M. S. Raghunathan
Discrete Subgroups
of Lie Groups
Springer -Verlag Berlin Heidelberg New York
1972

M. S. Raghunathan
Tata Institute of Fundamental Research
Bombay, India
AMS Subject Classifications (1970):
Primary 22E40
Secondary 32NXX, 20HXX
ISBN 3-540-05749-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-05749-8 Springer-Verlag New York Heidelberg Berlin
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To my parents

Preface
This book originated from a course of lectures given at Yale University
during 1968-69 and a more elaborate one, the next year, at the Tata
Institute of Fundamental Research. Its aim is to present a detailed
account of some of the recent work on the geometric aspects of the theory
of discrete subgroups of Lie groups. Our interest, by and large, is in a
special class of discrete subgroups of Lie groups, viz., lattices (by a
lattice in a locally compact group G, we mean a discrete subgroup H
such that the homogeneous space G\H carries a finite G-invariant
measure).
It is assumed that the reader has considerable familiarity with Lie
groups and algebraic groups. However most of the results used frequently
in the book are summarised in "Preliminaries"; this chapter, it is hoped,
will be useful as a reference.
We now briefly outline the contents of the book. Chapter I deals with
results of a general nature on lattices in locally compact groups. The
second chapter is an account of the fairly complete study of lattices in
nilpotent Lie groups carried out by Malcev. Chapters III and IV are
devoted to lattices in solvable Lie groups; most of the theorems here are
due to Mostow. In Chapter V we prove a density theorem due to Borel:
this is the first important result on lattices in semisimple Lie groups.
The next two chapters are somewhat of a digression from the main trend
of the book and may be omitted without any break in continuity (though
occasionally one of the theorems proved in Chapter VT is used elsewhere).
Chapter VI contains some general theorems on finitely generated
subgroups of Lie groups while Chapter VII is on cohomological results for
discrete subgroups of Lie groups (work of Mostow on the cohomology of
solv-manifolds and some results of Weil and Matsushima-Murakami on
compact locally symmetric spaces). Chapter VIII plays a central role; it is
indispensable for the rest of the book. The main result here is a theorem
due to Zassenhaus which has proved to be basic for the entire theory.
Among other results proved here are theorems due to H. C. Wang and
L. Auslander which show, among other things, that, at least up to a point,
the study of lattices in general Lie groups can be split into studying those

VIII
Prefac
in solvable and semisimple groups separately. A maximality property
due to H. C. Wang for lattices in semisimple groups is the subject matter
of Chapter IX. Chapter X is on reduction theory in GL(«) and the
compactness criterion: we follow here the methods of Mostow-Tamagawa.
The next chapter is devoted to some work of fundamental importance
due to Kazdan-Margolis (proofs of conjectures of Selberg and Siegel).
It also contains some unpublished work of the author. After a quick
survey of results on algebraic groups in Chapter XII, Chapter XIII takes
up the problem of constructing "good" fundamental domains for a
special class of discrete subgroups which we call rank-1 discrete
subgroups. It is shown here further that the class is big enough to include
"irreducible" lattices in groups with a rank-1 factor as well as arithmetic
subgroups of 2-rank-l algebraic Q-groups. A new feature here is the
simultaneous treatment of lattices on the one hand and arithmetic
groups on the other. The results on arithmetic groups in this chapter are
due to Borel-Harish-Chandra and Borel while those on general lattices
are generalisations of theorems due to Garland and the author. The final
chapter is on a theorem of Borel which guarantees the existence of
(arithmetic) lattices in semisimple Lie groups.
Two topics are perhaps conspicuous by their omission: the extensive
work on SL(2) and non-arithmeticity questions. However, elaborate
expositions of the former topic are available in existing literature while
the latter requires far more preparatory material than is described in
"Preliminaries". While the book lays no claim to great originality the
knowledgeable reader may recognise many proofs as new and hopefully
more transparent than the original ones.
I wish to thank Howard Garland for reading the initial chapters of
the book and offering many helpful and constructive comments on
them. My thanks are also due to Gopal Prasad and V. J. Lai who read
the proofs and gave me suggestions for improving the exposition. Mr.
K. K. Thilakan and Mr. P. Joseph typed the manuscript between them;
I extend my thanks to them for their painstaking job. Finally I would
like to take this opportunity to express my gratitude to my colleagues
M. S. Narasimhan and S. Ramanan; they were not directly involved
with this book but were largely responsible for the shaping of my
mathematical career in its formative period.
Bombay, March 1972
M. S. Raghunathan

Table of Contents
Preliminaries 1
1. Lie Groups and Lie Algebras 1
2. Algebraic Groups 7
3. Group and Lie Algebra Cohomology 12
4. Principal Bundles 14
Chapter I. Generalities on Lattices 16
Chapter II. Lattices in Nilpotent Lie Groups 29
Chapter III. Lattices in Solvable Lie Groups 43
Chapter IV. Polycyclic Groups and Arithmeticity of Lattices in
Solvable Lie Groups 56
Chapter V. Lattices in Semisimple Lie Groups: The Density
Theorem of Borel 78
Chapter VI. Deformations 89
Chapter VII. Cohomology Computations 105
1. deRham's Theorem 105
2. Hodge's Theory for Local Systems 108
3. Discrete Subgroups in Lie Groups Ill
4. Solvable Lie Groups 115
5. Semisimple Groups (Weil's Rigidity Theorem) 123
Chapter VIII. Discrete Nilpotent Subgroups of Lie Groups ... 139
Chapter IX. Lattices in Semisimple Lie Groups - A Theorem of
Wang 153
Chapter X. Arithmetic Groups: Reduction Theory in SL(n) and the
Compactness Criterion 159
Chapter XI. The Results of Kazdan-Margolis 172
Chapter XII. Semisimple Algebraic Groups (Summary of Results) . 189
Chapter XIII. Fundamental Domains 196
Chapter XIV. Existence of Lattices 215
Bibliography 223
Index 227

Preliminaries
A summary of results on Lie groups and algebraic groups
This book presupposes considerable familiarity with Lie groups and
algebraic groups. The present chapter is a summary of some of the
relatively deeper standard results that are often used in the book; it is
therefore to be treated as a convenient reference. No proofs are given.
Some of the notation for the entire book is fixed in § 1.1 below.
1. Lie Groups and Lie Algebras
1.1. Notation. As usual Z denotes the ring of integers, Q(resp. R, C) the
field of rational (resp. real, complex) numbers. Except occasionally all
Lie algebras considered are over R or C.
Lie groups are denoted by Roman capital letters and their Lie algebras
by the corresponding gothic lower case characters. Gothic lower case
characters usually denote Lie algebras.
If Vis a vector space over R(resp. C), GL(V) denotes the group of all
R-(resp. C-)linear automorphisms of V. End(V) denotes the Lie algebra
of all R-(resp. C-)linear endomorphisms of V. As usual when V=R"
(resp. C),GL(V) is denoted GL(«,R) (resp. GL(n, C)) and identified
with the group of all n x n non-singular matrices with entries in R (resp. C).
Also M(n,R) = End(R") and M(n,C)=End(C").
If G is a Lie group and g its Lie algebra exp denotes the exponential
map exp: fl—>G.
The adjoint representation of a Lie group G (resp. Lie algebra h) on
its Lie algebra g (resp. on itself) is denoted Ad (resp. ad). The Killing
form on a Lie algebra h is denoted A{ , ) (when there is no ambiguity
about the Lie algebra we are speaking about). It is the symmetric bilinear
form defined by
A (X, Y) = trace (ad X o ad Y)
for X, Yeq.
Often in this book, we identify Lie algebras of Lie subgroups of a
Lie group G with the corresponding subalgebras of the Lie algebra
A of G.

2
Preliminaries
For subgroups H1,H2 (resp. Lie subalgebras t^, h2) of a Lie group G
(resp. Lie algebra g), [JJu-ffJ (resp. [h^hj]) denotes the subgroup
(resp. subalgebra) generated by {aba-1 b~1\aeH1,beH2} (resp. [A,B],
Ae h, Beh2). The derived series of G (resp. g) is the descending sequence
of normal subgroups D* G (resp. ideals D* g) defined inductively by setting
D°G=G, DkG = [Dk~1G,Dk-1G] (resp.Z)0g = g;Z)tg = [Z)t-1g,Z)t-1fl]).
Similarly the sequence C° G=G, Ck G=[G, C*_1G] (resp. C°g = g
Cfc9 = [g> C*-1g]) is called the descending central series of G (resp. g).
A Lie group G (resp. Lie algebra g) is nilpotent if Ck G=(e) (resp. Ck g = 0)
for all large k (e denotes the identity element of G). G (resp. g) is solvable
if D* G={e) (resp. 0*^=0) for all large k.
If G is a connected Lie group and g its Lie algebra, then Z)* G (resp. Ck G)
is a Lie subgroup of G, the corresponding Lie subalgebra of g being
D*g (resp. C*g). In particular G is solvable (resp. nilpotent) if and
only if its Lie algebra is solvable (resp. nilpotent).
1.2. Solvable and nilpotent Lie algebras. We begin with the well known
Engel's theorem. Let g e End V be a Lie subalgebra consisting of
nilpotent endomorphisms. Then there exists veV, vj=0 such that Xv=0
for all Xeq. Any lie subgroup of GL(V) consisting of unipotent
automorphisms of V has an invariant vector v + 0.
As consequences we have the following, (i) A Lie algebra g is nilpotent
if and only if ad X is nilpotent for all Xeg. (ii) The normalizer of a proper
subalgebra a in a nilpotent Lie algebra contains a properly.
In a nilpotent Lie group, the normaliser of a proper Lie subgroup H
strictly contains H.
Note that the normaliser of a Lie subgroup of a Lie group is closed.
Let G be a Lie group and U a normal subgroup. Let p be a completely
reducible representation of G such that p{x) is unipotent for xe U. Then
p(x) = e for xeU.
Lie's theorem. Let G (resp. q) be a connected solvable Lie group
(resp. Lie algebra). Let p be a finite dimensional representation of G
(resp. q) on a complex vector space V. Then we can find a basis e±,...,eN
of V such that with respect to this basis, p(g) (resp. p{X)) is an upper
triangular matrix for all geG (resp.XeQ). If p is irreducible it is 1-
dimensional. If p is an irreducible representation ofG (resp. q) over R, p(G)
(resp. p(g)) is abelian.
We continue to assume that G is connected and solvable. If
p: G-»GL(h,K) (resp. p: g—► M{n, K)) K=R or C is any representation
the set {xeG\p(x) unipotent} (resp. {XeQ\p{X) nilpotent}) is a closed
normal subgroup of G (resp. an ideal of g).

