter \\\.Q Groupoids of Local Isometries
The purpose of this chapter is to prove a general result concerning the developability
of groupoids of local isometries. We shall show that if such a groupoid Q is Hausdorff
and complete (in a suitable sense, 2.10), and if the metric on the space of units of Q is
locally convex, then Q is equivalent to the groupoid associated to the proper action of
a group of isometries on a complete geodesic space whose metric is (globally) convex
in the sense of (II. 1.3). This result unifies and extends several earlier developability
theorems, as we shall now explain.
Groupoids of local isometries appear naturally in several contexts, for instance
as holonomy groupoids of Riemannian foliations and as the groupoids associated to
complexes of groups over MK-polyhedral complexes (C.4.16 and 4.17). In the first
case, the developability theorem which we prove (Theorem 2.15) generalizes a result
of Hebda [Heb86], and in the second case it generalizes results of Gromov [Gro87,
p. 127-128], Gersten-Stallings [St91], Haefliger [Hae90,91], Corson73 [Cors92] and
Spieler [Spi92]. In its outline, our proof of Theorem 2.15 follows the Alexander-
Bishop proof of the Cartan-Hadamard Theorem, which was explained in Chapter
II.4. The proof of our main lemma (4.3) is also adapted from their proof.
In order to smooth the reader's introduction to the concepts of covering,
equivalence and developability in the general setting of etale groupoids, we shall dedicate
section 1 of this chapter to a discussion of these concepts in the more familiar context
of differentiable orbifolds. Such an orbifold is said to be developable if it arises as
the "quotient" of a differentiable manifold Μ by the faithful, proper action of a group
of diffeomorphisms. We shall use the notion of covering to prove that an orbifold
with a geometric structure (for instance an orbifold of constant curvature) is always
developable; this was first proved by Thurston [Thu79].
The developability of complete Riemannian orbifolds of non-positive curvature
is a deeper theorem; it is closely related to the Cartan-Hadamard theorem and was
first discovered by Gromov. This result is subsumed by our main theorem (2.15), and
although we do not give a separate proof in the special case of orbifolds, we advise
the reader to keep this special case in mind while reading the proof.
In section 2 we define the basic objects of study in this chapter: etale groupoids,
equivalence, and developability. We also define Hausdorff separability and
completeness for groupoids of local isometries. By the time that we reach the end of section 2,
73 The results of Gersten-Stallings [St91 ] and Corson [Cors92] also apply to certain non-metric
2-complexes where Theorem 2.15 does not apply.

1. Orbifolds 585
the reader will be able to understand a precise statement of the developability theorem
(2.15).
In section 3 we introduce the notions of fundamental group and covering for
etale groupoids. Our approach to the fundamental group of a topological groupoid
Q is parallel to the usual description of the fundamental group of topological spaces
in terms of homotopy classes of continuous paths. We first define the notion of an
equivalence class of Q-paths (a notion which appears already in [Hae58]), these
classes play the role that continuous paths do in the topological case. We then define
the notion of homotopy of S-paths, and define the fundamental group to be the group
of homotopy classes of S-loops at a base point. In both sections 2 and 3 we illustrate
the main ideas with exercises and examples.
The proof of the main theorem is given in section 4. In order to follow the proof,
the reader will need to understand the various definitions in sections 2 and 3, and the
basic facts established in (2.14), (3.14) and (3.19).
1. Orbifolds
The notion of orbifold was introduced by Satake ([Sat56] and [Sat57]) under the
name of V-manifold. It was rediscovered in the seventies by Thurston [Thu79] who
introduced the term "orbifold". Thurston also defined the notion of covering of
orbifolds and derived from it the notion of fundamental group of an orbifold. In
this section we shall consider coverings of orbifolds, but postpone discussion of the
fundamental group to section 3 where we work in the more general framework of
etale topological groupoids. (As indicated above, our treatment of the fundamental
group, unlike Thurston's, is based on an appropriate notion of homotopy of paths.)
Basic Definitions
1.1 Definitions. A (differentiable) orbifold structure74 of dimension и on a Hausdorff
topological space Q is given by the following data:
(i) An open cover (VOie/ of Q indexed by a set /.
(ii) For each i e /, a finite subgroup Г, of the group of diffeomorphisms of a
simply connected и-manifold X,- and a continuous map qt : X,- —» V;, called a
uniformizing chart, such that qt induces a homeomorphism from ГДХ,- onto Vj.
(iii) For all x-, e X,- and xj e Xy such that #,·(*,·) = qj(xj), there is a diffeomorphism h
from an open connected neighbourhood W of jc, to a neighbourhood of x/ such
that qj oh = q^w- Such a map h is called a change of chart; it is well defined
up to composition with an element of I) (see exercise 1.5(1)). In particular if
i = j then h is the restriction of an element of Г,.
74 To simplify, we shall consider only differentiable orbifolds in this section, because the
analogue of exercise 1.5(1) is more difficult to prove in the topological case (cf. remark
1.6(2)).

586 Chapter Ul.Q Groupoids of Local Isometries
The family (X,·, g,)ie/ is called an atlas (of uniformizing charts) for the orbifold
structure on Q.
By definition, two such atlases of uniformizing charts (X,-, #,),·<=/, and (Xh #,),<=/,
define the same orbifold structure on Q if (X,-, g,)/,u/2 satisfies the compatibility
condition (iii).
Note that if all of the groups Γ, are trivial (or more generally if they act freely on
Xi), then Q is simply a differentiable manifold.
The orbifold structure on Q is said to be Riemannian (complex analytic, etc)
if each X, is a Riemannian (complex analytic, etc.) manifold and if the changes of
charts are accordingly Riemannian isometries (complex analytic, etc.).
If the orbifold structure on Q is Riemannian, there is a natural pseudometric on
Q, namely the quotient of the Riemannian length metric on the disjoint union of the
X,. The assumption that Q is Hausdorff implies that this pseudometric is actually a
metric and induces the given topology on Q. This metric will be called the quotient
metric on Q.
1.2 The Pseudogroup of Changes of Charts. Let X be the disjoint union of the X,.
We identify each X; to a connected component of X and we call q the union of the
maps qi. Any diffeomorphism h of an open subset U of X to an open subset of X
such that q = qh on U will be called a change of charts. The change of charts form a
pseudogroup75 Ti of local diffeomorphisms of X called the pseudogroup of changes
of charts of the orbifold Q (with respect to the atlas of uniformizing charts (X,, #,),<=/).
It contains in particular all the elements of the groups Γ,. If h : U —» У is a change
of charts such that U and V are contained in the same X, and if U is connected, then
h is the restriction to U of an element of Г, (see 1.5(1)).
Two points x, x1 e X are said to be in the same orbit of Ti if and only if there
is an element h e Ti such that h(x) = x1. This defines an equivalence relation on
X whose classes are called the orbits of Ti. The quotient of X by this equivalence
relation (with the quotient topology) will be denoted Ti\X. The map q : X -» Q
induces a homeomorphism from Tl\X to Q.
1.3 Developable Orbifolds
Let Г be a subgroup of the group of diffeomorphisms of a manifold Μ and
suppose that the action of Γ on Μ is proper. Let / : Af —> Q be a continuous map
that induces a homeomorphism from Γ\Μ with the quotient topology to Q. Note that
β is Hausdorff (1.8.4(2)).
75 A pseudogroup Η of local homeomorphisms of a topological space X is a collection Ti of
homeomorphisms h : U —>- V of open sets of X such that:
(1) if h : U -» V and ti : U' -» V belong to Ti, then their composition hti : ti~~'(U П
V) -> h(U П V) belongs to Ti, as does h~\
(2) the restriction of h to any open set of X belongs to Ti,
(3) the identity of X belongs to Ti,
(4) if a homeomorphism from an open set of X to an open set of X is the union of elements
of Ti, then it too belongs to Ti.

1. Orbifolds 587
The definition of a proper action ensures that one can find a collection of open
balls Xj с М such that:
(i) the subgroup Γ, = {y e Γ | y.X,- = X,} is finite and if y.X,- ПХ, φ 0, then
Υ e Γ,,
(ii) the open sets V,- = /(X) cover β.
Let g, be the restriction of/ to X,·. Then the atlas (X,, q{) defines an orbifold
structure on Q. This structure depends only on the action of Γ and not on the choice
of the open sets X,; it is called the orbifold quotient of Μ by the proper action ο/Γ.
We shall say that an orbifold structure on Q is developable {good in the terminology
of Thurston [Thu79]) if it arises from an action in this way.
1.4 Examples
(1) Examples of Orbifold Structures on the 2-Sphere S2. Identify S2 with CU{oo};
let Vo = С С S2 and let Voo = S2 \ {0}. Given two positive integers m, n, we define
an orbifold structure on S2 using the two uniformizing charts go : С —> Vo and
qoo : С —> Voo defined by qo(z) = z"1 and qooiw) = 1/w". These two charts define
a complex orbifold structure denoted S2m n) on S2: the group Го (resp. Гто) consists
of all the rotations of order m (resp. n) of С fixing 0; the changes from the chart go
to the chart q^ are local determinations of the multivalued holomorphic function
тш = (l/z)m/n.
This orbifold is developable if and only if m = n. Indeed if m = n, then the
map ζ (-> ζ"1 gives an isomorphism from the quotient of С U {oo} by the action
of the group generated by ζ н* emlmz to the orbifold S2n n). Conversely, if S2m n) is
developable, then there exists a connected manifold M, a subgroup Г of the group
of diffeomorphisms of Μ that acts properly, and a Γ-invariant continuous map / :
Mh>52 = CU {oo} inducing the orbifold structure S2m n) on S2. The restriction of/
to Μ' = Μ ν/-'({Ο, oo}) is a connected covering of С* = С \ {0} whose number
of sheets must be equal to both m and n, hence m = n.
(2) Construction of 2-Dimensional Hyperbolic Orbifolds. Consider a convex
polygon Ρ in the hyperbolic plane И2 such that the angle at each vertex υ, of Ρ is
Tt/rij, where n-, > 0 is an integer. One can define an orbifold structure on Ρ such that
the uniformizing charts are defined on open subsets of И2 and the changes of charts
are restrictions of hyperbolic isometries. One of these charts is the inclusion qo of
the interior of Ρ into P; and for each vertex v-, there is a uniformizing chart q, defined
on an open ball Β{υ,, ε) of small radius. The group Γ,· associated to this latter chart
is generated by the reflections of И2 in the sides of Ρ adjacent to υ,, and qt maps
χ e B(vh ε) to the unique point of Ρ which is in its Γ,-orbit. Similarly, if у is a point
of Ρ in the interior of a side, a uniformizing chart is defined over a ball B(y, ε) and
maps χ e B(y, ε) to itself if χ e Ρ and otherwise to its image under the reflection
fixing the side containing it. A classical theorem of Poincare (see for instance [Rh71],
[Har91], [Rat94]) asserts that the subgroup of Isom(H2) generated by the reflections
fixing the sides of Ρ acts properly on И2 with Ρ as a strict fundamental domain; it
follows that the orbifold structure defined above on Ρ is developable. This will also
be a consequence of Theorem 1.13.

588 Chapter ΙΙΙ.ί? Groupoids of Local Isometries
The same considerations apply to convex polygons Ρ in E2 and §2 with vertex
angles я/и,. Note that in these cases, however, there are very few possibilities for
such a polygon. (Exercise: List them.)
Let Q be the 2-sphere obtained by gluing two copies of Ρ along their common
boundaries. There is an obvious way to define a hyperbolic (resp. Euclidean or
spherical) orbifold structure on Q with conical points corresponding to the vertices
of P. (Exercise: Describe an atlas of uniformizing charts giving this structure.)
1.5 Exercises
(1) Let Γ be a finite subgroup of the group of diffeomorphisms of a connected,
paracompact, differentiable manifold M. Let q : Μ —> Γ\Μ be the natural projection.
Let/ : V —> W be a diffeomorphism between connected open subsets of M. Show
that if q o/ = q\v then/ is the restriction of an element g e Г.
(Hint: Start from any Riemannian metric on Μ and by an averaging process
prove the existence of а Г-invariant Riemannian metric. Using the local exponential
maps for this metric, show that one can introduce local coordinates around each
point χ e Μ such that, with respect to these coordinates, the action of the isotropy
subgroup of χ is orthogonal. Show that the open set Mo с Μ on which the action of Γ
is free is everywhere dense. The restriction of/ to У Π Mo is a Riemannian isometry,
hence by continuity/ itself is a Riemannian isometry. Given a point χ e U Π Μ0,
there is a unique g e Γ such that g.x= f(x). Then/ is equal to g on a neighbourhood
of x, hence the two isometries/ and g\u are equal.)
(2) Let Γ be a group acting properly by diffeomorphisms on a smooth paracompact
manifold M. Using a suitable partition of unity on M, show the existence of а Г-
invariant Riemannian metric on M.
(3) Show that every differentiable orbifold structure on a paracompact space Q
carries a compatible Riemannian orbifold structure.
1.6 Remarks
(l)In the definition (1.1) of an orbifold, it is common to assume that the sources
Xi of the uniformizing charts are open balls in W. This leads to the same definition
as ours. Equivalently we could have assumed that the sources X, of the uniformizing
charts g, : X-, —»■ V,· are differentiable manifolds on which the groups Γ, (not supposed
to be finite) act effectively and properly so that the maps q, induce homeomorphisms
Γ/\Χ, —»■ Vj. And one could say that a connected orbifold is developable if and only
if it can be defined by an atlas consisting of a single uniformizing chart.
(2) Topological orbifolds are defined in the same way as differentiable orbifolds:
in definition (1.1) one replaces diffeomorphism by homeomorphism everywhere. It
is still true that the changes of charts defined on a connected open set are well defined
up to composition with an element of a finite group (see 1.1 (iii) and 1.5(1)), but this
is harder to prove than in the differentiable case. A proof of this fact, communicated
to us by Bob Edwards, is based on the following basic theorem of Μ. Η. A. Newman:
if a finite group acts effectively by homeomorphisms of a connected manifold, then
the set of points with trivial isotropy is open and dense ([New31] and [Dre69]).

Coverings of Orbifolds 589
Coverings of Orbifolds
1.7 Definition of a Covering of an Orbifold. Consider an orbifold structure on
a space Q defined by an atlas {qt : X,- —» Vt} of uniformizing charts inducing
homeomorphisms ГДХ,- —> У,. We assume that theX, are disjoint; letX be the union
of the Xj and let q : X —» Q be the union of the qt. Let Ή be the pseudogroup of
changes of charts (1.2).
Let ρ : X —» X be a covering of X. On X there is a unique differentiable structure
such that ρ is locally a diffeomorphism. We assume that for each change of charts
h : U -> U' in W, there is a diffeomorphism й : p~l(U) -> p~l(U') commuting with
/j which is such that, for all h, h' e W,
Ш = hh' and h-~l = /г'.
In particular this gives an action of each group Г, on p~' (X,) that projects to the given
action of Г, on X,.
Let Ή be the pseudogroup generated by the diffeomorphisms h. (More precisely
the elements of Ή are the restrictions of the elements h to the open subsets of X and
all unions of such restrictions.) Then Η is the pseudogroup of changes of charts of an
orbifold structure on the quotient Q of X by the equivalence relation whose classes
are the orbits of H. We shall now define this orbifold structure. Let q : X —>- Q be the
natural projection. There is a continuous map ρ : Q —» Q which makes the diagram
X < X
ρ
(1-7-D 4 4
Q = H\X < Q = H\X
ρ
commutative.
The uniformizing charts of Q are the restrictions of q to the connected
components of p~~l (Xj). The group acting on such a component is the group formed by the
restriction of those y, with γ e Γ,·, that leave this component invariant. The only
point to check is that Q with the quotient topology is Hausdorff. This follows easily
from the hypothesis that Q is Hausdorff and the fact that the quotient of the open set
p~l(Xi) by the action of the finite group Γ, is Hausdorff.
A covering of the orbifold β is a commutative diagram like (1.7-1). The orbifold
Q is called a covering orbifold of the orbifold Q and the map ρ : Q —> Q is the
corresponding covering map.
1.8 Exercise. Show that, up to natural isomorphism, the notion of coverings for
orbifolds is independent of the choice of atlas for the orbifold structure.