1. Lie Groups and Lie Algebras
3
The identity component N of the group {xeG|Adx is unipotent}
(resp. the ideal LYeg| Ad X is nilpotent}) is the unique maximal normal
connected nilpotent subgroup of G (resp. the maximal nilpotent ideal
in g).
Let g be any Lie algebra and p a finite dimensional representation
of g. Let S be the set of all ideals acg with the following property: p(X)
is nilpotent for all Xea. Then all the ideals in S are contained in a unique
maximal ideal rt e g. For X en and Y€Q,p{X)p( Y) is nilpotent. Moreover
rt can be characterised as the ideal
{XeQ\tiacep(X)p(Y)=0 for all 7eg and p(X) is nilpotent}.
1.3. Levi decomposition. We recall that a Lie algebra is semisimple
if it has no proper solvable ideals. A Lie group is semisimple if its Lie
algebra is semisimple. The radical of a Lie algebra (resp. Lie group) is
its maximal solvable ideal (resp. maximal connected normal solvable
subgroup). Any Lie algebra (resp. connected simply-connected Lie group)
is a semidirect product of a semisimple subalgebra m (resp. subgroup M)
and its radical s (resp. S). Moreover M and S are simply-connected.
If G is not simply-connected we can still write G=M -S, where S is the
radical, M is connected and Mr\S is discrete.
1.4. Ado-Iwasawa theorem. Let g (resp. G) be a Lie algebra (resp.
connected simply-connected Lie group). Let s (resp. S) be the radical of g
(resp. G). Let rt (resp. N) be the maximal nilpotent ideal of g (resp.
connected normal nilpotent subgroup of G). Then g (resp. G) admits a
faithful (resp. locally faithful) finite dimensional linear representation
p: g —*M(ji, R) (resp. p: G—*GL{n, R)) such that p{n) (resp. p{N)) consists
of nilpotent (resp. unipotent) matrices. (A representation p of G is locally
faithful if the kernel of p is discrete.) Also p is faithful on N.
Moreover we can choose p in such a way that p{G) is closed in
GL(«,R).
In particular a connected simply-connected nilpotent Lie group
admits a faithful representation by upper triangular unipotent matrices.
1.5. Cartan decompositions. Let g be a semisimple Lie algebra. Its
Killing form A( , ) is non-degenerate.
A Cartan involution 6: g—>g is a Lie algebra automorphism of g
such that (i) 02 = Identity and (ii) {X, Y} = -A(X, 0(7)) for X, 7eg is a
positive definite form.
Cartan involutions exist on any semisimple Lie algebra. Moreover
any two Cartan involutions are conjugates under an inner automorphism
of g. (An inner automorphism of g is an automorphism of the form

4
Preliminaries
Adg, geG where G is any connected Lie group having g for its Lie
algebra.)
Let 9: g-»g be a Cartan involution. Let 1={X€q\6{X)=X} and
p = {X\9{X) = —X}. Then f is a subalgebra and g = f©p. For Xet and
Yep, A{X, Y)—{X, 7> = 0. A{ , ) restricted to f (resp. p) is negative
(resp. positive) definite.
In this book when we say that "g = f ©p is a Cartan decomposition
with f as the algebra" we mean that the map 6: g—>g defined by setting
0(X+Y)=X-Y
for Xet, Yep is a Cartan involution.
We have also a "global" decomposition.
Let G be a connected Lie group with g as Lie algebra. Let K be the
subgroup of G corresponding to f.
Theorem. The natural map
ft: Kx p—*G
defined by fe(k,X) = k-expX is an analytic isomorphism (of analytic
manifolds). In particular K is closed in G and exp maps p onto a closed
analytic submanifold of G.
Corollary 1. Any element geG can be written uniquely in the form
g = fe(g)- expX(g) with k(g)eK and X(g)ep.
Corollary 2. 9: g—*q is induced by an automorphism of G. In fact
0(g) = k(g) • exp — X(g) is the required automorphism of G.
If G is linear (i. e. admits a faithful linear representation) then K is a
maximal compact subgroup of G. Moreover any maximal compact
subgroup of G is conjugate to K by an inner automorphism. In particular
a maximal compact subgroup K' determines a Cartan decomposition
g = f'©p', !' being the algebra as follows: !' is the Lie subalgebra
corresponding to K' and v' = {X<zq\A{X, Y) = 0 for all 7ef'}.
Let p be a finite dimensional representation of G on a vector space F
over R. We let p stand for the induced representation of g as well. Then
there is a positive definite scalar product < , >F on F with respect to
which p(k) (resp. p(X)) is orthogonal (resp. symmetric) for all keK
(resp. .Yep). In particular p(X) is a semisimple endomorphism with all
eigen values real (cf. 2.6 below).
1.6. The representation of K on p. We retain the notation of § 1.5. The
subspaces f and p of g are Ad K-stable. Let a denote the adjoint action
of K as well as f on p.

1. Lie Groups and Lie Algebras
5
g is determined by f and a. More precisely, let g = f © p and g' = f © p'
be Cartan decompositions of semisimple Lie algebras g and g'. Let a'
denote the action off on p'. Let q>: f —*l' and ij/: p—>p' be isomorphisms
such that
(i) cpilX, Y]) = [q>{X\q>{Y)] for all X, Yet and
(ii) iP(a(X)-(Y)) = a'((p(X))(il/(Y))i0T Xet, Yev.
Then q> extends to an isomorphism of g on g'.
The Lie algebra g is simple (i.e. has no proper ideals) if and only if cr
is irreducible.
1.7. Iwasawa decompositions. Let g = f ©p be a Cartan decomposition
of a semisimple algebra g. An abelian subalgebra a of g contained in p
and maximal with respect to this property is a Cartan subspace of p. Any
two Cartan subspaces of p are conjugate under an element of K. The
dimension r of a Cartan subspace is called the rank of G (or g).
Let acp be a Cartan subspace. For Aea*( = Hom(a,R)) let
gA = {X'e g | [AT, A"] = X{X) ■ X' for all Xea}.
Let <P = {/.e a* |gA=t=0, A + 0}. Then g decomposes into a direct sum
9=LIgA©g0.
A<=4>
Introduce a lexicographic ordering on a* (this is done as follows: let
Yj,..., Yr be a basis of a; Aea* is greater than zero; if A=|=0 and if i =
inf{fc|;.(yfc)=|=0}, then A(y,)>0). Let <P+ = {Ae<P|/L>0} and n= LIgA.
A<=a>
Then rt is a nilpotent subalgebra of g. Let A (resp. N) be the Lie subgroup
of G corresponding to a (resp. rt). Then A and N are closed. Moreover the
maP I.KxAxN^G
defined by I(k,a,n) = kan is an analytic diffeomorphism of analytic
manifolds. G = K A ■ N is called an Iwasawa decomposition of G.
Let p be a finite dimensional representation of G (resp. g). Then for
xeN (resp. Xert) p(x) (resp. p(X)) is unipotent (resp. nilpotent). Moreover
if p is a faithful representation and xeG (resp. Xeq) is such that p(x)
(resp. p(X)) is unipotent (resp. nilpotent), x (resp. X) has a conjugate in N
(resp. rt). In particular if an element xeG (resp. Xeg) is such that p(x)
(resp. p(X)) is unipotent (resp. nilpotent) for a faithful representation p,
then the same holds for all representations.
Finally we have the following characterisation of semisimple and
unipotent elements in a connected linear semisimple Lie group G: xeG
is semisimple (resp. unipotent) if its orbit under inner conjugation is
closed (resp. contains the identity element in its closure). Also for Xeg,
ad X is nilpotent if and only if there exists //eg such that [//, Ar]=Ar.