590 Chapter Ш.^ Groupoids of Local Isometries
1.9 Galois Coverings. We maintain the notation of 1.7. Let G be a group acting
on X so that the action is simply transitive on each fibre of ρ (in other words ρ is a
Galois covering with Galois group G), and assume that the liftings h commute with
the action of G. Then we say that Q is a Galois covering of Q with Galois group G.
The action of G on X gives an action of G on Q and Q = G\Q. We shall see shortly
that this action is proper.
As the sources X,- of the uniformizing charts are assumed to be simply connected,
the covering ρ is trivial. After choosing a section X,- —> /?~"'(Χ,·), we can identify
p~x(Xi) with G χ X,; thus /j becomes the projection on the second factor, and the
action of G is by left translations on the first factor and is trivial on the second factor.
The lifting γ of у e Γ, is of the form (g, x) м> (g#>,(y)~', y.x), because γ commutes
with the action of G. Moreover у м> φ-,{γ) gives a homomorphism φί : Γ, —» G.
LetX, := {1} χ X, с ρ-1 (X,·) and let Vt := q{X{). The connected components
of X are the open sets g.X,-, indexed by (g, i) e G χ /, and the uniformizing charts
for Q are the maps g.X,- —> g.V,- which are the restrictions of q. Let Γ^,,) be the
group of diffeomorphisms of g.X,- formed by the restrictions to g.X,- of the elements
y, where у e ker#>,- (in this way Γ(5>[) is isomorphic to ker#>,)· The uniformizing
charts g.Xi —> g.V-, induce homeomorphisms Γ(^,)\§.Χ; —> g.V,.
WehaveQ = (J, ;) g.V,. The subgroup φ, (Г,) с Gleaves У, invariant. Moreover
g.ViDVi ф 0 if and only if g e φ,(Γ,). Indeed if g.V, П У, / 0, then there exists
ie V, and γ e Γ,· such that γ (χ) e у. V;, which implies that g = φ~χ (у). This shows
that the action of G on Q is proper.
1.10 Lemma. With the notations above, if the homomorphisms ψι : Γ,; —> G
associated to the Galois covering are injective, then Q is naturally a differentiable manifold
on which G acts properly by diffeomorphisms and the orbifold structure on Q is the
quotient ofQ by this action.
Proof. The restrictions of q to the open sets g.X; are homeomorphisms g.X, —> g.V,
which define on Q the structure of a differentiable manifold (because the changes
of charts are the elements of Η and therefore are differentiable). Clearly G acts
on Q by diffeomorphisms. The restrictions pi : Vt —> V; of /> to the open sets
Vi are the uniformizing charts for the quotient orbifold structure on Q. For each
i e /, let/; : X,- —» \/, be the diffeomorphism sending χ to the image under q of
the point (l,x) e X,. This map is a diffeomorphism which is φ,-equivariant, i.e.
fi(y.x) = <Pi(y).fi(x) for all у е Г,·. Indeed,
f(y.x) = q(y~l.(l, y.x)) = q{{q>i{y), χ)) = ψί(γ)Μγ)-
Moreover pt(ft(x)) = qi(x). Therefore, via the isomorphisms/;, the atlases (X,, qt)
and (V,, p^ are the same. D

Orbifolds with Geometric Structures 591
Orbifolds with Geometric Structures
1.11 (G, JO-Geometric Structure. Let G be a group acting by diffeomorphisms on
a simply connected differentiable manifold Y. We assume that the action is quasi-
analytic; this means that if the restriction of an element g e G to an open set U of Υ
is the identity of U, then g is the identity element of G.
A (G, Y)-geometric structure (or simply a (G, y)-structure) on a differentiable
manifold X is defined by a maximal atlas Λ made up of charts which are
diffeomorphisms ψ from open sets of X to open sets of Υ such that:
(i) the sources of the charts cover X,
(ii) for all ψ, ψ' e A, the change of charts ψ'ψ~λ, where it is defined, is locally
the restriction of an element of G to an open set of Y.
Given (G, y>structures on X\ and X2, with maximal atlases Αι and Аг, a diffeo-
morphism of an open subset of X\ to an open subset of X2 is called an isomorphism
of (G, y)-structures if the composition of h with any chart of A2 is a chart of Αι.
For instance, a Euclidean, spherical or hyperbolic manifold of dimension η is an
и-manifold with a (G, ^-structure, where Υ is respectively E", §", W and G is the
full group of isometries of Y. An и-manifold with an affine structure is, by definition,
a manifold with a (G, y)-structure, where Υ is the Euclidean space W and G is
the group of affine transformations of W. Interesting examples of (G, ^-structures,
where Υ is the 2-sphere §2 and G the group of Mobius transformations, can be found
in [SuTh83] (see also [Sco83]).
1.12 Orbifolds with a (G, y)-Geometric Structure. Consider an orbifold structure
on a space Q given by an atlas of uniformizing charts qt : Xt -> V,. A (G, Y)-
geometric structure on this orbifold is given by a (G, F)-geometric structure on each
Xi which is left invariant by the changes of uniformizing charts (in particular by the
groups Γ,·).
If У is a Riemannian manifold and if G is a subgroup of the group of isometries of
Y, then an orbifold with a (G, Y)-structure inherits a compatible Riemannian metric
for which the charts of the (G, y)-structure are Riemannian isometries.
Typical examples of such orbifolds are obtained by taking the quotient of Υ by
a subgroup of G acting properly on Y, or by taking the quotient of an open subset
Yo С Υ Ъу а subgroup of G that leaves Y0 invariant and acts properly on Yq. For
interesting examples of hyperbolic orbifolds, see [Thu79, chapter 11] and [Rat94].
1.13 Theorem. Let Q be a connected orbifold with a (G, Y)-geometric structure.
Then
(1) Q is developable. More precisely, there is a subgroup Г of G acting properly
on a connected manifold Μ endowed with a (G, Y)-structure such that Q with
its (G, Y)-structure is naturally the quotient of Μ by this action ο/Γ. Moreover
there is a map D : Μ —> Υ which is Y-equivariant and which is locally an

592 Chapter Ol.Q Groupoids of Local Isometries
isomorphism of(G, Y)-structures. The map D is called the development of Q
and Γ is called the holonomy group; both Ό and Γ are well defined up to
conjugation by an element ofG.
(2) Assume that Υ is a Riemannian manifold and that G is a subgroup of the group
of isometries of Υ (so the orbifold structure on Q is Riemannian). Assume also
that Q is complete with respect to the quotient metric (this is the case if Q is
compact). Then there is a natural identification of Μ with Υ under which D
becomes the identity map.
Before giving the proof of this theorem, which is very similar to the proof of
(1.3.32), we follow a digression concerning the spaces of germs of continuous maps.
1.14 Spaces of Germs. Let X and Υ be topological spaces. Consider the set of pairs
(f, x), where/ is a continuous map from an open subset U с X to Υ and χ e U. We
introduce an equivalence relation on this set: (f, x) ~ (/', x1) if and only if χ = χ1 and
/ is equal to/' on some neighbourhood of x. The equivalence class of (f, x) is called
the germ off at x. The point χ (resp. f{x)) is called the source (resp. the target) of the
germ of/ at x.
Let M(Y, X) be the set of germs of continuous maps from open sets of X to Υ
and let a : M(Y,X) -> X (resp. ω : M(Y, X) -> Y) be the map associating to a
germ its source (resp. its target). On M(Y, X) there is a natural topology, a basis of
which consists of the subsets Uf which are the unions of the germs of continuous
maps/ : U —> Υ at the various points of U. The projections α and ω are continuous
maps and α is an etale map, i.e. a maps open sets to open sets and its restriction
to any sufficiently small open set is a homeomorphism onto its image. Note that in
general M(Y, X) is not a Hausdorff space.
Consider a topological space Z, continuous maps/ : U —> Υ and/' : U' —> Z,
where U and U' are respectively open sets of X and Y, and a point χ e U such that
f(x) e U'. The germ of/'/ : f~l(U') -> Ζ at χ depends only on the germ g of/ at
χ and of the germ g' off at/(jc) and is called the composition g'g of those germs.
Let M(Z, Y)xY M(Y, X) be the subspace of M(Z, Υ) χ Μ(Υ, X) formed by pairs
(g\ g) such that a(g') — co(g). The composition of germs gives a continuous map
M(Z, Y)xY M(Y,X) -► M(Z, X).
Proof of Theorem 1.13. To a (G, F)-structure on an orbifold Q we shall associate a
Galois covering with Galois group G satisfying the hypothesis of Lemma 1.10.
We can assume that the sources Xt of the uniformizing charts qt : X, —> V, are
disjoint and are the sources of charts т/г,- : X, -> ^-(Х)) С У for the given (G, Y)-
geometric structure. Let X be the union of the X, and let q : X —> Q be the union of
the uniformizing charts. Let X be the space of germs of all the charts from X to У
defining the (G, F)-structure on X (this is an open subspace of the space of all germs
of continuous maps from open sets of X to Y). The group G acts by diffeomorphisms
on X: if g e G and χ is the germ of a chart ψ at x, then g.x is the germ of the chart
g ο ψ at χ.

Orbifolds with Geometric Structures 593
We have two projections ρ : X —> X and Δ : X —> Υ associating to each germ,
respectively, its source and target. Clearly ρ is G- invariant and Δ is both G-equivariant
and locally a diffeomorphism.
We first claim that ρ : X —> X is a Galois covering with Galois group G, i.e. G
acts simply transitively on the fibres of p. To see this, letX, be the union of the germs
of ψί at the points χ e X,-. It follows from condition (ii) in (1.11) йшХр~1(Х{) is the
union of the open sets g.X,, where g e G, and the condition of quasi-analyticity on
the action of G on Υ implies that these open sets are disjoint.
Let A : U —> У be an element of the pseudogroup Η of changes of uniformizing
charts. Such a map has a canonical lifting A : p~x{U) -> ί/ Π p~x{U')\ if i is the
germ at jc e i/ of a chart т/г for the (G, y)-structure on X, then A(i) is the germ at
h(x) of the chart ψΚ~'. We haye^A"> = A~', and if /г' : ί/' -> V" is another orbifold
change of charts, then hh' = hh' on p~l(h~l(U'). Note that each A commutes with
the action of G.
Let Ή. be the pseudogroup of local diffeomorphisms of X generated by the
elements A. Let Q be the quotient of X by the equivalence relation whose classes are the
orbits of H- Let q be the natural projection X -> Q and let ρ : <2 -> Q be the map
such that да = до.
According to (1.8), these data define an orbifold structure on Q which can be
considered as a covering of Q with covering projection ρ : Q —> Q. Moreover, as
the elements A commute with the action of G on X, we can consider Q as a Galois
covering of Q (see 1.9); the group G acts properly on Q with quotient Q.
As Δ : X —> У obviously commutes with the action of W, we get a continuous
map D : Q—*■ Υ such that A = D oq. Therefore we have the commutative diagram
X < X —^ Υ
ρ
4 4 4
Q ^— Q = П\Х ^^ Υ
ρ
It follows that the restriction of q to each of the open sets X, is a homeomor-
phism onto its image (denoted Vt с Q) because Δ is a local 18отофЬ18т. Also the
restriction of q to g.X, is а ЬотеотофЫвт onto g. V,. SinceX is locally diffeomor-
phic to Y, this implies that β is a manifold. It also implies that the homomorphisms
φι : Γ,- -> G described in (1.9) are injective. According to lemma 1.10, the orbifold
Q is the quotient of the manifold Q by the proper action of G. Moreover Q is endowed
with the (G, F)-structure for which D is locally an isomorphism of (G, y)-structures.
The connected components of Q are permuted by the elements of G. Fix a
connected component Μ of Q. Let Γ be the subgroup of G leaving Μ invariant and let
D be the restriction of D to M. As Q is connected, it is the orbifold quotient of Μ by
the action of Γ. This proves (1).
Suppose that G leaves invariant a Riemannian metric on Y. Then there is on X a
unique Riemannian metric such that all of the maps тД, are isometries. This metric

594 Chapter III.C? Groupoids of Local Isometries
is invariant by the pseudogroup Ti of changes of uniformizing charts. X also has a
unique Riemannian metric such that Δ is locally an isometry; it is invariant by the
action of G and the pseudogroup Ti. Assume that the metric on Q which is the quotient
of the metric on X associated to this Riemannian metric is complete. This metric is
the same as the quotient metric on Q associated to the action of Γ by isometries of
the manifold M. As the action of Γ is proper, it follows that the metric on Μ is also
complete. According to (1.3.28), this implies that Ό : Μ —» У is a covering. And as
Υ is assumed to be simply connected, this implies that D is an isometry. This proves
(2). D
2. Etale Groupoids, Homomorphisms and Equivalences
Etale Groupoids
2.1 Definition of an Etale Groupoid. A groupoid (Q, X) is a small category Q (see
C. A. 1) whose elements are all invertible; X is the set of its objects, identified to the
set of units of G by the map associating to χ e X the unit \x e Q. The projection
Q —> X that associates to each element of Q its initial object (resp. its terminal object)
will be denoted a (resp. ω) (these maps were denoted i and t in C). Given g e Q, the
object a(g) (resp. co(g)) will be called its source (resp. its target).
Given χ e X, the set Qx — {g e Q \ a(g) = w(g) = x] is a group called the
isotropy group of x. The subset Q.x = {y e X \ 3g e G, cc{g) — x, co(g) = y} is the
G-orbit or simply the orbit of x. The ^-orbits form a partition of X and the set of
^-orbits will be denoted G\X-
A topological groupoid (G, X) is a groupoid with a topology on G and X such that
the natural inclusion X —> G is a homeomorphism onto its image, and the projections
α, ω, as well as composition and passage to inverses, are continuous. The space of
orbits G\X is endowed with the quotient topology.
We say that a topological groupoid is etale if the maps a and ω are etale maps,
i.e. are locally homeomorphisms.
Associated to each etale groupoid (G, X) there is a pseudogroup of local
homeomorphisms of X. The elements of this pseudogroup are the homeomorphisms
h : U —> У of the form h — ω ο s, where s : U —> G is a continuous section
of α over U, i.e. a(s(x)) = x, Vjc e £/. If X is a differentiable manifold and if the
elements of the associated pseudogroup are diffeomorphisms, then (G, X) is called
a differentiable etale groupoid.
2.2 The Groupoid Associated to a Group Action. Let Γ be a group acting by
homeomorphisms on a set X. The etale groupoid (Γ κ Χ, Χ) associated to this action
is the category whose space of objects is the space X and whose space of elements is
G = Γ χ X, where Г is endowed with the discrete topology. The projections a and
ω are defined by a(y, χ) = χ and ω(γ, χ) — γ.χ. The composition (у, χ)(γ\ χ1) is
defined if χ — γ' .χ1 and is equal to (yy\ x1). The inverse of (y, x) is (y~\ Y-x)-

Etale Groupoids 595
We describe a series of other basic examples.
2.3 Examples
(1) Among the etale groupoids, we have two extreme cases, namely:
(a) X is a single point and Q is just a discrete group;
(b) X is a topological space and all the elements of G are units (equivalently, the
inclusion X —> G is a bijection). We say that this is the trivial groupoid X.
(2) The Etale Groupoid Associated to a Pseudogroup of Local Homeomorphisms.
Let Τι be a pseudogroup of local homeomorphisms of a topological space X. Its
associated etale groupoid (G, X) is the groupoid of all the germs of the elements
of Τι, with the germ topology. (This is an open subspace of the space of all germs
of continuous maps from open sets of X to X.) The natural inclusion of X into G
associates to χ e X the germ \x at χ of the identity map of X. The projections a and
ω associate to a germ its source and target, and the composition is the composition
of germs. From G one can reconstruct TL as the pseudogroup associated to G-
In general, if TL is the pseudogroup associated to an etale groupoid (G, X) (see 2.1),
then the etale groupoid of germs associated to Τι is a quotient of (G, X)- (Consider
for instance example 2.3(la).)
(3) The Etale Groupoid Associated to an Atlas of Uniformizing Charts for an
Orbifold. This is the etale groupoid (G, X) associated to the pseudogroup of changes
of charts (cf. 1.2); it determines the orbifold structure on Q if one identifies Q with
the quotient G\X of X by the equivalence relation whose classes are the orbits of G-
(4) The Etale Groupoid Associated to a Complex of Groups. Let G(y) =
(Ga, ψα, ga,b) be a complex of groups over a scwol У as defined in (C.2.1). We
construct an etale groupoid (G(y), Y) canonically associated to G(3O as follows (see
fig. C.13). We use the notations of (C.4.9, 4.13, 4.14). The space of units У will be
the disjoint union of the spaces st(cr), where σ e У(У). The groupoid G(y) will be
the disjoint union of the subspaces Gx,a of elements with sources in st(cr) and targets
in st(f). There are three cases to consider.
First case: σ = τ. In this case (Ga,a, st(cr)) will be the groupoid
(Ga к st(a), st(a))
associated to the action of Ga on st(cr).
Second case: There is an edge in Е(У) with initial vertex σ and terminal vertex τ.
In this case
Gr,a = \J Gr X St(fl),
a
where a e Е(У) with i(a) = σ and t(a) = τ. Recall that the subspaces st(a) are
disjoint in st(cr) (exercise C.4.6). The projection a (resp. ω) maps (g, x) € GT χ st(a)
to χ (resp. g.fa(*)), where/a is as defined in (C.4.14)). For this reason, it will be more
suggestive to use the notation (g,fa, x) for the element (g, x) e Gr χ st(a).
If (h, y) e Ga,a is such that* = h.y, the composition (g,fa, x)(h, y) is defined to be
(gfa(h),fa,y). If (g',-0 e ΰτ,τ withx' = g.fa(x), the composition {g', x!)(g Ja, *)

596 Chapter III.C? Groupoids of Local Isometries
is equal to (g'g,fa,x). If (a, b) e £(2)СУ) with i(b) = ρ and if (g,fa,x) e Gz.a
and (h,fb, y) € Οσ,ρ with χ = h.fb(y), the composition (g,fa, x)(h,fb, y) is equal to
(giSa(h)ga,bJab,y)-
As a set Qax is equal to Gz.a but the projections α and ω are exchanged, thus its
elements are (a priori formal) inverses of the elements of Gz.a- Thus to each element
(g,fa, x) e Gz.a is associated a symbol (g,fa, x)~l e Ga.z such thata((g,/a, x)~1)) =
g.fa(x) and w{{g,fa, x)) = x. The composition is defined accordingly. For instance, if
(g\y') e ft,r with g'y = g.fa{x\ then (g,/e,jc)-4g',y) = (*'""'*,/«. *)"'> and if
(A, jc) e £σ?σ, then (/г, x)(g,fa, x)~x — (gisa(h~l),fa, h.x)~l. Now assume that a, a'
are two distinct elements of Е(У) with t(a) — t(a') = τ and i(a) = σ, ι'(α') =
σ'- Let (g,fa,x) e ΰτ,σ and (g'Ja^x1) e Gr,a> be such that g./a(jc) = g'.faix1)-
Then {g',fa,,x')~l{g,fa,x) and (g,/a, л:)~1(#',/а<,х/) should be defined. The natural
projections from st(cr) and st(a') to \У\ map χ and xf to a point in st(a) Π st(a'); it
follows (see exercise C.4.6) that there is a unique b e £(У) such that either a' = ab
or α = a'b; exchanging the roles of a and a' we can assume that a — a'b. We
then define the composition (g',fa', x')~1(g,fa, x) as the unique element (h,fb, x) e
Gc',a such that (g,fa, x) = (g\fa'< x?)(h,fb, x), where h is defined by the equation
g'iO'(h)ga,tb = g. The composition (g,/a, x)_1 (g',/as x') is defined to be (h,fb, x)~{.
One proceeds similarly to define, for instance, the composition (g',fa>, x)(g,fa.x)~}
when i(a) = i(a').
Third case: σ and τ are distinct and not joined by an edge. In this case Gz,a is empty.
We leave the reader to check that the law of composition defined above is
associative (one needs to refer to (C.2.1) and (C.4.14)).
(5) The Holonomy Groupoid of a Foliation. A foliation Τ of codimension и on a
manifold Μ of dimension m can be defined by a family of maps/; : £/, —> K", where
{{/,},·,=/ is an open cover of M. The maps/), called local projections, are submersions
from Uj onto open sets V, of W with connected fibres (i.e. f[~\x) is a connected
submanifold of U-, for each χ e Vi). The following compatibility condition must be
satisfied: for each у e ί/,· Π Ц there is an open neighbourhood Щ с {/, Π £7, and a
homeomorphism Ajj ~-МЩ) -> Ж^) such that^ = Ajj o/)- on i/f.. Two such families
of submersions define the same foliation Τ if their union still satisfies the analogous
compatibility conditions. The leaves of Τ are the (m — «)-dimensional submanifolds
of Μ which are the connected components of Μ endowed with the topology whose
basis is the set of open sets in the fibres of the submersions/;.
Let X be the disjoint union of the open sets V,-. The holonomy pseudogroup of T,
defined by the family of submersions {/y : f/, —> V, С X}, is the pseudogroup of local
homeomorphisms of X generated by the germs of the Ц-г The holonomy groupoid of
Τ is the groupoid of germs associated to its holonomy pseudogroup. The holonomy
groupoid associated to a different family of compatible submersions that defines the
same foliation will be equivalent to the previous one in the sense of the definition
that follows.

Equivalence and Developability 597
Equivalence and Developability
2.4 Homomorphisms and Equivalences of Groupoids. A homomorphism (φ,/)
from an etale groupoid (G, X) to an etale groupoid (Q\ X') consists of a continuous
functor φ : Q —> Q' inducing a continuous map/ : X —» X'. We say that (φ,/) is an
etale homomorphism if/ is an etale map.
(φ,/) is an equivalence if/ is an etale map and if the functor φ is an equivalence
in the sense of (C.A.3), i.e.
(1) for each χ e Χ, φ induces an isomorphism from Qx onto Gt,xy
(2) f-.X^-X' induces a bijection G\X -* G'\X' of the orbits sets.
In this case we say that the etale groupoids (G, X) and (G', X') are equivalent.
This generates an equivalence relation among etale groupoids.
If (G, X) and (G', X') are differentiable etale groupoids (see2.1), ahomomorphism
(φ,/) : (G, X) —> (G1, X') is called a differentiable equivalence if it is an equivalence
and/ is locally a diffeomorphism. The equivalence generated by this relation is called
differentiable equivalence.
. 2.5 Remark. One can check (see exercise 2.8(1)) that two etale groupoids (G, X)
and (G\ X') are related by the above equivalence relation if and only if there is an
etale groupoid (£", X") and etale homomorphisms (φ,/) : (Q", X") -> (G, X) and
(ψ'J') ■ (G", X") -> (G\ X') which are equivalences.
There is a more general way of describing directly an equivalence from (G, X)
to (G', X'). We shall explain this through a series of exercises (see 2.8(3)).
2.6 Definition of Developability. An etale groupoid (G, X) is developable if it is
equivalent to the groupoid (Γ χ X, X) associated to an action of a group Γ by home-
omorphisms of a space X (see 1.3).
2.7 Examples
(1) Let (G, X) be an etale groupoid and let U с X be an open subset meeting
all the ^-orbits. Let (G\u, U) be the restriction of (G, X) to U, i.e. the subgroupoid
formed by the elements with source and target in U. Then the natural inclusion
(G\u, U) —> (G, X) is an equivalence.
(2) Let Г be a group acting freely by homeomorphisms on a topological space X so
that the natural projection p:X^- Y\X = X is a covering map. Let π:Γχϊ^ϊ
be the map sending (γ, χ) to p(x) = p(y.x)- Then the etale homomorphism (π, ρ)
from the etale groupoid associated to the action of Γ on X to the trivial groupoid X
is an equivalence.
More generally, if (G, X) is an etale groupoid such that the natural projection of
X onto G\X with the quotient topology is etale, and if Qx is trivial for all χ e X, then
(G, X) is equivalent to the trivial groupoid G\X. For instance the groupoid of germs
of all the changes of charts of an atlas defining a given structure on a manifold Μ is
equivalent to the trivial groupoid M.

598 Chapter III.С? Groupoids of Local Isometries
(3) The etale groupoid of germs of changes of charts of an orbifold is developable
if and only if the orbifold structure is developable.
An orbifold structure on a topological space Q could be defined as a differ-
entiable equivalence class of differentiable etale groupoids (G, X) together with a
homeomorphism from G\X onto Q, such that:
(i) Q\X is Hausdorff, and
(ii) each point of X has a neighbourhood U such that the restriction of (G, X) to U
is the groupoid associated to an effective action of a finite group on U.
(4) The Developability of Complexes of Groups. A complex of groups G(y)
over a scwol У is developable in the sense of (C.2.11) if and only if its associated
groupoid {0{У), У) is developable.
To see this, let us first assume that G(3O = (GCT, ψα, ga,b) is developable: thus
there is a group G acting on a scwol X such that У = G\X and G(y) is the associated
complex of groups with respect to some choices (we use the notations of C.2.9(l)).
Let X be the geometric realization of X and let G x X —> X be the corresponding
action. For each σ e У(У), there is a canonical isomorphism/σ : st(cr) -> st(o)
which is GCT-equivariant (see (C.4.11)). There is a unique etale homomorphism (φ,/) :
(0(У), Ϋ) -> (G к X, Χ), where/ : Ϋ -> Χ is the union of the maps/CT, and φ is
defined by:
<?(<?>*) = (g,/aW) if (g,*) € GCT X St(cr) = £σ,σ
<p(g,x) = (gha,fna)(x)) if C?,*) e G,(a) χ st(a) с QmAa).
These formulae define a functor φ : £(3Ό —> G к X. To check that this is the case,
note that for each χ e st(a), we have ha.f^a){x) = ft(a)(fa(x)) and we use C.2.9(l) and
C.4.14. It is clear that (φ,/) is an equivalence.
Conversely, assume that (0(У), Ϋ) is equivalent to the etale groupoid associated
to the action of a group G acting by homeomorphisms on a topological space X. Let
Υ be the geometric realization of У. We can identify Υ to Q(y)\Y and to G\X.
By exercise 2.8(3)(e), there is an etale homomorphism (φ,/) : (0(У), Ϋ) —>
(G к X, Χ) which is an equivalence (because each connected component of Υ is
simply connected), and / : Υ —> X induces the identity on Y. Therefore, for each
σ e V(3O, the ЬототофЬ18т φ maps (g, jc) e Ga χ st(cr) to an element of the form
(<οσ (#),/(*)) e G χ X, and the map g м> ^CT(g) is an isomorphism onto the isotropy
subgroup of σ = /(σ). If α e £(У) and jc e st(a), then φ(1((α), jc) e G χ Χ is of
the form (<p(a),f(x)). It is easy to check that the injective homomorphisms φσ and
the elements φ{α) € G define a morphism G(y) -> G, hence G(3O is developable
by (C.2.15). It is also easy to see that X is G-equivariantly homeomorphic to the
geometric realization of the development of У associated to this morphism (cf.
C.2.13).
(5) It is a remarkable fact that any connected etale groupoid of germs associated
to a pseudogroup of analytic local diffeomorphisms of an analytic one-dimensional
manifold is always developable. Indeed such a groupoid is equivalent to the groupoid
associated to a quasi-analytic action of a group Г on a simply connected analytic

Equivalence and Developability 599
manifold of dimension one (this manifold is not Hausdorff in general). This applies
in particular to the groupoid (classically denoted Tf) of all germs of analytic
local diffeomorphisms of K, or to the holonomy groupoid of an analytic foliation of
codimension one. This follows from the main lemma in [Hae58], its interpretation
in [Hae70] and the criterion for developability given in exercise 3.20(2). The
corresponding statement is not true in general if "analytic" is replaced by "differentiable".
(6) Let Τ be a foliation on a connected manifold Μ defined by a family of
submersions/; : U,- —» V,- С W and let (Q, X) be the associated holonomy groupoid
of Τ (see 2.3(5)). The induced foliation p~{(Jr) on a covering ρ : Μ —» Μ of
Μ can be defined by the family of submersions obtained by composing/; with the
restrictions of ρ to the connected components of p~{(Ui).
The developability of the holonomy groupoid (G, X) of Τ can be interpreted in
terms of foliation theory. One can show that (G, X) is equivalent to the groupoid
associated to an action of a group Γ on a connected manifold76 X if and only if there
is a Galois covering p:M^-M with Galois group Γ and a Γ-equivariant submersion
/ : Μ —» X with connected fibres which are the leaves of the foliation p~l (F).
2.8 Exercises
(1) Let {Q\, X\) and (Q2, X2) be two equivalent etale groupoids. Show that there
is a groupoid (G, X) and two etale homomorphisms (φι,/,) : (G, X) -> (Gi, X;),
i= 1,2, which are equivalences.
(Hint: If (ψι, hi) : (Gi, Xj) —> (G', X') are two etale homomorphisms which are
equivalences, let Q = {(gu gi) \ gi € Gi, f\(g\) = fiigi)} and letX = {(xux2) \
χ-, e Xj, hi (χι) = h2(x2)}- Show that (G, X) is naturally an etale groupoid and that
the natural projections (G, X) —> (Gi, Xi) are equivalences.)
(2) Given an etale groupoid (G, X) and an open covert = {i/,-};e/ of X indexed by
a set/, let U be the disjoint union of the £/,, namely the set of pairs (i, x), with χ e i/,.
Fori,/ e /, let ^y be the set of triples (i, g,j) where g e G and a(g) e Ц, a)(g) e i/,·.
Let Gu be the disjoint union of the Gij- Define on (Gu, U) the structure of an etale
groupoid such that the natural projection (г, g,j) м> g gives an etale equivalence to
(G, X). We say that (Gu> U) is the groupoid obtained from (G, X) by localization on
the cover U.
Show that two etale groupoids (G, X) and (G', X') are equivalent if and only if
there exist open covers U and W of X and X' such that the localizations (Gu, U) and
(Gu,, U') are 18отофЬю.
(3) In this series of exercises, we present a more direct definition of equivalence
between two topological groupoids (G\, Χι) and (Gi, Xi)- This definition can be found
in Jean Renault (see [Ren82]); it is equivalent to the definition given in (2.4).
Consider a topological space £ with two surjective etale maps a : £ —> X\
and ω : £ —> X2. A right action of G\ on £, over a : £ —> X\, is a continuous
This manifold is not Hausdorff in general but the action is always quasi-analytic.