6
Preliminaries
1.8. Maximal compact subgroups. Let G be a connected Lie group.
Let K' c G be a compact subgroup. Then G admits a maximal compact
subgroup K containing K'. The space G/K is analytically isomorphic to
a Euclidean space R". Further G contains a closed analytic submanifold
V analytically isomorphic to R" such that the map F: V x K —* G given by
(v, k) i-> v ■ k, ve V, keK is an analytic isomorphism (of analytic manifolds).
Moreover any two maximal compact subgroups of G are conjugates of
each other in G.
When G is nilpotent, G possesses a unique maximal compact
subgroup Gt which is central in G. This group is moreover stable under all
automorphisms of G.
Let G be a connected solvable group and N its maximal connected
nilpotent normal subgroup. Let K be a maximal compact subgroup of G.
Then Kr\N is a maximal compact subgroup of N. Moreover K r\N is
central in the group G
1.9. The exponential map. As mentioned earlier throughout this book
exp denotes the exponential map exp: g—>G of the Lie algebra g of the
Lie group G into G When G is the group GL{n,K), K=R or C, we have
a canonical identification of g with M (n, R) and under this identification
we have for XeM{n, R)
expX = l + X+X2/2\ + - + Xk/k\ + -.
Let V be a finite dimensional vector space over a field k. We denote
by GL(K) the group of all fc-automorphisms of V. Let fc = R or C and
p: G^GL{V) be a representation of G on V. We often denote by p also
the induced representation of g on V. Then the diagram
GL(V)
EndV
is commutative (note that End V is canonically isomorphic to the Lie
algebra of GL(K)). It follows from this that if A is an eigen value of p{X),
XeQ, then ex is an eigen value of p(exp^).
If G is connected simply connected and nilpotent, exp is an analytic
isomorphism of analytic manifolds. Consequently the Lie subgroup H
of G, corresponding to any Lie subalgebra of the Lie algebra of G, is
closed in G.
More generally if G is simply connected and nilpotent and if
Q=V1®V2

2. Algebraic Groups
7
is any decomposition of g as a direct sum of subalgebras, the map
VtxV2 >G
given by (X.,..., Xn) h-» exp X1 ■ exp X2, is an analytic isomorphism.
For a solvable simply connected group G the exponential map is not
in general an isomorphism. However if rtcg is the maximal nilpotent
ideal of the Lie algebra g of G and Fcg is any supplement to rt in g, the
given by(X, Y) \-> exp X ■ exp YforXen, Ye Vis an analytic isomorphism.
In particular if G is of dimension n, simply connected and solvable,
then G is homeomorphic to R" (hence contractible).
References
For basic material assumed at the outset our reference is Chevalley [1].
Results in §§1.2-1.4 can be found in any standard work on Lie algebras.
(Jacobson [1, Chapters II, III, VI] or Bourbaki [1, §§4-7] or Seminaire
Sophus Lie exposes 2-8.) Helgason [1] gives an account of the material
in §§1.5-1.7: Chapter III of that book contains an account of Cartan
decomposition, Chapter VI deals with Iwasawa decompositions and the
results in § 1.7 are proved in Chapter IX. The final result stated in § 1.5
is an immediate consequence of conjugacy of Cartan involutions and the
main result of Mostow [1]. An account covering partially the results in
§ 1.8 is contained in Seminaire Sophus Lie, expose 22; the results in § 1.8
are due to Iwasawa [1]. The contents of §1.9 (apart from the elementary
facts about the exponential map) are contained in, for instance, Seminaire
Sophus Lie, expose 22.
2. Algebraic Groups
Recall that GL(n, C) acquires the structure of an algebraic variety
through the imbedding GL(n, C)r-^C"2 + 1 defined by
g^(gu,det(gy)-1).
The ring of regular functions on this algebraic variety is evidently
c^.fdetpry))-1].
Regular functions are also referred to as polynomials.
By an algebraic matrix group we mean a subgroup G of GL(n, C)
which is an algebraic subvariety i. e. the set of zeros of an ideal of regular
functions on GL(n, C). G is said to be defined over a subfield ItcCorG

s
Preliminaries
is said to be a fc-group if the ideal of all regular functions vanishing on G
admits a system of generators belonging to the subring
^.(detpg)-1].
For a subring Be C, we set GB = G(B)= G n GL(n, B).
Throughout this book by an algebraic group we mean a subgroup
G<=GL(K), the group all C-linear automorphisms of a finite dimensional
vector space V which gets identified with an algebraic matrix group when
we identify GL(K) in GL(n, C) (dim V=n) through a basis of V. The ring
of regular functions on G is denoted C [G]. If V has a fc-structure (i. e.
Kis obtained by extension of scalars from a fc-vector space Vk), we will say
that G is defined over k if the algebraic matrix group associated to G
through the choice of a basis of Vk is a fc-group. In this case the subring
of regular functions defined over k is denoted fc[G].
Let G and G' be algebraic groups defined over k. A fc-morphism q>
(or a morphism defined over k) of G in G' is a homomorphism of groups
q>: G-^G'
such that/°(/)6fc[Gj for all/efe[G'].
2.1. Representation. A /c-representation (or a representation defined
over k) of a fc-group G is a fc-morphism
p: G->GL(K)
where V is a fc-vector space. A character defined over k is a fc-morphism
of G in GL(1).
Let G be a fc-group and p a representation defined over k in a fc-vector
space KThenp(Gk)c=GL(i;).
If p is a faithful ^-representation of G, the every ^-representation of G
is contained in a tensor power of p © p* where p* is the dual of p.
Let xeGeGL(K), G a fc-group. Then x can be written uniquely in
the form x = xsxu where xs (resp. xj is a semisimple (resp. unipotent)
matrix in GL(K)and xs-xu=xu-xs. Then x„xueG. If xeGk, xs,xueGk.
Let G be a fe-group and p a ^-representation of G on a fc-vector space V.
Lett)6Kk.Then
// = {geG|p(g)r6Cr}
is a fc-subgroup of G.
Conversely every fc-subgroup of G can be realised in this manner.
Let G be a fc-group. The Lie algebra g (over C) of G considered as a
complex Lie group has a natural fc-structure. For this fc-structure the
adjoint representation Ad of G in g is defined over k. Let h <= g be a vector
subspace. Then the group B = {geG|Adg(h)c=h} is an algebraic
subgroup. Note that if h is a Lie subalgebra and H the corresponding Lie

2. Algebraic Groups
9
subgroup B is the normaliser of H. B is defined over k if h is a fc-subspace
ofG.
The normaliser in G of a fc-subgroup H is a fc-subgroup of G.
2.2. Abelian and unipotent fc-groups. Let G be a /c-group. We define for
non-negative integers r the subgroups Dr G and (7 G of G inductively as
follows:
D°G=C°G=G
DrG = lD-1G,D'-1G], C'G=[G,C'-lG].
The groups ZX G and C G are normal fc-subgroups. In fact these groups
are stable under all automorphisms of G.
Let G be a fc-group and H a normal fc-subgroup. Then the group G/H
can be identified with a fc-group G' (unique upto isomorphism) such that
the natural map
7t: G->G/H( = G')
is a /c-morphism.
In particular the groups ITG/Dr+1G and C'G/C,+1G are abelian
/c-groups. A k(-split) torus is a fc-group T, (k-) isomorphic to a product of
copies of GL(1). A connected abelian fe-group is a direct product of a
lorus and a unipotent fc-group.
A unipotent abelian fc-group is isomorphic to a fc-vector space.
A fc-group G is unipotent if it consists entirely of unipotent elements.
If G is a unipotent fc-group, and H a normal fc-subgroup, H and G/H are
again unipotent fc-groups. In particular if G/H is abelian it is isomorphic
to a k-vector space.
If G is a unipotent fc-group and C G are defined as above, C G/Cr+1 G
= Hr is a unipotent abelian fc-group. Hence Hr is isomorphic to a fc-vector
space.
Another fact that is needed is the following. Let p be a representation
of a unipotent algebraic group [Zona vector space V. Let veVbe any
vector. Then the orbit {p(u)(v)\ue U} of v under 1/ is a closed subset of V
in the Zariski topology.
Let l/cGL(n,R) be a unipotent Lie subgroup and ucM(n,R) its
Lie algebra. Let ueM(n, C) be the C-span of u and U be the
corresponding (complex) Lie subgroup. Then U is the Zariski closure of U
inGL(n,C)andUR=l/.
2.3. Euclidean and Zariski topologies. Let G be an algebraic /c-group.
Then the connected components of G in the Zariski topology are the
same as the connected components of G in its topology as a Lie group.
(The latter topology is referred to as the euclidean topology in the sequel.)
Wc can therefore speak of connected components of G without specific
reference to the topology used on G.

10
Preliminaries
An algebraic fc-group G has finitely many connected components.
The identity component G° of G is an algebraic fc-subgroup of G. If xeG
is unipotent it is in G°. If G is unipotent it is connected.
If G is connected the set of fc-rational points Gk of G is dense in G in
the Zariski topology.
If H c Gk is any subgroup of Gk the closure H of H in G is an algebraic
fc-subgroup of G. In other words if H is an algebraic subgroup of G such
that H=H n Gk is dense in H in the Zariski topology, then H is a fc-sub-
group.
2.4. Real points. If G<=GL(n, C) is a fc-group, GR is a Lie group
with finitely many connected components. In general GR need not be
connected even if G is. However if G is unipotent GR is connected.
A subgroup of the form GR in GL(n, R) is a real algebraic group. We
refer to the topology on GR (or its identity component GR) induced by the
Zariski topology on G as the Zariski topology on GR (or GR). Let q>:
G—>G' be a fc-isomorphism of fc-groups with fceR; GR—S-^GR is an
isomorphism of groups which is a homeomorphism for both the Zariski
and euclidean topologies on GR.
Closure in the Zariski topology is often referred to as "Zariski
closure". Similarly "Zariski dense" means dense in the Zariski topology.
Let G be a connected linear semisimple Lie group and (/>: G —►GL(n, R)
be a faithful representation. Let G be the Zariski closure of q>(G) in
GL(n, R). Then GR = <p{G). The topology on G induced through q> by the
Zariski topology is independent of q>. Thus Zariski topology on G has an
intrinsic meaning.
Let G be a fc-group and H a normal fc-subgroup and
n: G^G/H = G'
the natural fc-morphism.
Assume that k e R and let GR (resp. GR°) denote the identity component
of GR (resp. GR). Then n defines an isomorphism (of Lie groups)
G°/G°nH + GR°.
In general the map 7tR: GR-^>G'R need not be surjective, though G{|—>GR°
is surjective. nR is surjective if GR is Zariski dense in G. In fact, in this case
the composite map GR-^G-^G'-^G7G'° is also surjective.
2.5. A structure theorem. Let G be a fc-group. Then G has a unique
maximal connected solvable algebraic subgroup S; S is defined over k.
It is the radical of G considered as a Lie group. The set N of unipotents in
S is the unipotent radical of G. N is an algebraic fc-subgroup.
A fc-group G is reductive if every representation of G is completely
reducible.

2. Algebraic Groups
11
G is semisimple if the radical of G is trivial.
A fc-group G is reductive if and only if its identity component is
reductive. A connected fc-group is reductive if and only if the identity
component of the centre is a torus T and G=MT where M is a connected
semisimple fc-subgroup of G.
Theorem. Let G be a k-group. Then we can find a reductive k-subgroup
M ofG such that G=MN where N is unipotent radical ofG and M r\N—(e).
Moreover if G = M' N where M' is a k-subgroup such that M' r\N=(e),
then we can find xeNk such that xM x_1 = M'. IfG is solvable, the identity
component of M is a torus.
From the theorem above one can deduce easily an analogous
statement for real algebraic groups. Let G be a fc-group with fceR. Let G{J be
the identity component of GR. Let N be the unipotent radical of G and M
a reductive fc-subgroup of G° such that G°=M ■ N. Let M% be the identity
component of MR. Then GJJ is the semidirect product G^ = MR- NR.
2.6. Reductive algebraic groups and real semisimple Lie groups. Let
G <= GL(n, R) be a real semisimple Lie group. Let g be the subalgebra of
M(n, R) corresponding to G and fl the C-span of g in M(n, C). Let G be
the (complex) Lie subgroup of GL(n, C) corresponding to g. Then G is
an algebraic R-subgroup of GL(n, C) and G is the identity component
ofG,.
As usual for a matrix AeM(n, R), A* denotes its conjugate transpose.
An algebraic subgroup stable under A\-*A* is reductive.
Let G1cG2...Gmc:GL(n, Q be reductive fc-subgroups of GL(n,C)
where fc=R or C. Then we can find geGk such that for l^i^m if
AegGig-Sthen^egGig-1,
The map A i-> A* of GL(n, C) restricted to Gt(k) is a Cartan involution.
One deduces easily from this result the following. Let G be a reductive
algebraic group. Let HcGbea reductive algebraic subgroup. Then the
centraliser of H in G is reductive algebraic.
References
General references for algebraic groups are A. Borel [1], Chevalley [2]
and A. Borel and J. Tits [1]. The result on orbits of unipotent groups is
proved for instance in M. Rosenlicht [2]. Zariski density of fc-rational
points is established by M. Rosenlicht [1]. A comparison of euclidean
and Zariski topology is given in A.Weil [1,Appendice]. Finiteness of
the number of connected components for real algebraic groups is proved
in Mostow [3]. For the structure theorem in § 2.5, see G.D. Mostow [2]
and Borel-Serre [1, Proposition 5.1]. Results in §2.6 are due to G.D.
Mostow [1].

12 Preliminaries
3. Group and Lie Algebra Cohomology
The groups Ext and Tor. Let M, N be left-modules over a ring A with unit.
A projective (resp. injective) resolution of M (resp. AT) is a complex
P={..._^_^_l_..._Po_0}
(resp./ = {0-/o-/1—■—/.-/.+1,...})
such that Pk (resp. Ik) is projective (resp. injective) and Hk{P) (resp.
Hk{I))=0 for k =t= 0, together with an isomorphism
p: H°(P)->M (resp. i: N-^H°(I)).
If (P, p) (resp. (J, 0) is a projective (resp. an injective) resolution of M
(resp. N), the cohomology groups r/'(Hom^(P, N)) (resp. /J*(Homx(M, /)))
of the complex Horn^P, N) (resp. Hom^M, /)) are independent of the
resolution chosen. Moreover we have also for each k, a canonical
isomorphism Hk(HomA(P, Nj) ~ Hk(HomA(M, /)). This group is denoted
ExtJ(M, N). For fixed M (resp. N\ ExtA(M,N) is a covariant (resp.
contravariant) functor on the category of ^-modules into the category
of abelian groups. In particular, the centre B of A acts on M (resp. N) and
this defines a B-module structure on Ext^ (M, N). The B-module structures
obtained from M and N coincide. Thus Ext$(M, N) is in a natural way a
B-module. In particular, if A is commutative Ext$(M, N) is an ^-module
in a natural fashion.
3.1. Lie algebra cohomology. Let g be a Lie algebra over a field k and
l/(g) the universal enveloping algebra of g. If p: g-^End* V is a
representation of g on a vector space V over k, then K can be regarded as a
[/(g)-module. In particular k can be made into a [/(g)-module through
the trivial representation of g on k. We define Hp(q, p) (= H"{q, V) if
there is no possibility of confusion) as the group Ext£(9)(fc, V) and call it
the p-th cohomology of g with coefficients in p.
The standard complex S of g is a convenient projective resolution
of k considered as a l/(g)-module. We refer to Cartan-Eilenberg [1,
Chapter XIII] for the definition of S. We need the following important
consequence of the definition of S. Let p be a representation of g on a
vector space V over k. Then HomD(g)(5, V)= C(g, V) is naturally
isomorphic to the complex described as follows: the p-th graded component
Cp of C(g, V) consists of all alternating ^valued p-forms on g; the
operator d: Cp—► Cp+1 is given by the following formula: for ae Cp and
■^i» ■■■»-^p+i69>
da(Xl,...,Xp+1) = "J:(-irip{XMX1,-,Xl,...,Xp+1)
j=l
+ Yl(-iy+><x<LXlxJixt,...,Xi,...,Xj,...,xp+1).

3. Group and Lie Algebra Cohomology
13
On the spaces Cp we make the Lie algebra g act as follows: for a.eCp
and Xeq, and Xt, ...,Xp€Q,
.=1
X\-+9(X) is representation of g on Cp. For Xeq, let i{X): Cp-^Cp_1 be
defined as follows: for aeCp and Xx, ...,Xp_leQ,
l(X)aL(Xl,...,Xp_l)=a(X,Xu...,Xr_l).
We then have for Xeq and aeCp,
(*) {i{X)od+doi{X)){a)=e(X){<x)+p{X)oa.
Suppose now h <= g is a subalgebra and we set
Cp(V)={<x\<xeCp(V),i(X)<x = 0, -0(X)a = p(X)°a for Xeh}.
Then the homogeneous subgroup £ C'p(V) of £ Cp(K) forms a sub-
complex which we denote C(g,h, V). (This is easily checked using the
formula (*).) The cohomology groups of C(g, h, V) are denoted H"{q, h, V)
and are called the cohomology groups of g relative to b with coefficients
in V.
We will have occasion to use the following well-known result (see for
instance Seminaire Sophus Lie, expose 5).
Theorem (Whitehead's Lemma). Let Qbea semisimple Lie algebra over
a field k of characteristic 0. Let p be a finite dimensional representation
of q. Then H\q, p)=0.
3.2. Group cohomology. Let T be a group and Z(F) its group-ring. An
action of T on an abelian group M makes M into a Z(r)-module.
Conversely on any Z(r)-module T acts in an obvious fashion. We consider Z
as a Z(r)-module through the trivial representation of T on Z. The
cohomology groups HP(T, M) are defined to be the groups Ext£(r)(Z, M).
An effective method for the computation of these groups for a wide
class of groups F is achieved by means of the standard (inhomogeneous)
complex S of T: this is a projective (in fact free) resolution of Z. We refer
to Cartan-Eilenberg [1, Chapter X, § 4] for the definition of S. For our
purposes we need only the following facts. Let V be an abelian group on
which r acts. We denote by p the map of F in Endz(K). Let Hom(S, V)=
C(T, V). We will describe the p-th graded components Cp of C(T, V)
and the differential operator d: Cp-^CP+1 for p<2.For p>0,Cp(r, V)
is the set of all maps of Gx-xG into V. C0{r, V)=V. The operators
p times
<l: C((K)->Ci+1(K), i=0,1 are given by the following formulae: for

14
Preliminaries
ve Vfe Ct(K) and x, yeT, we have
dv(x)=p(x)v — v
dftx, y)=f(x)+p(x)f(y)-f(x ■ y).
A 1-cocycle on F with coefficients in p (or if there is no ambiguity about p,
simply a K-valued 1-cocycle) is an element of Cj(K) such that df=0 i.e.
it is a map/: F—► Ksuch that for x, yeF,
f(x-y)=f(x)+p(x)f(y)
References
H. Cartan and S. Eilenberg [1].
4. Principal Bundles
Recall that a differentiable (resp. topological) principal fibre bundle is a
differentiable manifold (resp. topological space) P together with an
action (on the right) of a Lie group G with the following properties; the
quotient B of P for the action of G carries a natural structure of a (Haus-
dorff) differentiable manifold (resp. topological space) such that the
natural projection n: P-^B is differentiable. Moreover for each beB, we
can find an open neighbourhood Ub of b in B and a diffeomorphism (resp.
homeomorphism) <P: Ux G-^n~l(Ub) such that no4>(x,y)=x for xeUb,
geG and for xe Ub, g, heG,
«P(x,gfc)=«P(x,g)fc.
G is called the structure group, B the base space and P the total space of
the bundle. If n: P-^B is a differentiable (resp. topological) principal
bundle with G as structure group and A—L-+B is a differentiable (resp.
continuous) map the fibre product offand 7t is in a natural way a principal
differentiable (resp. topological) fibre bundle over A called the induced
bundle.
We have the obvious notion of an isomorphism of differentiable or
topological principal bundles with the same base and structure group.
4.1. Homotopic maps and induced bundles. The assertions made in
this paragraph are valid for both topological and differentiable principal
bundles. We will use simply the term bundle or principal bundle for both.
Let 7t: P —► B be a bundle over B with structure group G. Let f,g:A-^B
be two homotopic maps. Then the bundles induced by / and g over A
are isomorphic. (We assume that A is paracompact.)
Suppose now the total space P of the bundle n: P—> B is contractible.
Let n'\ P'-*A be any principal bundle with structure group G. Assume