600 Chapter Ul.Q Groupoids of Local Isometries
map £ xXl Q\ -> £, written (e, g,) м> e.g,, where £ xXl G\ is the subspace of
£ χ Q\ consisting of pairs (e, gi) such that a(e) = cu(g\). This action must satisfy
the following conditions: if e e £, g\, g{ e G\ are such that a(e) = w(g\) and
a(gi) = a)(g[), then (e.gi).g{ = e.(gig',); and e.l„(e) = e.
This action is said to be simply transitive with respect to ω : £ —> X2 if each
point of X2 has an open neighbourhood U with a continuous section j : U —> £
of ω over i/ such that the map (и, gi) м> s(u).g\ of ί/ χ χ, £i to ω~'(ί/) is a
homeomorphism, where ί/ χ χ, ί/ι is the subspace of ί/ χ £i consisting of pairs
(u,g}) with a(s(u)) = o)(gl).
Similarly, a left action of G2 on £, over ω : £ —> X2, is a continuous map
(g2, e) i-> g2.e of ^2 Хл-2 £ to £, where a(g2) = a)(e), and, if g2, g'2 e £2, e e 7 are
such that a(g2) = ω(#2) anc* ^(§2) = ω(β)> then one requires gi-{g2-e) — {gig'i>-e
and 1Ш(е).е = e.
An equivalence from (Gi,Xi) to (Οι,Χι) is a space £, with two etale maps
a : £ —* X\ and ω : ε —> X2, a right action of £1 on ί over α : £ —> Xi and a left
action of Gi on £ over ω : £ —> X2 such that:
(i) the action of G\ commutes with the action of G2,
(ii) the action of G\ (resp. Gi) is simply transitive with respect to ω : ε —> X2 (resp.
α :£->X,).
A typical example to bear in mind is the following. Let Τί\ and Ή2 be the
pseudogroups of changes of uniformizing charts of two atlas A\ and Λ2 defining
the same orbifold structure on a space Q. Let (G\, X\) and {Gi, Χ2) be the associated
groupoids of germs. Let ε be the space of germs at the points of Xi for the changes
of charts from the atlas A\ to the atlas A2, with the source (resp. target) projections
a : ε —> Χι (resp. ω : ε —> X2). The groupoid G\ acts on the right on ε over a by
composition of germs, and similarly Gi acts on the left over ω. Clearly ε gives an
equivalence from {G\, X\) to (G2, X2)·
(a) Let £2,1 be an equivalence from the etale groupoid (Gi, Xi) to the etale
groupoid (G2, ^2)· Show there is an equivalence £1?2 from (G2, X2) to (G\, Xi). (Hint:
take £1?2 = £2,1 and reverse the roles of α and ω.)
(b) Let £32 be an equivalence from (G2, X2) to the etale groupoid (G3, X3).
Construct an equivalence £3il from (G\,X\) to (G3, X3).
(Hint: Consider the subspace £32 Χχ2 £2,1 of £3,2 x £2,1 consisting of pairs (e, e')
with a(e) = ω(ε'), and define £31 as the quotient of £3,2 Χχ2 £2,1 by the equivalence
relation which identifies (e.g2, e') to (e, g2-e'), where gi e £2·)
(c) Let (φ,/) : (G\, Xi) —> (02, X2) be а homomorphism which is an equivalence
in the sense of (2.4). Let £2,1 = G2 xx2 Xi be the subspace of G2 x ^1 formed by the
pairs (g2> *) such that a(g2) = /(■*)■ Define a(g2, χ) = χ and co(g2, x) = co(g2). The
map ((g2, *), gi) i-> (g2<p(gi), oc(gi)), where g, e G\ with w(g,) = jc, defines a right
action of G\ on £21 over a. Similarly the formula (g'2, (g2, x)) (-> (g2g2> x) defines
a left action of G2 over ω. Show that £2?] defines an equivalence from (G\,X\) to
(&,X2).

Groupoids of Local Isometries 601
The map s : Χχ —> £2,1 sending χ to (lf(X),x) is a section of a. Show that
conversely an equivalence £2,1 is associated to a homomorphism (φ,/) if and only
there is a continuous section s : Χι —> £2,1 of a.
(d) Using exercise (1), show that (G\, X\) is equivalent to (£2, -X2) in the sense of
(2.4) if and only if there is an equivalence £2,1 from (G\, Χι) to (£2, -X2)·
(e) Assume that (G2, X2) is the etale groupoid (Г к Хг> -X2) associated to an action
of the group Г by homeomorphisms of X2. Let £2,1 be an equivalence from (G\, Χι)
to (G2, ^2)· Show that a : £2,1 —> ^1 is a Galois covering with group Г.
Using (c) and (d), show that this implies the following: let (G\, Χι) be an etale
groupoid which is equivalent to the groupoid associated to an action of a group Г on
a space X2; if each connected component of X\ is simply connected, there is an etale
homomorphism {φ J) :(5iJi)^ (ГкХ2, Хг) which is an equivalence.
Groupoids of Local Isometries
2.9 Definition. A groupoid (G, X) of local isometries is an etale topological groupoid
with a length metric on its space of units X that induces the given topology on X and
is such that the elements of the associated pseudogroup (see 2.1) are local isometries
ofX.
Equivalence among groupoids of local isometries is the equivalence relation
generated by etale homomorphisms (φ,/) : (G, X) —> (G1, X') which are equivalences
and are such that/ : X —> X' is locally an isometry.
2.10 Hausdorff Separability and Completeness. A groupoid (G, X) of local
isometries is said to be Hausdorff if G is Hausdorff as a topological space and for every
continuous map с : (0, 1] —> G, if lim,_,.o aoc a°d lim,->.o ^oc exist, then lim,^0 c{t)
exists.
A groupoid (G, X) of local isometries of X is said to be complete if X is locally
complete (i.e. each point of X has a complete neighbourhood) and if the space of
orbits G\X with the quotient pseudometric is complete.
2.11 Remark. To justify the second condition in the definition of Hausdorff
separability, consider the following example. Suppose that the metric space X is just the
Euclidean space E" and that G is generated by an isometry/ from an non-empty open
subset U of Ε" to a disjoint open subset V of E" (i.e. the elements of G are the germs
of the identity map of E" and the germs of/ and its inverse at the points of U and
V respectively). Then (G, X) is a groupoid of local isometries which is complete but
not Hausdorff: the second condition in the definition of Hausdorff separability is not
satisfied. Note that the quotient X' of E" by the equivalence relation which identifies
χ e U tof(x) e V is a non-Hausdorff space and that (G, X) is equivalent to the trivial
groupoid X'.
In general, if (G, X) is Hausdorff, then the space of orbits G\X need not be
Hausdorff in general (see the first example below).

602 Chapter ΠΙ.Q Groupoids of Local Isometries
2.12 Examples
(1) If a group Γ acts by isometries on a length space X, then the associated
groupoid (Γ κ X, X) is a groupoid of local isometries, and it is Hausdorff. It is
complete if and only if X is a complete metric space.
(2) The groupoid (G, X) of germs of changes of uniformizing charts of a Rie-
mannian orbifold is a groupoid of local isometries, and it is Hausdorff (in this case
Hausdorff being equivalent to the condition that the orbit space Q is Hausdorff). The
quotient pseudometric on Q is always a metric, and (Q, X) is complete if and only if
Q is complete.
(3) Let G(y) be a complex of groups over a scwol У whose geometric realization
\У\ is a Μκ-simplicial complex with finitely many shapes (see C.4.16). Then the
associated etale groupoid {0(У), Υ) is a groupoid of local isometries; it is always
Hausdorff and complete. Indeed the quotient pseudometric on the space of orbits \У\
is the length metric associated to the given M^-structure, which is always complete
by (1.7.13).
(4) Assume that X is a Riemannian manifold and that (G, X) is an etale groupoid
such that the elements of the associated pseudogroup are Riemannian isometries.
Then (G, X) is a groupoid of local isometries, called a groupoid of local Riemannian
isometries.
The holonomy groupoid of a Riemannian foliation (see [M0I88]) is a typical
example of a groupoid of local Riemannian isometries. A Riemannian foliation
fona Riemannian manifold Μ is by definition a foliation such that the leaves
are locally at constant distance; equivalently, if Τ is given by local submersions
f-, : Ui —> V-, С К", there should exist on each V,- a Riemannian metric such that/;
is Riemannian submersion (meaning that, for each point χ e U-„ the restriction of
the differential of/ to the subspace of TXM orthogonal to the tangent space of the
leaf through χ is an isometry onto the tangent space of V; at/)(*)). It follows that the
associated holonomy groupoid is a groupoid of local Riemannian isometries. If the
Riemannian manifold Μ is complete, then the holonomy groupoid is Hausdorff and
complete.
For a systematic study of closures of orbits in Hausdorff and complete groupoids
of local Riemannian isometries, see [Hae88], where examples of such groupoids that
have a single orbit but are non-developable are exhibited.
2.13 ^-Connectedness. An etale groupoid {G, X) is said to be connected
(equivalently X is "G-connected") if for any two points x,yeX there is a sequence of points
(x\, у ι,..., Хк, у к) such that хг = χ, ук = у, the point x-, is in the £/-orbit of yi+\ for
i— 1,... ,k— 1 and there is a path joining jc, to y, for each / = 1,... ,k.
To illustrate the notion of Hausdorff separability and completeness, we prove the
following lemma, which will be useful later.
2.14 Lemma. Let (G, X) be a groupoid of local isometries which is Hausdorff. Let
χ & X and ε > 0 be such that the closed ball B(x, ε) is complete. Given a path

Statement of the Main Theorem 603
с : [a, b] —> X of length < ε and an element g e Q with c(a) = a(g) and w(g) = x,
there is a unique continuous map t м> g(t)from [a, b] to Q such that g(a) = g and
a(g(t)) = c(t),forall t e [a, b].
Thus, if a Q-orbit is a pseudodistance less than ε from the Q-orbit of x, then it
meets B(x, ε).
Proof. Let / be the maximal subinterval of [a, b] containing a on which g(t) can be
defined; note that g(t) is unique because Q is a Hausdorff space. Because Q is etale,
the interval / is open in [a, b]. The length of the path t м> co{g{t)) is equal to the
length of the path с restricted to /, hence is smaller than ε. Using the hypothesis that
G is Hausdorff and B(x, ε) is complete, we see that / is also closed, hence equal to
[a, b].
To prove the second assertion of the lemma, note that if in G\X the orbit of у is
within a pseudodistance ε of the orbit of x, then there is a sequence (x\,y\,... ,xk, Ук)
of points of X such that x} — x, у к — у, the points jc, and yi+\ are in the same orbit
and there are paths c, joining jc, to y,- such that the sum of the lengths of the paths c,
is smaller than ε. Thus we can apply the first part of the lemma successively to the
paths с,-. D
Statement of the Main Theorem
The aim of this chapter is to prove the following result.
2.15 Developability Theorem. Let (G, X) be a connected groupoid of local isome-
tries which is Hausdorff and complete. If the metric on X is locally convex11, then
(G, X) is developable.
The conclusion of the theorem is that (G, X) is equivalent to the groupoid
associated to an action of a group Γ by isometries on a metric space X. The group Γ will
be the fundamental group of (G, X) and the space X will be the space of equivalence
classes of ^-geodesies issuing from a base point xq e X with free endpoint. (All
of these terms will be defined in the next two sections.) The space X has a natural
length metric making it locally isometric to X. As X is complete and simply
connected, its metric is globally convex by the Cartan-Hadamard theorem (II.4.1). If X
is non-positively curved, then X is a CAT(O) space.
As a special case of 2.15 we have:
2.16 Corollary (Gromov). Every complete Riemannian orbifold of non-positive
curvature is developable.
2.17 Corollary. Every complex of groups of non-positive curvature (see C.4.16) is
developable.
in the sense of Busemann, see II. 1.18 and II.4.1.

604 Chapter Ul.Q Groupoids of Local Isometries
This follows from (2.15) because non-positive curvature implies that the metric
is locally convex and the developability of a complex of groups is equivalent to the
developability of its associated etale groupoid (see 2.7(4)).
By definition, a Riemannian foliation Τ is transversally of non-positive curvature
if it is defined by local Riemannian submersions/i : i/,· —> V-„ where the Riemannian
metric on V-, is of non-positive curvature. The proof of the following corollary is left
as an exercise (see 2.7(6)).
2.18 Corollary (Hebda). Let Τ be a Riemannian foliation on a complete
Riemannian manifold Μ which is transversally of non-positive curvature. Then the foliation
induced on a suitable Galois covering ρ : Μ —> Μ is defined by a Riemannian
submersion with connected fibres f : Μ —> X, where X is a Riemannian manifold of
non-positive curvature.
3. The Fundamental Group and Coverings of Etale Groupoids
Equivalence and Homotopy of £-Paths
3.1 ^-Paths. Let (Q, X) be an etale groupoid. A Q-path с = (go. ci, g\,..., ck, gk)
over a subdivision a — to < ■ ■ ■ < tk = b of the interval [a, i»]cR consists of:
(1) continuous maps c, : [i,_i, i,] -> X,
(2) elements g, e Q such that ce(gj) = c1+i(i,) for i — 0, 1,... ,k — 1 and ω(#,) =
d(ti) fori = 1,... ,k.
The initial point of с is χ := co(go) and у := ce(gk) is its terminal point. We say
that с joins jc to y. If go = 1* and gk = 1^ are units, they can be dropped in the
notation for с (If с : [α, b] —> X is a path joining jc to y, it can be considered as the
α-path (l„c,l,).)
If (Q, X) is a groupoid of local isometries, then the length 1(c) of the £?-path с =
(go, C\, gi,..., сь gk) is the sum of the lengths of the paths c,. The pseudodistance
between the ^-orbits of χ and у is the infimum of the lengths of ^-paths joining χ
toy.
:/y> «л Я у8
Fig.g?.lA^-path
3.2 Equivalence of £/-Paths. Among ^-paths parameterized by the same interval
[a, b] we define an equivalence relation generated by the following two operations.

3. The Fundamental Group and Coverings of Etale Groupoids 605
(i) Subdivision: Given a Q- path с = (go, C\, .. ., gk) over the subdivision a =
to < t\ < ■ ■ ■ < t/c = b, we can add a new subdivision point ή e [ί,_ι, ί,·]
together with the unit element g\ = \Citf.) to get a new sequence, replacing c,
by c'i, g\, c", where c\ and d( are the restrictions of c, to [ί,_ι, ή] and [г^, ί,].
(ii) Replace the £?-path с by a new path d = (go, c\, g\, ...,dk,g'k) over the same
subdivision as follows: for each i — 1,... ,k, choose a continuous map h-, :
[ti~i,t;] -> 0 such that α(Α,·(ί)) = c,(i) and define c\\ : t н> ω(/ϊ,(ί)), g- =
Α;(ί/) gih~lx{ti) for г = 1,... Д - 1, g'0 = ^Г'(?о) and ^ = hk(tk)gk.
Fig. ^.2 Equivalent ^-paths
Note that if two £/-paths are equivalent, then under a suitable common subdivision
one can pass from the first to the second by an operation of type (ii). Note that two
equivalent £?-paths have the same initial and terminal points. We also note that if
(G, X) is Hausdorff, then the continuous maps /г, in (ii) above are uniquely defined
by с and d.
If a £?-path с joining χ to у is such that all the c, are constant maps, then the
equivalence class of с is completely characterized by an element g e G with a(g) = у
and a)(g) = x.
3.3 Examples
(1) Let Μ be a differentiable manifold whose differentiable structure is given by
an atlas of charts q, : X-, —> V), where the V,- cover Μ and theX, are disjoint. Let X be
the union of the X, and let q be the union of the q,. Let (G, X) be the etale groupoid
of germs of all the changes of charts. Then the set of equivalence classes of £/-paths
joining χ to у corresponds bijectively to the set of paths in Μ joining q(x) to q(y).
More generally, if (φ,/) : (G,X) —> (G',X')is an equivalence of etale groupoids,
then φ induces a bijection from the set of equivalences classes of £-paths joining χ
to у to the set of equivalence classes of £'-paths joining/^) to f(y).
(2) Let X be K2 with rectangular coordinates (x, y). Let Q be the half-plane
defined by у > 0 and let q : X -> Q be the projection (x, у) м> (χ, \y\). Let Γ be the
cyclic group of order two generated by the symmetry σ : (χ, у) \-> (х, —у). The map
q induces a homeomorphism Г\Х —> Q and can be considered as a uniformizing
chart defining an orbifold structure on Q. The pseudogroup Ti of changes of charts
is generated by σ; let G be the etale groupoid of germs of elements of Τι.