4. Principal Bundles
15
that A is paracompact. Then we can find a map/: /4—>B such that the
bundle induced by / is isomorphic to n': P'—>A Moreover if g: A-^B
is any other map, / and g are nomotopic if and only if they induce
isomorphic bundles.
4.2. Sections. A differentiable (resp. topological) principal bundle
7t: P—► B is trivial if it is isomorphic to the differentiable (resp. topological)
product bundle BxG.
A trivial differentiable (resp. topological) bundle n: P-^B admits a
section: i.e. a differentiable (resp. continuous) map a: B-^P such that
7t o a is the identity.
If 7t: P—► B is a differentiable bundle and M a closed C°° submanifold
of B and x: M—>P is a C°° map such that k°t(x)=x for all xeM, then t
is called a C00 section over M. A C00 section over M can be extended
to a neighbourhood Q of M as a C°° section.
Similarly if te: P—>B is a topological bundle a continuous section
t : M —► P over a closed subset M c B can be extended to a section over a
neighbourhood jQ of M in B.
4.3 Comparison. The classifications of topological and differentiable
bundles over a paracompact manifold B are identical. In other words,
every topological bundle is topologically isomorphic to the underlying
topological bundle of a differentiable bundle. Moreover two differentiable
bundles over B are isomorphic if and only if they are isomorphic as
topological bundles.
In particular if a differentiable bundle admits a continuous section,
it admits a differentiable section.
4.4. Lie groups. Let G be a Lie group and H a subgroup and G/H
the quotient homogeneous manifold. The map G—► G/H is a differentiable
principal bundle with H as structure group. In particular we can find
tin open set V in Rp (p=codim H in G) and a diffeomorphism onto an
open subset Q of G
4>: VxH-^Q
such that Qh = Q for heH and
&(v,hh')=$(v,h)-h'.
Refereuces
N. II. Steenrod [1].

Chapter I
Generalities on Lattices
In this chapter we collect together some general results on lattices in
locally compact groups. The chapter begins with an investigation of
when a homogeneous space admits an invariant measure. Later some
criteria, when a closed subgroup H of a locally compact group G intersects
a lattice T in G in a lattice in H are given.
Lemmas 1.1-1.7 are standard and quite elementary. Many among
the rest of the results are essentially due to A. Selberg though most of
them are not to be found stated explicitly in the available literature.
Some results proved in this chapter will not be used until much later.
Let G be a locally compact group and H a closed subgroup. Let
CC(G) (resp. Cc(H),resp. CC(G/H)) denote the space of continuous complex
valued functions on G (resp. H, resp. G/H) with compact support.
We fix left invariant Haar measures nG,nH respectively on G and H.
For feCc{G) (resp. CC(H)) we denote the integral of / with respect to
HG (resp. nH) by
ff<g)dg (resp.J/(*)itt).
G H
With this notation we now define a map /: CC(G)-^ CC(G/H) as follows.
Let feCc(G) and geG. Let g denote the coset of g in G/H. Then
IU)®=\ftgh)dh.
H
Clearly the map / is linear over C. Our first assertion is
1.1. Lemma. / is surjective.
Proof. Let n: G-^G/H be the natural map. For feCc(G/H) let/be
the continuous function f°n on G. Then it is easy to check that for
<peCc{G), we have I(q>f)=I(q>)-f. Now if ueCc(G/H) is any element,
we can find a relatively compact open subset Q of G such that u vanishes
outside n(Q). Let i//eCc(G) be a function such that i//(g)^0 for geG and
i//(Q)= 1. Then it is easily seen that for xen(Q\I(i//)(x)*0. Let veCc(G/H)

I. Generalities on Lattices
17
be defined as follows:
v(x)=I(il/)(x)~1u(x) if xen(Q)
= 0 if x$n{Q).
Then it is evident that u(x)=I(i//)(x)-v{x). Thus u=I(i//)-v=I{i//v).
Thus u is in the image of /. This proves the lemma.
Next let AG (resp. AH) be the modular function onG (resp. H)-AG
(resp. AH) can be defined as follows: for xeG (resp.H) and feCc(G)
(resp. CC(H)) we have
Ao(x)-ff<g)dg=ff<gx)dg
G G
(xesp.AH{x)-\f{h)dh=\f(hx)dh).
H H
It is then easy to prove the following
1.2. Lemma.
imdg=if(g-i)AG(g)dg.
G G
Similarly
£f{h)dh = £f{h-1)AH{h)dh.
From Lemmas 1.1 and 1.2 above we will now deduce
1.3. Lemma. Let u be a continuous function on G such that for geG
and heH,we have
u(gh) = u(g)-AG(h)AH(h-1).
Then if I (f)=0 for some fe CC(G), we have
Su(g)f(g)dg=0.
G
Proof Let XeCc(G). Then clearly
\\u{g)m\dg\\f{gh)\dh<n
G H
so that by Fubini's theorem, we have
Q=\u(g)X(g)dg\f{gh)dh
G H
= \dh\u{g)X{g)f{gh)dg
= ldh\AG(h)u(gh-l)X{gh-l)f{g)dg
H G

IS I. Generalities on Lattices
= \dh\AG(h)u{g)AG(h-l)AH(h)X{gh-l)mdg
H G
= \dg\X(zh-1)AH{fr)-f(g)u(g)dh
G H
= \dg{\X{gh)dh)f{g)u{g)dh
G H
= J/W(g)/(g)«(g)dg.
G
Now X can be chosen such that /(A) takes the values 1 on n(E) where E
is the support of / (see Lemma 1). In this case 7(A) is 1 on 7t_1 n(E) so
that 7(A) •/=/: Thus
0=fdg7(A)(g)/(g)u(g)
G
= J/(g)«fe)<*g.
G
This proves the lemma.
Corollary. Assume that G and H are unimodular. Then for feCc(G),
if l(f)=0, we have
}f(g)dg=0.
G
Now let n be a Borel measure on G/H and x'- G—► R+ a continuous
homomorphism. Then /* is said to be semi G-invariant with character x
if for every geG and measurable set E cz G/H, we have
n(gE)=x(g)-n(E).
In general a homogeneous space G/H does not admit a semi-invariant
measure (with any character). The necessary and sufficient condition is
given by
1.4. Lemma. The homogeneous space G/H admits a semi-invariant
measure if and only if the homomorphism AG Ag1 :H-^R+ can be extended
to a continuous homomorphism of all of G. Moreover given any
homomorphism u: G—>R+ such that u\H = AGAg1, G/H admits a semi-invariant
Borel measure with character u. Further, this measure is unique upto a
scalar multiple.
Proof. Assume that G/H admits a semi-invariant measure fi. Let
fi: CC(G/H)^> C be the linear functional
f(f)= \f-d».
G/H
Then fiol is a linear functional on G which again defines a semi-invariant
measure n' on G with character u (u is the character associated to n).

1. Generalities on Lattices
19
From the uniqueness of the Haar measure it follows that after modifying fi
by a scalar multiple we have
fi°nf)=$f(g)u(g)dg.
G
Let Rh denote the right translation by he//. Then we have for he//,
I(foRh) = AH(h)I(f).
It follows that
^o(h)if(g)u(g)dg
G
= y(gh)u(gh)dg
G
= \{f°Rh){g)-u{g)dgu{h)
G
= u{h)(n°I){f°Rh)
= u(h)AH(h)(jioI)(f)
= u{h)AH{h)\f{g)u{g)dg.
G
Thus
AG(h)Aj11(h) = u(h).
Conversely let u: G—> R+ be a continuous homomorphism such that
u(h) = AG(h)-Aj11(h)
for all he//. Then in view of Lemma 1.3 we have
j"(g)-/(g)dg = 0
G
for all/e CC{G) such that /(/) = 0. Thus the linear form u,/i-> J/(g) u{g)dg
of CC(G) in C factors through/: G
CC(G) —'—> CC(G/H)
Clearly u' defines a semi-invariant measure on G/H with character u.
The uniqueness of the measure follows from that of the Haar measure.
A semi-invariant measure on G/H is invariant if the associated
character is trivial. Evidently G/H admits an invariant measure if and
only if AG\H = An. We have also

20
I. Generalities on Lattices
1.5. Lemma. A semi-invariant measure \i on G/H if finite, is invariant.
Proof. In fact n(x ■ G/H)=n(G/H) so that since n(G/H)< oo, we have
X(x)= 1 for all xeG where % is the character associated to \i.
1.6. Lemma. Let G be a locally compact group and Hl and H2 closed
subgroups such that //, => H2 ■ If G/H2 carries a G-invariant finite measure
then G/Hi and HJH2 carry invariant finite measures and conversely.
Proof. If G/H2 carries a finite invariant measure so does G/Hl: we
can take the direct image of the measure under the map G/H2 * G/Hl.
Since G/Hl and G/H2 carry invariant measures,
^gIh, = ^h, and ^gIh2=^h2
so that AHi\H2 = AH2. ft f0^ows that HJH2 carries an Hl-invariant
measure. We denote the invariant measures on G/H2, G/Ht and HJH2
by X, n and v respectively. Consider now the linear functional L on
CC{G/H2) defined as follows: let feCc(G/H2); then
(1) L(/)= \dn J f(ghH2)dv,
G/H, H1/H2
It is then easily seen that L defines an invariant measure L on G/H2 and
because of the uniqueness of the invariant measure, L is a scalar multiple
of X. SinceA(G///2)<oo, we have J d\i \ dv< oo. By Fubini's theorem
G/H, H,/H2
\(HjH2)< oo. This proves one part of the lemma.
When HJH2 and G/H^ carry finite invariant measures evidently (1)
defines an invariant measure L on G/H2; that the measure is finite is an
immediate consequence of the definition.
1.7. Lemma. Let G be a locally compact group. Let U be a closed
subgroup of G and H a closed subgroup normalizing U. Assume further
that HU is closed in G. Then HU/H carries a HU-invariant finite measure
if and only if U/U n H carries a U-invariant finite measure.
Proof. Let <p: U/H n [/—► HU/H be the natural map. Evidently q> is a
homeomorphism. Moreover q> is compatible with the action of U on both
spaces. Thus if H U/H carries an H [/-invariant measure, U/UnH carries
a [/-invariant measure. Now assume that U/U n H carries a [/-invariant
finite measure. This measure defines a [/-invariant measure on HU/H
as well via (p. We denote this last measure by fi. For heH, define the
measure Lhn on HU/H by setting
LhniE)=n(LhE)