606 Chapter ΠΙ.^ Groupoids of Local Isometries
Consider the £-paths с = (ci, g\, C2) and d = (ci, gj, C2) parameterized by
[0, 1] joining (0, 1) to itself, where c\ maps t e [0, 1/2] to (0, 1 — 2t), c% maps
ί e [1/2, 1] to (0, 2i - 1) and g\ (resp. g[) is the germ at (0, 0) of the identity (resp.
of σ). The images under q of these paths look the same, but they are not equivalent
(not even homotopic in the sense of (3.5)).
3.4 Inverse 6-Paths and Composition. From now on, unless otherwise specified,
all g-paths will be parameterized by [0, 1]. Let с = (go, C\,..., gk) be a 6-path
over the subdivision 0 = to < ·■ ■, < tk = 1. The inverse of с is the £/-path
c~l = (g'0, c'[,..., g'k) over the subdivision 0 = f0 < ■ ■ ■ < fk = 1, where 7< = 1 —th
g'i = gu-i and eft) = Cjt-/+i(l — t). Note that the terminal point of с is the initial
point of c-1 and vice versa. The inverses of two equivalent paths are equivalent.
Given two £/-paths, с = (go, C\,..., gk) over the subdivision 0 = to < ■ ■ ■ <
tk = 1 and c' = (g'0, c\,..., g'k) over the subdivision ()=*(,<■■■< ip = l, sucn
that the terminal point of с is equal to the initial point of c', their composition с * с' (or
concatenation) is the £/-path c" = (g'0\ c",..., g'l+k.) defined over the subdivision
<o < ■ ■ ■ < 4+t, where
t1! = till if 0 < i < k, /f = 1 /2 + t'^/2 ifk<i<k + k',
c'!(f) = ct(t/2) for 1 < i < k; c'!(f) = c-_k(2t -l)fotk<i<k + k',
g'l = gi for 0 < i < k, g'l = gkg'o, g'i = glk fovk<i<k + k'.
Note that if с is equivalent to c, and c' is equivalent to c', then с * с' is equivalent
to с * с'.
3.5 Homotopies of Q -Paths. Two ^-paths с and c' (parameterized by the interval
[0, 1 ]) are homotopic78) if one can pass from the first to the second by a finite sequence
of the following operations:
(1) equivalence of £/-paths,
(2) elementary homotopies: an elementary homotopy between two £/-paths с and
c' is a family, parameterized by s e [sq,s\], of £/-paths cs = (g^, c\,..., gsk) over the
subdivisions 0 = f0 < f\ < ■ ■ ■ < fk = I, where ή, cf and gj depend continuously
on the parameter s, the elements g^ and g[ are independent of s and c'° = c, c" = c'.
The homotopy class of a £/-path с will be denoted [c]. Note that two equivalent
^-paths are homotopic. If с and d are composable £-paths, the homotopy class
of с * d depends only on the homotopy classes of с and d and will be denoted
[c * c'] = [c] * [c']. If c, d and c" are three £/-paths which are composable, then
[c * c'] * [c"] = [c] * [c' * c"]. Note that although (c * d) * c" is not equivalent to
d * (d * c"), the homotopy class of (с * с') * с" is equal to the homotopy class of
Always, implicitly, relative to their endpoints.

The Fundamental Group 607
Fig. Q.3 An elementary homotopy between £/-paths
c*(c' * c")\ it will be denoted [с] * [с'] * [с"]. All these properties are proved as in
the usual case of paths in topological spaces.
In the notation of example 3.3(1), there is an obvious bijection between the
homotopy classes of ^-paths joining two points x,yeX and the homotopy classes
of paths in Μ joining q(x) to q(y).
The Fundamental Group iz\{{Q, X), x0)
3.6 Definition. Let (G, X) be an etale groupoid. A G-loop based atxQ e X is a path
parameterized by [0, 1] joining xq to xq. With the operation of composition, the
homotopy classes of ^-loops form a group called Mas, fundamental group π\ ((G, X), xo)
of(G, X) based atxQ.
Let (φ,/) : (G,X) —> (G!, X') be a continuous homomorphism of etale groupoids.
Given a £/-path с = (go,C\,... ,gk) over a subdivision to < ■■ ■ < tk, its image under
(φ,/) is the £/'-path cp{c) = (<p(go),f ° C\,..., (p{gk)) over the same subdivision. It
is clear that if two ^-paths are equivalent (resp. homotopic), then their images under
any continuous homomorphism are equivalent (resp. homotopic). In particular (φ,/)
induces a homomorphism
*ι (@,Χ),*ο)->*,((£', ХО.Дло))-
The first part of the following proposition is proved as in the special case of
topological spaces. The second part is left as an exercise.
3.7 Proposition. Let (G, X) be an etale groupoid and let x$ eX be a base point. Let
a be a G-path joining xq to a point x\. Then the map that associates to each G-loop
с atXQ the G-loop {a~x * с * a) at jq induces an isomorphism from n\{{G, X), xq) to
nd(S,X),xu-
Let (φ,/) : (G, X) —> (G',X') be a continuous etale homomorphism of etale
groupoids. lf(<p,f) is an equivalence, then the induced homomorphism on the
fundamental groups is an isomorphism.
3.8 Definition. Recall that an etale groupoid (G, X) is connected (equivalently X is
^-connected) if any two points of X can be joined by a £/-path. It is simply connected
if it is connected and if щ ((G, X), xo) is the trivial group.

608 Chapter Ul.Q Groupoids of Local Isometries
3.9 Examples
(1) Assume that (G, X) is the etale groupoid associated to an action of a group
Γ by homeomorphisms on X. Then any £/-path (go, cx, g\,..., ck, gk) joining χ to у
defined over the subdivision 0 = to < ■ ■ ■ <tk = 1 is equivalent to a unique £/-path
of the form (c, g), where с : [0, 1] —> X is a continuous path with c(0) = χ and
8 = (У. У) with Υ-У = c(l)· Indeed if g,- = (y;, jc,) e Γ χ X, and we choose с to be
the path mapping t e [ί,-ь i,] to yo · · · y,-_i-c,-(/) and define γ = γο- ■ -Yk, then the
given £-path is equivalent to the 6-path (c, g), where g = (γ, Xk).
We describe the group n\((G, X), xo) in this special case. Let Xq be the path
component of xq and let Го be the subgroup formed by the elements γ e Γ such
that γ.χο e Xq- Every £-loop at xq is equivalent to a unique g-loop of the form
(c, g) where с : [0, 1] -> X0 is a continuous path with c(0) = jc0 and g = (γ, χ0),
with y.^o = c(l); therefore j/ e Γ0. Another such ^-loop (c', g') with g' = (y', xq)
represents the same element of πχ ((G, X), xo) if and only if у = γ' and с is homotopic
to d. Hence we have the exact sequence (compare with C.1.15):
1 -» 7Ti(Xo, Xo) -> 7Ti((^,X),JCo) -> Го-> 1.
(2) Consider an orbifold structure defined on a space Q by an atlas of uniformizing
charts, and let (G, X) be the etale groupoid of germs of changes of charts. The orbifold
fundamental group of Q based at xq e X is by definition the group ni((G, X), xo)- If
Q is connected, then up to isomorphism this group is independent of the choice of
base point and the choice of a compatible atlas of uniformizing charts (see (3.6) and
(3.7)).
3.10 Exercises
(0) Show that an equivalence^,/) : (G, X) —> (G1, X') of etale groupoids induces
a bijection from the set of equivalence classes of £-paths joining χ to у to the set of
equivalence classes of £/'-paths connecting/(jc) to f(y). Show that (φ,/) induces an
isomorphism n\((G,X),xo)) —> πΊ((£/',X'),/(*o))·
(1) What are the fundamental groups of the trivial examples 2.3(1)? Show that the
fundamental group of the orbifold S2mn described in 1.4(1) is a cyclic group whose
order is the greatest common divisor of m and n.
(2) Let (G, Щ be the etale groupoid of the germs (at all points of K) of the elements
of the group of diffeomorphisms of Μ that is generated by the diffeomorphisms
h\,... ,hn, where each /г, is the identity on some non-empty open set t/,. Show that
(G, K) is simply connected. (Consider first the case where η = 1.)
(3) Let G(y) be a complex of groups over a connected scwol У and let (в(У), Υ)
be the associated etale groupoid (see 2.3(4)). Show that the fundamental group of
G(y) as defined in C.3.6 is isomorphic to the fundamental group of (GiV), Ϋ).
(4) The Seifert-van Kampen Theorem. Let (G, X) be an etale groupoid and let
X, and X2 be two open subsets of X such that X = X, U X2. Let X0 = X, П X2. We
assume that the restriction (С?,·, X,) of (G, X) to X, is connected for г = 0, 1, 2. Let
xo e Xq be a base point. Show that n\{{Q,X),xo) is isomorphic to the quotient of

Coverings 609
the free product π\ ((G\, X\), xq) * П\ ((Gi, Хг), *o) by the normal subgroup generated
by the elements of the form j\(y)J2(y)~1, where γ e n\((Go, Xo),*o) andy, is the
homomorphism induced on the fundamental groups by the inclusion from (Go, Xo)
into (Gi, Xd, i = 1,2.
(5) Let Τ be a foliation on a connected manifold M, and let (G, X) be its holonomy
groupoid as described in 2.3(5). Show there is a surjective homomorphism from the
fundamental group of Μ to the fundamental group of (G, X)- (Thus (G, X) is always
simply connected if Μ is simply connected.)
(Hint: In the notations of example 2.3(5), let с : [0, 1] -» Μ be a loop in Μ
with c(0) = c(l) = yo e Ц0. Let 0 = to < h < ■ ■ ■ < i*. = 1 be a partition such
that each c([i,_ 1, i,]) is contained in some U}i for i = 1,..., к — 1; let у ι■ = c(i,) and
xi = j^Cft). xo = fj0(yo)\ let с = (g0, c\,...,ck, gk) be the £-loop at x0 defined as
follows: c,- : [ί,_ι,ί,] -* Vj; maps t to fj:(c(t)), gt is the germ of /$. +| at^,^,) andg^
the germ at jcq of /^- Show that the map associating to the homotopy class of с the
homotopy class of с is a surjective homomorphism from π\(Μ, yo) to n\((G, X), xo).)
Coverings
3.11 The Action of a Groupoid on a Space. Let (G, X) be an etale groupoid, let X
be a topological space and let ρ : X —» X be a continuous map. Let G = Χ Χχ G be
the subspace of Χ χ £ consisting of pairs (x, g) with /j(jc) = w(g). A r/g/ϊί action of
(^, X) on X over ρ is a continuous map (x, g) м> i.g of G to X, such that:
(1) p(x.g) = a(g),
(2) if a(g) = co(g'), then x.(gg') = (x.g).g',
(3) if χ ep~~l(x), thenxU = x.
To such an action is associated the etale groupoid (G, X) where ω((χ, g)) = χ
and a((x, g)) = a(g) = x.g; the composition (x, g)(xf, g') is defined if xf = x.g and
is equal to (x, gg')\ the inverse of (x, g) is (x.g, g~~l). The map (x, g) i-> (p(i), g) is
a continuous homomorphism denoted (π, ρ) : (G, X) —> (^, X).
Note that any right action of G on X over ρ can be converted into a left action of
G over ρ by defining g.i = x.g~{, where a(g) = p(x).
3.12 Definition of a Covering. In the preceding definition, if ρ : X —» X is a
covering, we say that (π, ρ): (<5, X) —> (^, X) is a covering of etale groupoids.
Moreover, if /? : X —> X is a Galois covering such that the action of its Galois
group Γ on X commutes with the action of G, then we say that (π, ρ) is a Galois
covering with Galois group Γ.
3.13 Examples
(1) Coverings of a Groupoid Associated to an Action of a Group. Let Γ be a group
acting by homeomorphisms on a simply connected topological space X. Let Го be a
subgroup of Г. Let X = Г/Г0 х X, where Г/Г0 has the discrete topology. The group
Г acts naturally on X by the rule γ.(γΤ0,χ) = (γγΤ0, γ.χ). Let ρ : Γ/Γ0 χΧ^Χ

610 Chapter ΠΙ.Q Groupoids of Local Isometries
be the natural projection. The functor π :ГхХн> Γ κ Χ mapping (γ, χ) to (γ, p(x))
gives a morphism (π,ρ) : (Γ κ Χ, Χ) —> (Γ κ Χ, Χ) which can be considered as
a covering. The natural inclusion Го x X —> Γ χ X sending (yo, *) to (yo, (Го, *))
defines an etale homomorphism (Го κ Χ, Χ) —> (Γ κ Χ, Χ) which is an equivalence.
In fact the equivalence classes of connected coverings of the groupoid (Γ κ Χ, Χ)
are in bijection with the conjugacy classes of subgroups of Г.
(2) Coverings of an Orbifold, Consider an orbifold structure on a space Q defined
by an atlas of uniformizing charts and let (Q, X) be the associated groupoid of germs
of changes of charts. Then any covering (G, X) of (G, X) is the groupoid of changes
of charts of an orbifold structure on the space of orbits Q = G\X, and β is a covering
orbifold of the orbifold Q, in the sense of (1.7).
(3) The Orientation Covering. Let (G, X) be the etale groupoid of germs of a
pseudogroup of local homeomorphisms of a manifold X. Let ρ : X —> X be the 2-
fold orientation covering of X: the points of the fibre p~' (x) are the two possible local
orientations of X at x. The groupoid (G, X) acts naturally on X. The corresponding
etale covering is called the orientation covering of (Q, X). The groupoid is defined
to be orientable if there is a continuous section s : X —> X which is invariant by G-
We considered the following lemma earlier in the special case of orbifolds (1.10).
λα
3.14 Lemma. With the notations of 3.12, let (π,ρ) : (G, X) —> (G, X) be a Galois
covering with Galois group Γ. If the natural projection q : X —> X := Q\X = X/G
is etale and if Gx — U for each χ e X, then (G, X) is equivalent to the groupoid
(Γ κ X, X) associated to the natural action of Г on X.
Proof. Let (Г x Q, X) be the etale groupoid defined as follows. The space Γ χ Q is the
subspace of the product Γ χ Χ χ G (where the topology on Γ is discrete) consisting
of the triples (γ, χ, g) with p(x) = w(g). The source and target projections a and ω
aiea((y,x, g)) = x.g Άηάω((γ, χ, g)) — γ.χ.. The composition (γ, χ, g)(y'' ,χ1', g')'i$,
defined if x.g = y'.xf and is equal to (γγ', γ'~χ .χ, gg'), and the inverse of (γ, χ, g)
\5{у-\у.Х^^-{).
The lemma follows from the following two assertions:
(1) The etale homomorphism (Г x§,X)-> (G, X) sending (γ, χ, g) to g and χ
to p(x) is an equivalence.
(2) The etale homomorphism (Γ χ G, X) ->■ (Γ κ X, X) sending (γ, χ, g) to
(γ, q(x)) and χ to q(x) is an equivalence.
To prove (1), we only use the hypothesis that (π,ρ) is a Galois covering. The
group Γ acts transitively on each fibre of p, hence the homomorphism induces a
bijection on the spaces of orbits. The isotropy group of a point χ e X with χ = p(x)
consists of the triples (γ, xf, g) such that g € Gx and χ = xf.g = γ.χ?; equivalently
it consists of the pairs (γ, g) with γ e Γ, g e Gx and γ.χ — x.g. Given g e Qx,
there is a unique γ satisfying this equality, hence the homomorphism induces an
isomorphism from the isotropy group of χ onto Gx.

Coverings 611
To prove (2), we observe that the etale homomorphism (Γ x Q, X) —> (Γ χ Χ, Χ)
induces a bijection (Γ χ G)\X —» Γ\Χ. The isotropy subgroup of q(x) is the set of
γ e Γ such that γ.χ = x.g for some g e Qx. The hypothesis that Qx is trivial implies
that this element g is uniquely determined by γ. Therefore the homomorphism
induces a bijection of the isotropy subgroups. D
The theory of coverings for etale groupoids is strictly parallel to the theory of
coverings for topological spaces, as the following sequence of results illustrates.
3.15 Proposition (Liftings of £-Paths). Let (π, ρ) : (Q, X) -» (Q, X) be a covering
of etale groupoids, letxo e X be a base point and letx0 = p(xo). For every Q-path
с = (go, c\, ..., gk) issuing from xq, there is a unique Q-path с issuing from xq such
that n(c) = с. If с' is a Q-path homotopic to c, then its lifting c' issuing from x.q is
homotopic to c.
Proof. Let с = (go,Ci,..., gk) be defined over the subdivision 0 = to < ■ ■ ■ <
tk = 1. The lifting с = (go, с ι, ..., gk) of c is constructed by induction as follows:
go = (xo, go), c, : [i;_i, tj] —> X is the unique continuous path such that c,- = pc,
and c,(i;_i) = a(g,_i), gi = (Ci(tj), gi). Each stage of the construction is uniquely
determined by the previous stage and go is uniquely determined. It is clear that the
liftings issuing from xq of two equivalent £/-paths are equivalent (?-paths, and the
lifting of an elementary homotopy is an elementary homotopy because ρ is a covering
projection. This proves the proposition. D
3.16 Corollary. The homomorphism K\((G,X),xo) —> n\((Q, X),xo) induced by
(π, ρ) is injective.
3.17 Proposition. Let (cp,f): (Q1, X') —> (Q, X) be a continuous homomorphism of
etale groupoids and let (π,ρ) : (Q,X) —> (Q, X) be a covering. Assume that X' is Q'-
connected and locally arcwise connected. Suppose that (Q1, X') is simply connected
and fix base points x'0 e X' and xq e X such thatf(x'0) = p(xo) = xo- Then there is
a unique continuous homomorphism (<p,f) : (Q1', X') —> (Q, X) such thatf = ρ of,
φ = π ο φ andf(x'0) = xq.
Proof. Let с be a £/'-path issuing from x'0. If φ exists, then ф(с) is a Q-path с issuing
from xo such that ^(c) = n(c). As such a path is unique (by (3.15)), this shows the
uniqueness of φ.
Given a point x' e X', there is a £/'-path с joining x'0 to x', because X' is assumed
to be <J'-connected. Moreover, as Q' is simply connected, the homotopy class of
such a path is unique. Let с be the (5-path issuing from xq whose projection by π is
<p(c). Let/ : X' —» X be the map associating to x' the end point of c. To check that
this map is continuous at x', choose a neighbourhood U of fix1) that is mapped by
ρ homeomorphically onto an open set U с X; as / is continuous and X' is locally
arcwise connected, /~'(U) contains an arcwise connected neighbourhood V of x';

612 Chapter III.С/ Groupoids of Local Isometries
we claim that/~'(i/) contains V. Indeed for each point ζ € V, we may choose a
continuous path сг in У joining x1 to z; its image under/ can be lifted uniquely as a
path c, in U issuing from/(У); considering cz as a £/'-path and сг as a g-path, the
lifting off(c * cz) is с * cz, hencef~l(U) contains V. Π
3.18 Corollary. Let (π, ρ): (§, Χ)-> (G, X) and (π', ρ'): (£', X') -> (G, X) be two
simply connected coverings of {G, X), and choose basepoints jcq e Xandx!0 e X such
that p(xo) = p'(%q). Then there is a unique isomorphism (<p,f) : (G, X) —> (G', X')
such that π = π' ο φ andf{xo) = χ?0.
By definition, an automorphism у of a covering (π, ρ) : (G, X) —> {G, X) is a
homeomorphism γ : X —> X that commutes with the projection /? and the right
action of G, namely ρ ο γ = ρ and for all g e G, χ £ X with /j(jc) = w{g) we have
y.(i.g) = (y.x).g.
3.19 Corollary. Lei (6, X) be an etale groupoid and assume that X is locally arcwise
connected and Q-connected. Any simply connected covering (π, f): (Q, X) —» (Q, X)
is naturally a Galois covering, with Galois group Γ = n\((G, X), xo)-
Proof. Fix a base point jcq e X. By Corollary 3.18, the group Γ of automorphisms of
(π, ρ) acts simply transitively on the fibre p~l(x0). Let xq e p~l(xo). Given γ e Γ,
let с be a g-path joining γ.χο to xq. Let π(γ) be the homotopy class of the £-1οορ
p(c) at x0; it is independent of the choice of с because (Q, X) is simply connected.
As in the case of topological spaces, one shows that the map Φ : Γ -» πλ({0, X), xo)
is an isomorphism; we leave the details as an (instructive) exercise. D
3.20 Exercises
(1) Construction of the Universal Covering. Let (G, X) be an arcwise connected
etale groupoid such that X is locally simply connected (this means that each point
χ e X has a fundamental system of neighbourhoods which are simply connected).
Construct a simply connected covering (π, ρ) : (Q, X) —> (G, X) using an analogue
of the classical construction of the universal covering of a locally simply connected
space.
(Hints: л will be the set of homotopy classes [c] of £/-paths с issuing from a base
point xo e X (all the paths are parameterized by [0, 1]); the projection ρ : X -» X
will map [c] to the endpoint of c. Define a topology on X as follows: given a simply
connected open set U in X and [c] e X with χ — p([c]) e U, there is a canonical
lifting i/[c] of U in X; namely to у е (7 one associates the homotopy class of the
composition of с with a path in U connecting χ to у (this homotopy class is well
denned because U is simply connected). Show, that the subsets of the form U[C]
constitute a basis for a topology on X and that ρ : X —> X is a covering.
Consider the map that associates to each ([c], g) e X x G with ω(#) = p([c]) the
element [c * g] e X. Show that this map defines a right action of (G, X) on X over
/?. Let [c'] e 7Γι((ί/, Χ), jc0). Check that ([с'], [с]) м> [с' * с] defines a left action of

Proof of the Main Theorem 613
Γ = 7Ti((G, X), xo) on X which is simply transitive on each fibre of ρ and commutes
with the right action of (Q, X).
Let (G, X) be the etale groupoid associated to the action of (G, X) on X overp, and
let ρ : G -+ Xbefhemapsending([c],g)to[c].g.Showthat(^,p) : (G,X) -> (G,X)
is a simply connected covering.)
(2) A Criterion/or Developability. Let (G, X) be a connected etale groupoid such
that X is locally simply connected. Show that the following conditions are equivalent.
(i) (G, X) is developable, i.e. equivalent to the etale groupoid associated to an
action of 7Ti((6, X), xq) on a simply connected space X.
(ii) Each point of X has a simply connected open neighbourhood U such that: if
g e G is not a unit and a(g), w(g) e U, and if с : [0,1) —> (7 is a continuous
path with c(l) = ω(#) and c(0) = a(g), then the homotopy class of the £/-loop
(c, g) is non-trivial.
(Hint for (ii) =>· (i): Using the notations of the preceding exercise, let X be the
quotient space of X by the equivalence relation which identifies [с] е G with [c].g,
where g € G and ω(#) = />([c]). Let q : X —» X be the map associating to [c] its
equivalence class. Check that if condition (2) is satisfied, then q is an etale map. (It
is sufficient to check that the restriction of ρ to an open subset of the form U[c] is
injective.) Then apply lemma (3.14).)
4. Proof of the Main Theorem
Outline of the Proof
We first outline the proof, which is adapted from the proof of the Cartan-Hadamard
Theorem given by S. Alexander and R. L. Bishop (see II.4). Given a groupoid
(G, X) of local isometries, we first define the notion of ^-geodesic. Assuming that
G is Hausdorff and that the metric on X is locally complete and locally convex, we
introduce a topology (see 4.4) on the set X of equivalence classes of ^-geodesies
issuing from a base point x0 e X; for this topology the projection ρ : X —> X sending
each ^-geodesic to its endpoint is etale.
The groupoid G acts naturally on the right on X over p. Let X := X/Q be the
quotient of X by this action. The natural projection q : X —> X is etale (4.5) and
the groupoid (G, X) associated to the action is equivalent to the trivial groupoid X.
Moreover the space X is contractible, in particular it is simply connected. Hence
the equivalent groupoid (G, X) associated to the action of G on X is also simply
connected.
If (G, X) is complete, then ρ : X —» X is a covering (4.7). Therefore (G, X) —>
(G, X) is a Galois covering with Galois group Γ = ni((G, X), xo)· The action of Γ
on X by covering automorphisms commutes with the right action of G, hence it also
acts on X. It then follows from lemma 3.14 that (G, X) is equivalent to the groupoid
associated to the action of Γ on X.

614 Chapter III.С? Groupoids of Local Isometries
^-Geodesies
4.1 ^-Geodesies. Let (G, X) be a groupoid of local isometries. A ^-geodesic is a
£?-path с = (go, c\, g\,..., ct, gk) over the subdivision 0 = ?o < · ■ ■ < h = 1 that
satisfies the following conditions for i = 1,... , k.
(1) c, : [ίί-ь ti] —> X is a constant speed local geodesic.
(2) If gi; : (7 —> £/ is a local section of α defined on an open neighbourhood of
ot(gi) such that gi(a(gi)) = g,, then for ε > 0 small enough, the concatenation
of c,|[if_M/] and cuog,o ci+\ |[ί,,ί,+ε] is a constant speed local geodesic.
Note that any £-path which is equivalent to a б-geodesic is a ^-geodesic. If
(G, X) is the groupoid associated to an action of a group Γ by isometries on a length
space X, then any ^-geodesic joining χ to у is equivalent to a unique ^-geodesic
of the form с = (c\, g\) where C\ : [0, 1] —» X is a constant speed local geodesic
issuing from χ and g\ = (γ, у) with γ.у = с(I) (see 3.9(1)).
4.2 Lemma. Lei (£?, X) be a groupoid of local isometries which is Hausdorff. Fix
g e G with χ = a(g), у = w(g) and let r > 0 be such that the closed balls B(x, r)
and B(y, r) are complete and the metric restricted to these balls is convex (see II. 1.3).
Then there is a unique continuous map g : B(x, r) —> Q such that g(x) = g andaog
is the identity ofB(x, r); moreover h := ω οg is an isometryfrom B(x, r) onto B(y, f).
Proof. Let cy : [0, 1] —> B(x, r) be the unique constant speed geodesic segment
joining χ to x1 € B(x, r). As (G, X) is Hausdorff and B(y, r) is complete, there is a
unique continuous lifting c^ : [0, 1] —> G such that a(cy(i)) = cy(i) and cy(0) = g
(see 2.14). We define g(x') = Cy(l); this gives a map g : B(x, r) —*■ G which
is continuous (because geodesies vary continuously with their endpoints in B(x, r)
where the metric is convex) and ω ο g is a local isometry from B(x, r) to B(y, r). The
same argument applied to the inverse of g shows that this map has an inverse which
is also a local isometry, Hence ω ο g is an isometry from B(x, r) onto B(y, r). D
The next lemma is the analogue of lemma II.4.3.
4.3 Main Lemma. Let (G, X) be a groupoid of local isometries which is
Hausdorff. Assume that the metric on X is locally complete and locally convex. Let
с = (go,c\, gi,... ,Ck, gk) be a G-geodesic joining χ to у over the subdivision
0 = to < · ■ · < tk = I. Then there is an ε > 0 such that:
(i) for each i there are unique continuous maps g, : B(a(gi), ε) —> G which are
sections of a, and are such that gi(a(gi)) = g-,;
(ii) foreachpairofpointslc,y e Xwithd(x,x) < 8,d(y,y) < ε, there is a unique Q-
geodesic с = (g0, c~i, g,,..., ck, gj.) that is defined over the same subdivision
as c, joins χ to y, and is such that d(c,(t), c,(i)) < ε and ~gt = gi(a(^i)) for
i = 0,... ,k and t e [0, 1]. Moreover
1(c) < 1(c) + d(x, x) + d(y, y).

Proof of the Main Theorem 615
Proof. The proof is an adaptation of the argument of Alexander-Bishop [AB90] used
in lemma II.4.3. In the proof of II.4.3, instead of dividing the interval [a, b] into three
equal parts, one can use any fixed partition of the interval.
Choose r > 0 small enough so that, for each i = 1,... ,k and each t e [ί,_ ι, i,],
the closed balls B(c-,(f), r), as well as the closed balls B(x, r) and B(y, r), are complete
and the distance function is convex on these balls.
It follows from (4.2) that, for each i = 0,... ,k, there are unique continuous
maps gi : B{a{g{), r) -> Q which are sections of a such that gi(a(gi)) = g, and
Ηχ — ωο gi is an isometry from B{a{gi), r) onto B(co(gj), r).
We first assume the existence assertion in (ii) and deduce from it the uniqueness
assertion in (ii) and the inequality for the lengths. Consider two 6-geodesics с and
c' defined over the subdivision 0 = ίο < ■ ■ · < h = 1, as in (ii), with endpoints
Зс, у and 3c', y'. Assume ε < r. The function mapping t e [ί,_ι, ί,] to d(c(t), c'(t)) is
continuous on the interval [0, 1] and locally convex, hence convex. This shows that if
с and c' have the same endpoints, then they are equal. The stated inequality concerning
lengths is proved like in lemma II.4.3. Also, provided ε < r, any Cauchy sequences
(3c„) in B(x, ε/2) and (yn) in B(y, ε/2) will converge to points 3c e B(x, ε/2) and
у € B(y, ε/2), and the above convexity argument shows that the sequence of unique
б-geodesics joining 3c„ to yn will converge (uniformly) to the unique 6-geodesic
joining χ to y.
To prove the existence of с for ε small enough, we shall argue by induction on
k. The case к = 1 follows from (II.4.3).
We may assume by induction that the analogue of the lemma is true for the Q-
geodesic d = (go, ■ ■ ■, c*-i > gn) obtained by restricting the 6-geodesic с of the
lemma to the subdivision 0 = fo < ■ ■ ■ < it-i of the interval [0, in] and for some
ε' smaller than r. Let 5 < 0 be small enough so that 5 < ε' and in — 5 > ί^_2·
Let c" : [tk~\ — δ, tk\ —> X be the local geodesic defined by c"(i) = c*(i) for
t e [tk-i,tk] and hk_x(c"(t)) = сы(г) for t e [i*_i - 5, i*_,]· Let z0 = c"(i*-i)
and ζ'ό = c"(tk-\ — δ). Inductively, we define sequences of points z), and z),' in
B(ck(tk-i), ε'). Assume z'n^ and z!,',_l are already defined. The first paragraph of the
proof shows that our inductive hypothesis on c' gives a unique б-geodesic c'n over
the subdivision ft < · · · < it-i joining 3c to z'n_l. Let d'n : [in — 5, tk] —> X be the
unique local geodesic joining zj_, to hk(y) which is r-close to c"(see II.4.3). Define
z'n = <(ft-i) andz;,' = t',«fe-i - δ)).
By convexity we have
d(C Ci) ^ ^^ d(Ci. z,',) for» > 1,
and
d(z'o,z") < d(x,x).
h-i
Similarly,
d(z'n, <+1) < /*7f*"' <*(£-..<,') for» > 1,

616 Chaptering Groupoids of Local Isometries
Fig. QA The proof of the main lemma
and
dtf0,z\)<
Choose a number λ > 1 /2 such that
h-i — δ δ к — in
d(y,y).
max
h-\ h~\ h — h-ι + δ tk — tk-i + 8
<λ < 1.
Assuming d(x, x) < e'andd(y,y) < ε', these inequalities imply that ύ?(ζ,',, z'n+l) <
λ"+'ε' and d(z',[, <+1) < λ"+'ε', hence d(z'0, z'n) and d{^, <) are both smaller than
-^ ε'. Therefore, if we assume that ε < -^ε', then the sequences (z'n) and (z,") that
we have defined inductively, are Cauchy sequences. As the closed balls B(c(t), ε)
are complete and the metric is locally convex, the sequence of б-geodesics c'n
converges uniformly to a б-geodesic c' = (g0, C|,..., gt_|) over the subdivision
0 = ίο < · ■ ■ < /fc_i; this б-geodesic joins χ to z' = limzj,, and g, = g,(c,(i,_i)).
Also, the sequence c"n converges to a local geodesic c" : [in — 5, it] —» X joining
z" = limzn' to hk(y). Moreover hk^(c"(t)) = с*_| (ί) for t e [in - 5, ί*-ι]·
Therefore, if we define ck = c"\[tt_ljt] andg* = gk(y), then с = (g0,..., си,й_,, ck, gk)
is the 6-geodesic joining χ to у that we were looking for. Π
The Space X of 6-Geodesics Issuing from a Base Point
Let X be the set of equivalence classes of S-geodesics parameterized by [0, 1] that
issue from a base point xq e X. Let ρ : X —> X be the map associating to each
equivalence class its terminal point. Let xq denote the point of X represented by the
constant б-geodesic at xq, namely (go, C\, g\), where go = g\ = 1*0 anc^ ci № = xo
for each ί e [0,1].