I. Generalities on Lattices
21
for a measurable set £ in HU/H. Here LhE denotes the image under
the action ofheH on HU/H. Then we have for ueU
{Lhn)(LuE)=n(LhLuE)
=n{LhLuL-hlLhE)
= n(LhE)=Lhn{E).
Thus Lh n is again [/-invariant. Since n is a homeomorphism, we have
Lhn=a.(h)n
for some scalar a(h)eR+. Now n is a finite measure so that we have
0<fi(HU/H)< oo. On the other hand,
«{h)n{HU/H) = L„n{HU/H)
= n{L„HU/H)=n{HU/H).
It follows that a(h)=1 for all heH. Thus n is //-invariant as well. Thus \i
is //[/-invariant. Hence the lemma.
Now consider the case when H is a discrete subgroup of G. In this
case AH is trivial so that
It follows that G/H admits a semi-invariant measure with character AG.
If this measure is finite, we have AG=1.
1.8. Definition. A discrete subgroup H of G is a /arrice if G/// carries
a finite invariant measure.
L9. Remark. If G admits a lattice //, //crker zJG=Gj; G/G, eR+ is
a group with finite Haar measure; thus G=Gl i.e. G is unimodular.
1.10. Definition. A subgroup HcG is called a uniform subgroup if
G/// is compact.
1.11. Remark. A discrete uniform subgroup, it is easily seen is a
lattice. As we shall see later a lattice is not necessarily uniform. Nor does
a compact homogeneous space in general carry an invariant measure.
For instance if G=SL(2, R) (=group of 2 x 2 matrices of determinant 1)
und BcG is the subgroup
B = lx\X=^ ^eSL(2,R)|
then it is not difficult to see that AG=l while AB$ 1 and moreover that G
does not admit any non-trivial continuous homomorphism into R+.
Also G/B it is easily seen, is compact.

22
I. Generalities on Lattices
Suppose that G is a locally compact group and rcG is a lattice.
We will now investigate when a closed subgroup H c G is such that H r\T
is a lattice in H. We will attempt to give some conditions for this to be
true. Towards this end we prove first the following.
1.12. Theorem. Let Gbe a locally compact group satisfying the second
axiom of countability and TaG a lattice. Let {xn}l^n<a0 be a sequence
in G. Let n: G-^G/T be the natural map. Then the sequence n(x,) has no
convergent subsequence in G/r if and only if there exists a sequence
*iVn}i<n<x in T such that yn^e for any n and xnynx~l converges to e as n
tends to oo.
Proof. Let {Bn}1gn< x be an increasing family of compact subsets of G
00
such that (J Bn=G. Let n denote a Haar measure on G as well as the
corresponding invariant measure on G/r. Then, since the measure of G/r
is finite, en=n(G/r—n(Bj) tends to 0. It is then easy to see that we can
find a fundamental system of compact neighbourhoods Vn of e such that
n(V^>sn. Let Wn= V~l Vn. Then Wn also form a fundamental system of
neighbourhoods. Now for each n, the set 7t(Kn_1 VnB,) is compact. Thus
if 7i (x„) has no limit points, there exist Xn such that
n(x^n{V-lVnBn)
for »i_t A„. We then claim that n(Vnxm)mt(VnB,) is empty; in fact if this
intersection were not empty, we can find v, v'eVn,beBn such that n(v xm)=
n(v'b) so that
vxm = v'by
for some yeT. But then xm=v_1 v' by so that 7t(xm)e7c(I^""1 Vn Bn). Thus
n{*{Vn xJ)<n{G/r-n{VnB$^n.
On the other hand ^(I^xJ>en. It follows that for m^.Xn we can find
yeT, y=|= e such that
vxny„=v'xn
for some v, v'e V„. Thus we have for m~^kn
xmymx-1 = v-iv'eWn.
Since Wn is a fundamental system of neighbourhoods of e the desired
result follows.
Conversely, suppose now that xneG is a sequence for which we can
find y„eT, yn=¥e such that xn y„ x"1 tends to e as n tends to oo. If possible

1. Generalities on Lattices
23
let 3i(x„) have a limit point n(z), zeG in G/T. By passing to a subsequence
if necessary we can assume that we can find 9ner such that x„ 0„ converges
to z. But then
w„=x„ yn x- 1=xnon e-1 y„ e„ e~' x-'
=x„e„(p„e-1x-1, q>„er
tends to e and xn9n tends to z. It follows that q>n = 9~l x~l w„x„0„
converges to e so that q>n = e for large n; but (p„ = Q^ly„Q„ and 7„=t=e, a
contradiction. Hence the theorem.
From Theorem 1.12 we will now deduce
1.13. Theorem. Let G be a second countable locally compact group and
F a lattice in G. Let H be a closed subgroup. Then if HnT is a lattice
in H, HT is closed in G; equivalently the natural injection
H/HnT^G/r
is proper. If H is normal in GorifT is uniform then HT is closed in G if
and only if H n T is a lattice in H.
Proof. Assume that H n r is a lattice in H. We will then show that
the natural map HjH n r—* GjT is proper. By the definition of the
topology on G/T it would follow that HT is closed in G. To prove that
the natural map is proper we need only to prove the following: if nH
(resp. 3iG) is the natural projection H—* H/Hn r (resp. G —* G/T) then for a
sequence x„ in H, nH(x„) converges to a limit if and only if 3iG(xn) converges
to a limit. To see this we note first that if nH(x„) converges so does nG(x„).
Suppose next that nH(xn) has no limit point, then we can find (according
to Theorem 1.12) elements yneTnH such that x„7„x~' converges to e
while y„ + e for all n. But then by Theorem 1.12 again -tg(x„) has no limit
points either. Hence the first assertion.
Assume now that HT is closed in G. If H is normal in G, then according
to Lemma 1.6, HT/r carries a finite //r-invariant measure. It follows
that H/HnT carries an //-invariant measure (Lemma 1.7) which is
finite.
Finally consider the case when HT is closed in G and G/T is compact.
Once again consider the natural map
i: H/HnT^G/r.
1 ,et E be the image of i. Since HT is closed in G and G/r is compact,
E is compact. Now if n: G^G/r is the natural map W1(E) -ȣ is a
covering and hence in particular open. Now n~l(E) is precisely HT.

24
I. Generalities on Lattices
The diagram
H >Hr=ii-1(E)
I I .
H/HnT >£
is commutative; the two vertical maps are covering maps and are hence
open. The inclusion H^HT is open since T is discrete. Thus the map
H/H nf->£ is open. It follows that this map is a homeomorphism.
Thus H/H n r is compact. This proves the theorem.
1.14. Lemma. Let G be a second countable locally compact group and
r<=-G a lattice. Let AcT be a finite set and let GA denote the centralizer
of A in G. Then GA T is closed.
Proof. Let xneGA and yner be sequences such that xnyn converges
to a limit z. We have to show that z belongs to GA r. To see this we note
that for yeA
z-1yz=iim(y-1x-1yxByB)
= limy„-1yyB.
Since T is discrete, it follows that for large n, say n ""> n0,
so that yn+iy„"1eGj for n^n0. Thus
for all «^n0. Hence
z„e GA. Then since G4 is closed in G, zn converges to a limit z'e G. Hence
z=z' y^eGj T. This proves the lemma.
1.15. Lemma. Let G be a Lie group and r a lattice in G. Let H be a
closed subgroup of G. Suppose that there exists a neighbourhood Q of e
in G such that for every pair of elements Tt, T2eH and weii such that
y=Tto) T2eT, we have yeHn T. Then Hr\T is a lattice in H.
Proof. Let V be a neighbourhood of e in G such that V1 Fcfl, Let
n: G —> G/H be the natural map. Assume that V is chosen to be so small
that there is a sections: n (V)^> G.Let V1=n(V). Consider the natural map
<P: KjXff^G
(v, h)\-*a(v)h.

1. Generalities on Lattices
25
Then (P is an open map and defines a map
<Pt: V^H/HnT^G/r.
We now claim that ^ is 1-1. In fact if we have.
a(v)h = a(v')h'y
where v, v'eV^ h,h'eH and yeT, it follows that
y = h'-lo(v')-lo(v)h
and aid)-1 o(v)eV-lVcQ so that yeHnT. Thus 4>j is 1-1. On the
other hand it is easily seen that a Haar measure on G/r when pulled back
to Vt x H/H n r, takes the form dfx x dv where dfi is a measure on Vt
and dv a Haar measure on H/H n T. Moreover since the measure of G/r
is finite and 4>t is an open map,
(dn x dv)(Vt x H/H n T) < oo.
It follows from Fubini's theorem that dv(H/H nf)< oo. This proves the
lemma.
Lemma 1.15 admits a partial converse.
1.16. Lemma. Let G be a locally compact group and r a lattice. Then
for a closed subgroup H, if H r\T is a uniform lattice in H, H has the
following property: There exists a neighbourhood Q of e such that if
T, o) T2er, Tt, T2eH, coeii, then TjW T2eH.
Proof. If possible let Tn, Sn be two sequences in H and co.eGa sequence
in G converging to e such that Tno)nSn=yner, yn$H. Since H/HnT is
compact, we can find 6n, <pneH n T such that T~l<pn and 6n S"1 converge
(after passing to subsequences of Tn and Sn if necessary). Then
<on= T'1 yn S-1 = T'1 <pm q>;1 yn B~l 6n S"1
converges to e. It follows that q>^lyn6~l converges to a limit in H.
Thus for all large n,
(Pnl Jn On' = ^+1 V. + l 9~^eH .
But then y„eH for all large n, a contradiction.
1.17. Definition. A closed subgroup H of a locally compact group
will be said to be strongly compatible with a lattice T in G if there exists
it neighbourhood Q of e in G such that if 7^ to T2eT, Tt, T2eH, coeii then
T, (o T2eH.
1.18. Remarks, a) Let G be a locally compact group and r a lattice
in G. Let H be a closed subgroup of G and K a closed subgroup of H.

26
I. Generalities on Lattices
Then if H is strongly compatible with T and K is strongly compatible
with HnT then K is strongly compatible with T.
b) If//, and H2 are strongly compatible with T then evidently Hl n //2
is strongly compatible with r.
c) Lemma 1.5 can now be reformulated to say the following: let G
be a Lie group and r a lattice; let H be strongly compatible with T;
then // n T is a lattice in H.
1.19. Lemma. Let G be a Lie group and r a lattice in G. Let //t be a
closed subgroup of G such that //t n T is a lattice in Hl. Let H2 be a closed
subgroup such that //2 n T is a uniform lattice in H2 ■ Then H1nH2nr
is a uniform lattice in H1nH2.
Proof. In fact we will prove that H1nH2 is strongly compatible
with r. If possible let coneG be a sequence converging to e such that
there exist T„, S„eHlc\H2 such that yn=Tn(onSn<=r but yn$H1nH2.
Now T~* and Sn may be assumed to converge modulo T (note that
H2/H2 n r is compact). In view of Theorem 1.13, this implies that we can
find 0n,<pn in rr\Ht such that Tn~18n and ^S'1 converge in //t. It
follows that
^„=rB-1M„-1y„<p„-V.A-1
converges to e. Thus Q~l yn <p~l converges to a limit in Ht. It follows that
O-'y^-'eH,
for all large n. Hence Tna>nSneH1nr for n large. On the other hand
since H2 is strongly compatible with T, TnconSneH2 for all large n,
a contradiction. This proves the lemma.
We close the chapter with a result due to Chabauty [1]. We need
for this some further definition. Let G be a locally compact group and Sn
a sequence of subsets of G. We will say that Sn converges to S if for every
compact set KcG, and neigbourhood U of e in G, there exists an integer
r0 = r0 {K, U) such that for all r^r0 and x e Sr n K, x U r\ S+0 and for all
yeSnK, y[inSP=|=0. With this notation we will now prove
1.20. Theorem (Chabauty [1]). Let G be a separable locally compact
group such that eeG has a fundamental system {Va\aeZ+} of
neighbourhoods with the following property: for aeZ+, the boundary of Vx has Haar
measure zero. Suppose rnis a sequence of lattices in G such that for some
open subset W of G with ee W, Wr\r„={e} for all n. Then a subsequence
rXii of r„ converges to a limit T. T is a discrete subgroup of G and if \i is a
right Haar measure on G,
H {G/r) ^ Lim inf fi{G/rj.

1. Generalities on Lattices
27
Proof. Let {£r}reZ+ be an increasing family of compact subsets of G
such that \J Er= G. Let Wl be a symmetric neighbourhood of e such that
W~l Wx<=. W. Then we can find finitely many points x,,x2> ...,xv(r)in £r
such that Er<= y X;W,
l£.£«>(r)
Now for any n, we claim that r„n £r contains utmost <p(r) points: in fact
if this intersection has more than q>{r) points two of them say y, y' + 7 will
be contained in some x; W, but then y = xt co, y'=xt co', co, co'e W, so that
y1 y' = co~1 co'er„n W, a contradiction since y~V + e-
Assume then that we have found subsequences {rin}, {r2n},..., {rkn}
of {rn} such that {rr+ln} is a subsequence of {/";„} with the following
property Er n /^„ has a fixed number ar of points which converge to some
limits when ordered in some way. Now consider Ek + lnrkn. This has
utmost cp(k +1) points. Hence by passing to a subsequence we can assume
that we have the same number ak + l of points in all these intersections.
The Bolzano-Weierstrass theorem now enables us to choose a
subsequence such that these points of intersection converge to some limits. Thus
rik + l)„ can be chosen to be a subsequence of rkn. The same method starts
the induction at k = \ as well. Setting rXn — rnn we obtain a convergent
sequence. The limit is easily seen to be a group r. Since for any open
neighbourhood V of e such that V<=. W, VnT = {e}, it follows that r is
discrete.
To establish the last inequality we proceed as follows. We first assert
that there exists an open subset Q of G such that if jt: G^>G/r is the
natural map, n is one-one on Q and G/T—n(Q) has measure 0. Let
{t/„|neZ + } be a collection of open subsets of G with the following
properties: (i) each Un is a left translate of an open set in {FJaeZ + },
(ii) the {U„I neZ +}, is a locally finite covering of G and (iii) Un is relatively
compact in G. Let Qn be the increasing family of open subsets of G defined
us follows: Ql = Ul; assume 0„_. defined; then Qn = Qn_l*u(Un — Q„_x F)
(for a set A in G, A is the closure of A). Let Q'n be defined by setting Q[ = U.;
when Q'n_l is defined, Q'n = Q'n_1yj(Un—Q'n_ir). Clearly Q' = {jQ'n is a
measurable set such that n(Q') — G/r and % restricted to Q' is 1-1. Let
i2=[jQ„. We then claim that ficO' and n(Q) = n(Q'). We will in fact
show that Q'n=>Qn and that fi(Q'n — Qn) = 0. We will prove inductively the
loll owing statement:
Qn<=Q'n; m(O;-O„)=M(O„-O„)=0.
The start of the induction when n — 1, is obvious. Assume the statement
proved for n<n0. Then we have
Q =Q ,kj(U —Q ~n
"no «o-l * no no-1 >
Q' =Q' ,yj(U -Q ,n.
"no "n0-l <■ n0 "no-1 '

28
I. Generalities on Lattices
It follows that Q^cO^. Since the {Un} form a locally finite family we
have fi^Tr=fiB0_1-r. Now
G being separable and T discrete, T is countable. Since r is countable and
by induction hypothesis n (fl^ _ t - Q^ _ t)=^(fi^ _t —i2no_i)=0,it follows
that n(ii'^ -QJ=0. Finally, we have
so that _ _ _
Q -Q <=(S2 ,-Q ,)u([/ ~U )T
"no "n0 "i"no — 1 *'no —1' v no no/1
and by choice ,«([/„„ — UKO)=0. Since T is countable and ^0f2„o-i — ^no-i)
= 0 by induction hypothesis, Q^—Q^ has measure zero.
We have thus shown the existence of an open subset Q such that
n{n(Q))=n(G/r) and the set {yer\Qyr\Q^<ji) = {e}. Suppose now that
K<=Q is any compact subset. We then assert that there exists an integer r0
such that for r^r0, K"1 KnTXr={e}. In fact if this were not the case
we can find 6reK~l KnrXr,6r+e for infinitely many integers r. But then 6r
will converge to a limit deT. Since 6ri W, 6$ W so that 6 *e. The limit 8
on the other hand belongs to K~1KnT, a contradiction. Thus K maps
one-one in G//^ for all large r. It follows that n{K)<n{G/rk) for all
large r. Since n(K) can be made to approximate n(S2) as closely as we
want, it follows that
MG/r)=Mfl)=Lim inf ^(G/rv).
Hence the theorem.
(Theorem 1.12 can in fact be deduced as a consequence of
Theorem 1.20 in many cases, notably when G is semisimple. We will however
not go into this.)
We now introduce a definition which will be useful later for concise
formulation of results.
1.21. Definition. A subgroup T of a locally compact group G is an
L-subgroup if it has the following Property L: let n: G^G/r be the
natural map; then for a sequence xneG, 7t(x„) has no convergent
subsequence if and only if there exists yner—{e} such that xnynx~1 converges
to e.
Note that if G is second countable, r is necessarily discrete.

Chapter II
Lattices in Nilpotent Lie Groups
This chapter is essentially a detailed account of some results of Malcev [1]
on nilpotent Lie groups. The arguments used are simple and often
involve—as is to be expected—an induction on the dimension of the
Lie group.
We recall a well-known result on euclidean groups. Let HcR'bea
closed subgroup of R". Then we can find a direct sum decomposition
R"= V® W of R" and a basis eu ..., ek of Wsuch that
H= V@Z et +Z et + ■ ■■ +Zer.
In particular the identity component of H is a subspace and G/H carries
it finite invariant measure if and only if r=k and then G/H is compact.
A representation t. N—> GL(n, R) of a Lie group is unipotent if t(x)
is unipotent for all xeN. Recall that if N is nilpotent and simply
connected N admits a faithful unipotent representation p: N—> GL(n, R).
If N denotes the Zariski closure of p(N) in GL(n, C\ p(N)=NR (cf.
Preliminaries §§ 1.4, 2.2).
2.1. Theorem. Let N be a simply connected nilpotent Lie group and
I'cN a closed subgroup. The following conditions on r are equivalent.
1) For some faithful unipotent representation p: N—>GL(n, R), p(N)
and p{T) have the same Zariski closure in GL(n, C).
2) N/r is compact.
3) N/r carries a finite invariant measure.
4) There are no proper connected closed subgroups of N containing f.
5) For any unipotent representation p: N—>GL(n,R), p(r) and p(N)
have the same Zariski closure in GL(n, C).
Proof. The scheme of the proof is 1) => 2) => 3) => 4) => 5) => 1).
1)=>2). We argue by induction on dimN. When dimN=0, there is
nothing to prove. Assume the result proved for all nilpotent Lie groups
ol'dim<p and that dim N=p. Now dim[N, N]<p and p\lN,m is faithful
and unipotent; moreover since p(N) and p(F) have the same Zariski
closure in GL(n, C), p([T, T]) and p([N, NJ) have the same Zariski