Proof of the Main Theorem 617
Given a 6-geodesic с = (go,, cx, ..., gk) joining xq to a point у over the
subdivision 0 = ίο < · ·' < h — 1, we shall write Xec to denote the set of g-geodesics
given by (4.3) that join xq to the various points of B(y, ε), and we write Xec to denote
its image in X. The projection ρ maps Xec bijectively onto B(y, ε). This shows that
the map Xec —>- Xec sending each 6-path to its equivalence class is injective.
Note that if we refine the subdivision of с we don't change Xec. Therefore there
is no loss of generality if we assume in what follows that the б-geodesic с =
(go, с ι,..., ck, gk) satisfies the following condition: there exists r > 0 such that the
closed balls of radius r centred at the sources and targets of the g-t, and at each point
Ci(f), are complete geodesic spaces on which the distance function is convex; we also
assume that c,([i,_b i,]) С B(c,(i,_i), r/3) (in particular each c, is a constant speed
geodesic segment). We shall consider sets Xec with ε < r.
4.4 Proposition. With the assumptions of lemma 4.3, the subsets Xec с X form a
basis for a Hausdorff topology on X, which respect to which ρ is etale.
Proof Consider two sets Xec and X^, which intersect. Without loss of generality,
we can assume that с and c' are defined over the same subdivision 0 = ίο <
• · · < tk = 1 and satisfy the condition preceding the statement of the
proposition. By construction p(Xec) апар(Щ) are balls B(y, ε) and B(y', ε') in X. Assume
that с = (go, ci,..., ~gk) e Xsc and c' = (g0, c\,..., ~g'k) e XSJ are two equivalent
6-geodesics. The image under ρ of the equivalence classes of с and c' is a point
z e B(y, ε) П В(У, ε').
The proposition is a consequence of the following assertion.
Claim: If δ is such that B(z, S) С B(y, ε) П В(У, ε'), then Β(ζ, δ) С р(Хес П χ£).
By definition, there are continuous maps h; : \t-,-\, t{\ —> Q such that a(hi(t)) =
Ci(t), ω(Α,·(0) = c;.(i)and/i;(i,-)g) = £Λ+ι(ί,·) fori = 1,... ,k-l,a\sog0 =]>'0^(0)
and g^. = hk(\)~gk. Using lemma 4.2, we can find liftings й, : Β(βι(ί^{), 5) -^ Q of
α such that fy(c(i;_i)) = /ϊ,(ί,_ι). As the projection α is etale and б is Hausdorff, for
each i = 1,... ,k — 1 and и е B(c,(i,), 5) we have
hi(co(gi(u))) gi(u) = g'^(hi+l(u))) hi+l(u),
also go(") = §о(ш(^1(и)) h\(u) and hk(w(gk(u))gk(u) = ^.(и) for each и in,
respectively, Β(β\ (0), 5) and B(ce(~gk), δ), because these two last equalities are true when и
is the centre of the ball. This implies that the /г, induce an equivalence from elements
of Xl С X^- to elements of Xi С Χί'. Thus Xl = Xt. This proves the claim. D
с с с с с с г
The Space Я = tlQ
Given an element χ e X represented by a 6-geodesic с = (go. Ci,..., #t) and an
element g e Q with ω(#) = p(x), we write jc.g for the element of X represented
by the 6-geodesic eg obtained from с by replacing its last entry gk with gkg. This
defines a continuous right action of Q on X over ρ : X —» X.

618 Chapter III.C? Groupoids of Local Isometries
Let (Q, X) be the etale groupoid associated to this action (see 3.11). The following
lemma is analogous to (II.4.5).
4.5 Lemma. Let X be the quotient ofX by the right action ofQ described above,
with the quotient topology. (X is also the space of orbits of the groupoid Q.) Then X is
Hausdorff and contractible. The natural projection q : X —> X is etale and induces
an equivalence from (Q,X) to the trivial groupoid X. In particular the groupoid
(Q, X) is simply-connected.
Proof. We first note that if g e Q is not a unit and if co(g) is the terminal point of c,
then eg is not equivalent to c, because Q is Hausdorff (see 3.2). More generally, if
g : B(a(g), ε) -* B(w(g), ε) is a lifting of α such that g(a(g)) = g, then Xec П Xecg =
0. This shows that q : X —» X is etale, that X is Hausdorff, and that the map
(x, g) ι-* q(x) gives an equivalence from the groupoid (Q, X) to the trivial groupoid
X.
It remains to check that X is contractible. Given s e (0, 1] and χ e X represented
by the б-geodesic с = (go, C\, ..., gk) over the subdivision 0 = t\ < · ■ · < i* = 1,
let rs(x) e X be the element of X represented by the 6-geodesic obtained from с
by restricting the parameter t to the interval [0, s] and reparameterizing by [0, 1].
(Note that if s = tj, then the б-geodesics (go, C\, ..., c,) and (go,C\, ... c,, g,) over
the subdivision 0 = ίο < · · · U = s of [0, s] (reparameterized by [0, 1]) represent the
same point of X.) Then (s, x) м> rs(x) is a continuous map [0, 1] x X —> X defining a
homotopy that connects the constant map from X to xq to the identity map of X. D
The Covering ρ :Χ^ Χ
4.6 Lemma. With the assumptions of Theorem 2.15, given χ e X and a rectifiable
path s ι-* γ in X parameterized by s e [0, 1], with y° = p(x), there is continuous
path s и-* у9 in X such that /?(ys) = ys and ψ = χ.
Proof. We assume that s i-> y9 is a constant speed parameterization. As ρ is etale,
the maximal interval containing 0 on which the lifting s i-> y5 can be defined is
open. So it will be sufficient to prove that if a lifting s i-> y5 is defined on [0, a), then
linij^a yr exists. Let £ be the sum of the length of the path s м> ys and the length of
a 6-geodesic representing y°.
Let q : X -+ Q\X be the natural projection. We endow Q\X with the quotient
pseudometric and its associated topology. For s e [0, a), if yr is represented by the
б-geodesic cs = (g^, c\,..., gsn) over the subdivision 0 = i^<--<f9=l, then
the map С : [0, 1] χ [0, α) -> б\Х sending (i, j), with i e [ί,_ι, ί,], to ?(cf(i)) is
well-defined. For ti,t2 e [0, 1] and si,S2 e [0, a), by (4.3) we have
а{С{ц, л), C(f2, i2)) < £(|fi - f2| + ki - J2|).
Therefore the map С extends to a continuous map [0, 1] χ [0, a] —> S\X, still
denoted C. By the compactness of the interval [0, 1], we can find a subdivision

Proof of the Main Theorem 619
0 = to < ■ ■ ■ < tk = 1, & number ε > 0 and points xq, x\ ,..., xk in X such that
q(xi) = C(ti, a), Xk = y(a), the balls B(x,, 3ε) are complete and geodesic with convex
distance function, and each (i/ — i,_i) is smaller than ε /I.
Let s\ e [0, a) be such that the length of the curve sh>/ restricted to [s\,d\
is smaller than ε and d{C{t,s\), C(t,a)) < ε for all t e [0,1]. Using (2.14) we
can find points z, e Β(χ,, ε) for i = 1,... ,k — 1 such that q{zf) = C(i,, si); let
Zo = *o and Zk — y" · Using (2.14) again, we can find a representative of y" of the
form c" = (go , с|',..., g^1) over the subdivision 0 = ίο < ■ ■ ■ < h = 1, where
cj'ft-i) = z,-i and gj is the unit lXo. We claim that we can also find, for each
s e [i|, a), a unique representative cf = (gs0, c\,..., gsk) of y5 defined over the same
subdivision 0 = ίο < · · · h = 1, where g- and cf vary continuously with s. By
lemma 4.3 this is possible for s close to s\. Suppose that cf can be defined on the
interval [sl, s2); the length of each path c] is smaller than ε, as is the length of each
path s м> cs(tj) (because its length is no greater than that of the corresponding path
s ι-* y(s), by lemma 4.3); therefore each path cf is contained in B(x-,-\, 3ε); as the
closure of these balls is assumed to be complete, we can define cf2 as the limit of
cs as s —> S2, and hence cs can be defined on the interval [s\, J2]· If ^2 < 1, we can
extend the construction of c5 to a larger interval using (4.2), and eventually to the
whole interval [s 1, a]. D
4.7 Corollary. If(G, X) is a groupoid of local isometries which is connected, Haus-
dorff, complete and such that the metric on X is locally convex, then ρ : X —> X is a
covering ofX.
Proof. Let с = (go, C\,..., g„) be a rectifiable Q-path defined over a subdivision
0 = so < ■ ■ ■ < s„ = 1 such that cu(go) = xq. Using the preceding lemma, we
can construct, by induction on n, a 6-path с = (go, C\,..., g„) defined over the
same subdivision such that go = (jcq, go), and c, : [i,~i, s(\ —> X is the unique path
in X such that/?(i) = c,-(j) and c,-(j;_i) = C;_|(i,_i).g,_i. In particular/? maps the
terminal point of с to the terminal point of c. As X is g-connected, this shows that ρ
is surjective. The preceding lemma implies that ρ is a covering over each connected
component of X. G
The End of the Proofofthe Main Theorem. Let я : Q —> Q be the map (jc, g) i-> g.We
have just seen that (π,ρ) : φ, Χ) -» {Q, X) is a covering. By (4.5), (Q, X) is simply
connected, therefore (π, ρ) is a Galois covering with Galois group Γ = ττι ((б, X), *ο)
(see (3.18). Thus the theorem follows from 3.14. D

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Index
D„ (dihedral group), 378
DK (diameter of MnK), 24
мпк,гъ
0(n, 1), 307
P(n,R), 314
P(n,R),,324
S(n,R), 314
i/(n, 1), 307
Κ"·\301
R-tree, 167, 399
£-paths, 604
A,„(G;H), 482
5-hyperbolic (see hyperbolic), 398, 407,
«-cones, 59
- curvature of, 188
0K(Q), 307
c^limit, 79
A(p,q,r), 158
π, -injectivity of local isometries, 200
ε-filling, 414
ε(χ), 100, 114
(G,Y)-stracture, 591
etale map, 42
1-2-3 Theorem, 488
3-manifold, 258, 496, 502, 510
4-point condition
- for CAT(re) spaces, 164
- for hyperbolicity, 410
action of a group
- cobounded, 137
- cocompact, 131
- extension to the universal covering, 3i
- on a scwol, 528
- on a space (terminology), 131
- proper, 131
- quasi-analytic, 591
- strata preserving, 372
action of a groupoid, 599, 609
Alexandrov's Lemma, 25
Alexandrov's patchwork, 199
algebraic group, 327
all-right spherical complex, 127, 210
amalgamated free product, 377, 497
- along a free subgroup, 503
- along an abelian subgroup, 500
amalgamation of metric spaces, 67
angle, 9
- «-comparison, 25
- Alexandrov (upper), 9, 173
- alternative notions of, 162
- between points at infinity, 280
- comparison, 8
- continuity properties, 184, 278
- in CAT(re) spaces, 184
- in the bordification, 278, 281
- notation for, 184
- Riemannian, 39, 173
- strong upper angle, 11
angular metric, 280
annular diagrams, 454
apartment, 337, 343
approximate midpoints, 30, 32, 160, 164
area, 414
- of a surface, 425
AreaE, 414
Arzela-Ascoli theorem, 36
ascending chain condition, 247
aspherical manifold, 213
asymptotic
- geodesic lines, 182
- geodesic rays, 260, 427, 428
axis (of an isometry), 231
Banach space, 5
barycentre, 116
barycentric
- coordinates, 124, 125
- subdivision, 116-118, 125

638 Index
Basic Construction, 381, 542
- properties of, 384, 545
bicombing, 471
Bieberbach Theorem, 246
bordification, 260
Borromean rings, 224
boundary at infinity
- isaCAT(l)Space, 285
- as an inverse limit, 263
- horofunction construction, 267
- oftf",90
- ofKH",309
- of a 5-hyperbolic space (see Gromov
boundary), 427
- of a product, 266
- of an R-tree, 266
Britton's Lemma, 498
building (spherical, Euclidean, thick), 343
Busemann function, 268, 273, 428
- in P(n, R), 335
Cartan-Hadamard Theorem, 193
CAT(re)
- 4-point condition, 163
- implies САТ(к/) if re < re', 165
- inequality, 158, 161
- limits of, 186
- space, 159
category
- associated to a group action, 574
- equivalence, 575
- small, 573
- without loops (scwol), 519
Cayley complex, 153
Cay ley graph, 8, 139
Cayley transform, 90
centralizers
- in hyperbolic groups, 462
- in semihyperbolic groups, 477
- virtual splitting, 234
centre of a bounded set, 178
characteristic map of a cell, 101
circle isometrically embedded, 202
circumcentre, 179
Clifford translation, 235
CN inequality, 163
coarse, 138
coboundary, 535
cocompact
- group action, 131
- space, 202
collapse, 219
combinatorial complex, 153
combinatorial map, 153, 217
combing line, 471
commensurable groups, 141
comparison angle, 8
- lower semicontinuity, 286
comparison point, 158
comparison triangle, 8, 158
completion of a CAT(0) space, 187
complex
- MK -polyhedral, 114
- (abstract) simplicial, 123
- cubed, 115
- cubical, 111,212
- metric simplicial, 98
- squared, 115,223
complex of groups, 535
- associated category, 538
- associated to an action, 539
- developable, 541
- fundamental group of, 548
- induced, 538
- non-developable, 379
- of curvature < re, 562
- simple, 375, 535
concatenation of paths, 12, 547, 606
cone
- asymptotic, 79
- Euclidean, 60
- over a metric space, 59
- simplicial, 124
cone topology, 263, 281, 429
cone type, 455
conjugacy problem, 441, 446, 453, 474, 489
convex
- r-convex, 4
- function, 175
- hull, 112
- metric, 159, 176, 193,603
- subspace, 4, 176
convexity
- of balls in CAT(re) spaces, 160
- of the metric, 176
covering, 42
- Galois, 46
- map, space, 42
- of a category, 579
- of a complex of groups, 566
- of scwols, 527
- of etale groupoids, 609
Coxeter
- complex, 393
- graph, group, system, 391
cross ratio, 82, 85

Index 639
cubical complex, 111
- of non-positive curvature, 212
- with specified link, 212
curvature
- < к (in the sense of Alexandrov), 159
- in the sense of Busemann, 169
- non-positive, 159
- of MK -polyhedral complexes, 206
- ofKH",304
- sectional, 171
Davis Construction, 213
decision problems, 440, 494
Dehn complex (of a knot), 222
Dehn function, 155, 444, 451, 487, 504, 507
Dehn presentation, 450
Dehn twist, 257
Dehn's algorithm, 417, 449
developable
- etale groupoid, 597
- complex of groups, 541
developing map, 46
development
- of a (G, y)-structure, 592
- of an m-string, 104
development D(y, φ), 542
disjoint union of metric spaces, 64
displacement function, 229
distance, 2
distortion, 488, 506, 509
divergence function, 412
doubling
- along a subgroup, 482, 498, 504
- along a subspace, 498
dunce hat, 115,533
edge path, 526
Eilenberg-MacLane space, 470
elementary homotopy, 154, 527, 577
Embedding Theorem, 512
ends of a space, 144, 430
equivalence
- ~ of functions, 415
- of £/-paths, 604
- of categories, 575
- of etale groupoids, 597, 600
equivariant gluing, 513, 518
Euclidean
- de Rham factor, 235, 299
- space, 15
excess of a triangle, 168
exponential map, 94, 170, 196, 316
fibre of a morphism, 568
fibre product, 488
filter, 78
finite presentation of a group, 135
finite state automaton, 456
finiteness conditions FPn,FLn, F„, 470, 481
Finsler metric, 41
first variation formula, 185
flag, 340
flag complex, 210
flat, 321
- half-plane, 290
- manifold, 246, 255
- polygons, 181
- sector, 283
- subspaces, 247, 278, 296
- triangle, 180
Flat Plane Theorem, 296, 459
Flat Strip Theorem, 182
Flat Torus Theorem, 244, 254
- algebraic, 475
foliation, 596
- Riemannian, 602
frame, 340
free face, 208
free group, 134
functor, 574
- homotopic, 575
geodesic, 4
- Mocal, 405
- closed local, 38
- local, 4, 160, 194
- parallel, 182
- ray (see asymptotic), 4
- regular, singular, 322
- terminology for, 4
- uniqueness in CAT(re) spaces, 160
- word, 452
geodesic extension property, 207, 237
geodesic metric space, 4
geodesic simplex, 98
geodesic space, 4
geometric realization
- of a poset, 370
- of a scwol, 522
- of a simplicial complex, 124
geometric structure, 591
germ, 45, 592
- topology, 46
girth, 210
gluing, 67
- equivariant, 355