30
II. Lattices in Nilpotent Lie Groups
closure. It follows from induction hypothesis that [N, N]/[r, F] is
compact. Now let U denote the Zariski closure of p(N). Then [U, U]
is the Zariski closure of p{[N, ND- Let n: U-> U/[U, U] be the natural
map. Then nop(r) is Zariski dense in the algebraic group U/[U, U].
U/[U, U] is an abelian unipotent algebraic group so that it is isomorphic
to a vector space V over C. Moreover if dim V=k under the identification
of U/[U, U] with \,nop(N) corresponds to a real subspace of
dimension k. Now nop{T) is Zariski dense in V and hence contains a basis
for V over C. Since %°p(T) is contained in n°p{N), nop(r) spans n°p{N)
considered as a vector space over R. It follows that we can find a compact
set CcJV such that n° p{Q) -no p(r)=no p(N). Since p is faithful and
[U, U] n p (N)=p ([N, N]),it follows that there exists a compact set Q a N
such that N=fi[N,N]r. On the other hand since [N,N]/[r, F] is
compact we can find a compact subset Q' <= [N, N] such that [N, N] =
fi' • [r, T]. Thus N=Q£iT. Hence N/r is compact.
2)=>3). We note that N as well as any closed subgroup of N are
unimodular. Hence N/r carries an N-invariant measure. Evidently this
has to be finite if N/r is compact.
3)=>4). Let H0 be a connected Lie subgroup of N containing T.
Suppose H0^N.N being nilpotent we can find a connected closed normal
subgroup H of N containing H0 and of codimension 1 (cf. Preliminaries
§1.2 and 1.9). Then N/H carries a finite N-invariant measure (Lemma 1.6).
On the other hand this invariant measure must be a scalar multiple of
the Haar measure on the 1-dimensional group N/H. Now N being simply
connected and H connected N/H is simply connected. Thus it is
isomorphic to R and hence the Haar measure cannot be finite, a
contradiction.
4)=>5). Let U be the Zariski closure of p(N), and V the Zariski
closure of p{T). Then V is a unipotent algebraic group. Moreover since
p(N)<=GL(n,R), UnGL(n,R)=p(N). Since VcU, Xnp{N)=XnGL{n,R).
Since V is unipotent algebraic V as well as V n GL(n, R) are connected
(see Preliminaries § 2.2 and 2.3). Thus Vn GL(n, R)=p{N). This proves
the assertion.
5)=>1) is obvious.
2.2. Remark. In the sequel we will say that a subset £ in a simply
connected nilpotent Lie group N is Zariski dense in N if for some (hence
every) faithful unipotent representation p: N—> GL(n, R), p(E) and p(N)
have the same Zariski closure in GL(n, C).
Thus the assertion 5) => 2) can be reformulated as
2.3. Theorem. A subgroup H of N is Zariski dense in N if and only
if N/H is compact.

II. Lattices in Nilpotent Lie Groups
31
Corollary 1. Define inductively Dk(N) (resp. Ck(N)) as the group
£>k(N)=[Dk_1(N),Dk_1(N)] (resp. Ck(N)=£N, C^JN)]) with C0(N)=
D0(N)=N. Then ifH isa uniform subgroup of^HnD^N^resp.HnQiN))
is a uniform subgroup Dk(N) (resp. Ck(N)).
Corollary 2. Let N be a simply connected nilpotent Lie group and H a
uniform subgroup. A connected Lie subgroup U of N is normal in N if
and only if it is normalized by H.
Corollary 3. Let H be a closed uniform subgroup of a nilpotent Lie
group N and H° the connected component of e in H. Then H° is a normal
subgroup of N.
(When N is not simply connected, we can pass to the universal
covering.)
2.4. Lemma. Let N be a connected simply connected nilpotent Lie
group. Let Ut and U2 be closed subgroups of N. If Ut and U2 are connected
so is t/j n U2 ■
Proof. The exponential map exp: tt^iV of the Lie algebra n of N
into N is a diffeomorphism. It follows from this that exp maps any
subalgebra diffeomorphically onto the corresponding closed subgroup
of N. Thus if ut and u2 denote respectively the Lie subalgebras
corresponding to t/j and U2, UlnU2=exp(ulnu2) and evidently utnu2 is
connected. Hence the lemma.
2.5. Proposition. Let N be a simply connected nilpotent Lie group
and H any subgroup. Then H is contained in a unique minimal connected
closed subgroup H of N. If H is closed then H/H is compact,
Proof The proposition follows from the lemma above and
Theorem 2.1.
2.6. Remark. H is Zariski dense in ft in the sense of Remark 2.2.
Thus the Zariski closure H of H is the minimal connected closed subgroup
ol N containing H.
We also need in the sequel the following
2.7. Theorem. Let Tbea finitely generated nilpotent group. Then every
subgroup of r is finitely generated.
Proof. For subsets A, B of T let {[A, B]} denote the set
{aba'1 b~1\aeA, beB}.
Let S be a finite set of generators for r and let S0=S and inductively
I'oi k>0, Sk = {[S,Sk_l']}. Since T is nilpotent, Sk — {e} for all large k. Let

32
II. Lattices in Nilpotent Lie Groups
Cr(iT) be the subgroup of T generated by \J Sk. We will now show that
Cr(r) is a normal subgroup of T: we argue by downward induction.
For r large Cr(r)=e. Assume then that Cr(r) is normal in T and consider
the natural map
n: r^r/c,{r).
Then for seS, s'eSk, k^r-l, [s,s-]eSk+1 and Sk+l<=Cr(r) so that if
s-e [j Sk,n{s') commutes with n{S) hence with r/Cr(r). Thus 7r(CP_i(r))
(tgr-1
is central in r/Cr(r) and since CP_1(r)=7t_1re(CP_1(r)), C^^/") is
normal in T. Moreover, it is clear that [T, C^JfYJc.C^r) so that
CP-i(r)/CP(r) is abelian. Setting rr= Cr{T) we see that each subgroup H
of r admits a filtration Hr=Hr\rr such that Hr_jHr is a subgroup of the
finitely generated abelian group rr_jrr. Thus each Hr_JHr is finitely
generated so that H is finitely generated. This proves the theorem.
2.8. Proposition. Let r be a finitely generated nilpotent group and
r=r0=>ri=>-=>rk=eandr=ro=>ri'=>-=>ri={e} be two filtrations of
r such that I] (resp. I]') is a normal subgroup ofrl_l (resp. I]'_i) and rt_jrt
(resp. Ii_JIi) is abelian. Ihen i]_i/7; {resp. r{_jr{) is finitely generated and
£ rank(/]_1//])= £ rank(rUV).
We omit the proof which results from standard arguments.
2.9. Definition. The integer £ rank(7]_i//D which is independent
of the filtration chosen is called the rank of the finitely generated nilpotent
group r.
2.10. Theorem. Let r be a discrete subgroup of a simply connected
nilpotent Lie group N. Let t be the Zariski closure ofTin N (cf. Remark 2.6).
Then r is finitely generated and
rank T=dim f.
Proof. We argue by induction on dimension of N. If dim f <dim N,
induction hypothesis applies. We may thus assume that t=Ni.e. that N/r
is compact. If N is abelian, N is isomorphic to a euclidean vector space
and r is necessarily the subgroup generated by a basis for this vector space.
Assume then that N is not abeliaa Let n: N—>N/[N, N] be the natural
map. Tn [N, N] is a lattice in [N, N] (Corollary 1 to Theorem 2.3) and
since dim [N, N] <dim N, fn [N, N] is finitely generated and
rank (fn [N, N])=dim [N, N].

II. Lattices in Nilpotent Lie Groups
33
Consider now n{r). Now n(r) is discrete in N/[N, N] (cf. Theorem 1.13)
and also (N/[N, N])/^(r) is compact. It follows that n(r) is a lattice in
N/[N, N] and hence n(r) is finitely generated. Now N/[Af, N] is
isomorphic as a Lie group to a real vector space R*(fc=dim N/[N, NJ)n(r)
must therefore be the Z-span of a basis of R*. Thus
rank(7t(r))=dim N/[N, N].
Since the sequence
{e}^rn [N, N]^r^7r(r)^ {e}
is exact, r is finitely generated and
rank T=rank fn [N, N] + rank 7t (/")
= dim [N, N] + dim NfcN, N]
=dimN.
This proves the theorem.
Corollary 1. Let N be a nilpotent simply connected Lie group and H
a closed subgroup. Let H° be the connected component of e in H and H
the Zariski closure of H in N. Then
dim H = dim H°+rank H/H°
(H/H° is a finitely generated nilpotent group).
Proof. H° is normal in H and H/H° is a lattice in H/H°. The result
now follows from Theorem 2.10.
Corollary 2. Let N be a connected nilpotent Lie group (not necessarily
simply connected). Let r<=.N be any discrete subgroup. Then r is finitely
generated.
We can now deduce
2.11. Theorem. Let N and V be two nilpotent simply connected groups
and let H be a uniform subgroup of N. Then any continuous homomorphism
p. H^*V can be extended in a unique manner to a continuous
homomorphism p: N^>V.
Proof. Let H'ciVxFbe the subgroup
{{x,p(x)\xeH}.
Let U be the Zariski closure of H' in N x V. Let nt (resp. n2) denote the
projection ofNxVonN (resp. V). Since nt{U) is connected and contains
//, 7t| (U)=N. Thus ni\u is onto. We claim that rc^y is 1-1. Since U and N

34
II. Lattices in Nilpotent Lie Groups
are nilpotent simply connected groups, it suffices to prove that dim U =
dim N. Now by Corollary 1 to Theorem 2.10
dim N=dim H°+rank H/H°
dim t/=dim //'°+rank H'/H'°
where H° (resp. H'°) is the connected component of