640 Index
- of CAT(re) spaces, 347
- using local isometries, 351
Gluing Lemma for Triangles, 199
graph
- Cayley, 8
- combinatorial, 7
- metric, 7
- of groups, 498, 534, 554
graph product of groups, 390
Gromov boundary, 427
- as a set of rays, 427
- as classes of sequences, 431
- topology, 429
- visual metric, 435
Gromov polyhedra, 394
Gromov product, 410, 432
Gromov-Hausdorff (see also limit)
- distance, 72
- pointed convergence, 76
groupoid, 594
- etale, 594
- associated to a complex of groups, 595
- associated to an action, 594
- complete, 601
- connected, 602, 607
- Hausdorff, 601
- of local isometries, 601
- simply-connected, 607
- topological, 594
- trivial, 595
groups
- amalgam of, 376
- as geometric objects, 139
- direct limit of, 376, 386
- hyperbolic, 509
- n-gon of, 379
- nilpotent, 149
- poly cyclic, 149,252,479
- semihyperbolic, 472
- triangle of, 377
growth of groups, 148, 391
- polynomial, 148
- rational, 457
Hadamard space, 159
Hausdorff distance, 70
Hubert space, 47
Hirsch length, 149
HNN extension, 492, 497, 552
- trivial, 498
holonomy
- pseudogroup, 596
- group, 46, 592
- groupoid, 596
homogeneous coordinates, 81, 302
homology manifold, 209
homomorphism of groupoids, 597
- etale, 597
homotopy of £/-paths, 606
Hopf link, 224
Hopf-Rinow theorem, 35
Hopfian groups, 513
horoball, 267
horofunction, 267
horosphere, 267, 428
- inKH",310
horospherical subgroup, 332
hyperbolic (see also 5-hyperbolic)
- (<5)-hyperbolic, 411
- group, 448
- metric space, 398
- quasi-isometry invariance, 402, 412
hyperbolic space (H" and KH"), 19, 302
- ball model, 310
- embedding in P(n, R), 330
- hyperboloid model, 18
- Klein model, 82
- parabolic model, 310
- Poincare ball model, 86
- Poincare half-space model, 90
hyperplane
- bisector, 16, 18, 21
- inE", 15
- inS", 18
- inH",21
incoherence, 227
injectivity radius, 119, 202
internal points of a triangle, 408
inversion, 85
isometries, 2
- elliptic, 229
- finite groups of, 179
- hyperbolic, 229, 231
- ofM^,26
- of H", 229
- ofKH",307
- of a compact space, 237
- parabolic, 229, 274
- Riemannian, 41
- semi-simple, 229-231, 233, 331
isomorphism problem, 441
isoperimetric inequality, 414
- linear, 417, 419, 449
- quadratic, 416, 444
- sub-quadratic, 422

Iwasawa Decomposition of GL(n, R), 323
Jacobi vector field, 171
join
- curvature of, 190
- of morphisms, 532
- of posets, 371
- of scwols, 531
- simplicial, 124
- spherical, 63
Klein bottle, 246
knots and links, 220
lattice, 142, 249, 300, 362
law of cosines
- Euclidean, 8
- hyperbolic, 20
- inM",24
- in KH", 303
- spherical, 17
length
- of a chain, 65
- of a curve, 12
- of a string, 99
- Riemannian, 39
length metric, 32
- induced, 33
- on a covering, 42
length space, 32
limit
- 4-point, 186
- Gromov-Hausdorff, 72, 186
- of CAT(re) spaces, 186
link
- geometric, 102, 103, 114
- lower, 532
- simplicial, 124
- upper, 370, 532
link condition, 206
- for 2-complexes, 215
local development, 388, 558, 565
local groups, 535
local homeomorphism, 42
loxodromic, 332
manifold of constant curvature, 45
map
- etale, 42
- simplicial, 124
mapping class groups, 256
Markov properties, 455
membership problem, 488
Index 641
metric
- convex, 159
- Finsler, 41
- induced, 2
- induced length metric, 33
- locally convex, 169
- Riemannian, 39
metric space, 2
- geodesic, r-geodesic, 4
- locally uniquely geodesic, 43
- proper, 2
- totally bounded, 74
- uniquely geodesic, 4
Min(7), 229
monodromy of a covering, 572, 581
morphism
- homotopy of, 537
- non-degenerate, 371, 526
- of complexes of groups, 536
- of posets, 371
- of scwols, 526
- simple, 376
- strata preserving, 372
Moussong's Lemma, 212
Mobius group, 85
nerve of a cover, 129
no triangles condition, 210
non-positively curved, 159
norm, 5
- £p, 6, 47, 53
- of a quaternion, 301
normal closure, 134
orbifold, 585
- change of charts, 585
- developable, 587
- Riemannian, 586
orbit of a point
- under a groupoid, 594
parabolic subgroup, 332
parallelogram law, 48
ping-pong lemma, 467
Plateau's problem, 426
points at infinity, 260
poset, 368
- affine realization, 370
pre-complex of groups, 576
pre-Hilbert space, 47
presentation of a group, 134
product decomposition
- and the Tits metric, 291

642 Index
- ofMin(7), 231
Product Decomposition Theorem, 183,
291
product of CAT(re) spaces, 168
product of metric spaces, 56, 58
projection (orthogonal)
- onto convex subspaces, 176
projective space, 81, 302
proper
- group action, 131
- metric space, 2
pseudogroup, 586
- associated to an etale groupoid, 594
pseudometric, 2
- positive definite, 2
q.m.c. property, 445
quasi-centre, 459
quasi-conformal structure, 436
quasi-geodesics, 142
- in hyperbolic spaces, 401
- taming, 403
quasi-isometric embedding, 138, 507
quasi-isometries, 138
- invariants of, 142
- preserve hyperbolicity, 402, 412
quasi-projection, 463
quasiconvexity, 460, 475
quaternions, 301
quotient pseudometric, 65
rank
- of a symmetric space, 299
- of an abelian group, 247
rays (see geodesic, asymptotic)
- quasi-geodesic, 427
realization
- affine, 124, 125,370,371
- all-right spherical, 127
- Euclidean, 127
rectifiable curve, 12
reflection through a hyperplane, 16, 18, 22,
85
regular geodesic, 335
regular language, 457
residually finite group, 511
reverse-inclusion, 371
Rips complex, 468
Rips construction, 224, 488
ruled surface, 426
Sandwich Lemma, 182
scalar product, 47
scwol, 521
- associated to a poset, 521
- connected, 521
- geometric realization, 522
- opposite, 521
- vertex and edge of, 521
Seifert-van Kampen Theorem
- for complexes of groups, 552
- for etale groupoids, 608
semi-direct product of groups, 52
semi-simple
- isometry, 229, 331
- matrix, 331
semihyperbolic, 472
Shapes(K), 98
simple complex of groups, 375
- of curvature < «,388
- simple morphism of, 376
simple groups, 512
simplicial complex
- abstract, 123
- metric, i.e. piecewiseMK, 98
small cancellation, 216, 220
Solvable Subgroup Theorem, 248
source, 594
- of a germ, 592
space of directions, 190
spherical join, 63
- as the boundary of a product, 284
Splitting Theorem, 239, 291
- algebraic, 480
Stallings-Bieri groups, 483, 485
star (open, closed), 99, 114, 125, 369, 555
stereographic projection, 84
stratified
- MK -polyhedral complex, 369
- complex, set, space, 368
strict fundamental domain, 372, 373
strictly developable, 377, 389
string, 99
- taut, 106
subembedding of a 4-tuple, 164
subgroup
- abelian, 244, 249, 475
- central, 234
- distortion, 488, 506
- finite, 459
- finite index, 511
- free, 467
- quasi-isometrically embedded, 461
- reductive, 327
- retract, 504
subscwol, 521

Index 643
support, 99, 113
Svarc-Milnor Lemma, 140
symmetric space, 299
- of non-compact type, 299
- rank of, 299
systole, 202
- of MK-polyhedral complex, 203
tangent cone, 190
tangent vectors in KH", 302
target, 594
- of a germ, 592
tesselationsofH2, 384
Tits boundary, metric, 289
- isCAT(l), 289
torsion groups, 250
torus bundles, 141,502
totally geodesic submanifold, 324, 334
totally real subspace, 306
tower, 217
translation length
- algebraic, 464, 466
- of an isometry, 229, 253
transvection, 315
tree, 7
- Bass-Serre, 355, 358, 554
triangle
- S-slim, 399
- <5-thin,407
- comparison, 24, 158
- geodesic, 158
- hyperbolic, 20, 303
- insize of, 408
- internal points of, 408
- spherical, 17
triangulation, 414
truncated hyperbolic space, 362
twisting element, 535
ultrafilter (non-principal), 78
ultralimit, 79, 186
uniform structure, 433
uniformly compact, 74
universal development
- of a complex of groups, 553
van Kampen diagram, 155, 505
Vandermonde determinant, 330
virtual properties of groups, 245
visibility
- local, uniform, 294
- of hyperbolic spaces, 400, 428
- spaces, 294
visual boundary, 264
visual metric, 434
weak topology, 125, 368
Weyl chamber, 322, 337
word
- geodesic, 452
- reduced, 134
word metric, 139
word problem, 441, 442, 449, 474, 487

Grandlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
A Selection
219. Duvaut/Lions: Inequalities in Mechanics and Physics
220. Kirillov: Elements of the Theory of Representations
221. Mumford: Algebraic Geometry I: Complex Projective Varieties
222. Lang: Introduction to Modular Forms
223. Bergh/Lofstrom: Interpolation Spaces. An Introduction
224. Gilbarg/Tradinger: Elliptic Partial Differential Equations of Second Order
225. Schiitte: Proof Theory
226. Karoubi: K-Theory. An Introduction
227. Grauert/Remmert: Theorie der Steinschen Raume
228. Segal/Kunze: Integrals and Operators
229. Hasse: Number Theory
230. Klingenberg: Lectures on Closed Geodesies
231. Lang: Elliptic Curves. Diophantine Analysis
232. Gihman/Skorohod: The Theory of Stochastic Processes III
233. Stroock/Varadhan: Multidimensional Diffusion Processes
234. Aigner: Combinatorial Theory
235. Dynkin/Yushkevich: Controlled Markov Processes
236. Grauert/Remmert: Theory of Stein Spaces
237. Kothe: Topological Vector Spaces II
238. Graham/McGehee: Essays in Commutative Harmonic Analysis
239. Elliott: Probabilistic Number Theory I
240. Elliott: Probabilistic Number Theory II
241. Rudin: Function Theory in the Unit Ball of Cn
242. Huppert/Blackburn: Finite Groups II
243. Huppert/Blackburn: Finite Groups III
244. Kubert/Lang: Modular Units
245. Cornfeld/Fomin/Sinai: Ergodic Theory
246. Naimark/Stern: Theory of Group Representations
247. Suzuki: Group Theory I
248. Suzuki: Group Theory II
249. Chung: Lectures from Markov Processes to Brownian Motion
250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations
251. Chow/Hale: Methods of Bifurcation Theory
252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations
253. Dwork: Lectures on p-adic Differential Equations
254. Freitag: Siegelsche Modulfunktionen
255. Lang: Complex Multiplication
256. Hormander: The Analysis of Linear Partial Differential Operators I
257. Hormander: The Analysis of Linear Partial Differential Operators II
258. Smoller: Shock Waves and Reaction-Diffusion Equations
259. Duren: Univalent Functions
260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems
261. Bosch/Giintzer/Remmert: Non Archimedian Analysis - A System Approach
to Rigid Analytic Geometry
262. Doob: Classical Potential Theory and Its Probabilistic Counterpart
263. Krasnosel'skiiyZabreiko: Geometrical Methods of Nonlinear Analysis
264. Aubin/Cellina: Differential Inclusions
265. Grauert/Remmert: Coherent Analytic Sheaves
266. de Rham: Differentiable Manifolds
267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I
268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol.11

269. Schapira: Microdifferential Systems in the Complex Domain
270. Scharlau: Quadratic and Hermitian Forms
271. Ellis: Entropy, Large Deviations, and Statistical Mechanics
272. Elliott: Arithmetic Functions and Integer Products
273. Nikol'skn: Treatise on the Shift Operator
274. Hormander: The Analysis of Linear Partial Differential Operators ΙΠ
275. Hormander: The Analysis of Linear Partial Differential Operators IV
276. Liggett: Interacting Particle Systems
277. Fulton/Lang: Riemann-Roch Algebra
278. Barr/Wells: Toposes, Triples and Theories
279. Bishop/Bridges: Constructive Analysis
280. Neukirch: Class Field Theory
281. Chandrasekharan: Elliptic Functions
282. Lelong/Graman: Entire Functions of Several Complex Variables
283. Kodaira: Complex Manifolds and Deformation of Complex Structures
284. Finn: Equilibrium Capillary Surfaces
285. Burago/Zalgaller: Geometric Inequalities
286. Andrianaov: Quadratic Forms and Hecke Operators
287. Maskit: Kleinian Groups
288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes
289. Manin: Gauge Field Theory and Complex Geometry
290. Conway/Sloane: Sphere Packings, Lattices and Groups
291. Hahn/O'Meara: The Classical Groups and K-Theory
292. Kashiwara/Schapira: Sheaves on Manifolds
293. Revuz/Yor: Continuous Martingales and Brownian Motion
294. Knus: Quadratic and Hermitian Forms over Rings
295. Dierkes/Hildebrandt/Kiister/Wohlrab: Minimal Surfaces I
296. Dierkes/Hildebrandt/Kiister/Wohlrab: Minimal Surfaces II
297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators
298. Berline/Getzler/Vergne: Heat Kernels and Dirac Operators
299. Pommerenke: Boundary Behaviour of Conformal Maps
300. Orlik/Terao: Arrangements of Hyperplanes
301. Loday: Cyclic Homology
302. Lange/Birkenhake: Complex Abelian Varieties
303. DeVore/Lorentz: Constructive Approximation
304. Lorentz/v. Golitschek/Makovoz: Construcitve Approximation. Advanced Problems
305. Hiriart-Urruty/Lemarechal: Convex Analysis and Minimization Algorithms I.
Fundamentals
306. Hiriart-Urruty/Lemarechal: Convex Analysis and Minimization Algorithms II.
Advanced Theory and Bundle Methods
307. Schwarz: Quantum Field Theory and Topology
308. Schwarz: Topology for Physicists
309. Adem/Milgram: Cohomology of Finite Groups
310. Giaquinta/Hildebrandt: Calculus of Variations I: The Lagrangian Formalism
311. Giaquinta/Hildebrandt: Calculus of Variations Π: The Hamiltonian Formalism
312. Chung/Zhao: From Brownian Motion to Schrodinger's Equation
313. Malliavin: Stochastic Analysis
314. Adams/Hedberg: Function Spaces and Potential Theory
315. Burgisser/Clausen/Shokrollahi: Algebraic Complexity Theory
316. Saff/Totik: Logarithmic Potentials with External Fields
317. Rockafellar/Wets: Variational Analysis
318. Kobayashi: Hyperbolic Complex Spaces
319. Bridson/Haefliger: Metric Spaces of Non-Positive Curvature
320. Kipnis/Landim: Scaling Limits of Interacting Particle Systems
321. Grimmett: Percolation
322. Neukirch: Algebraic Number Theory
323. Neukirch/Schmidt/Wingberg: Cohomology of Number Fields
324. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes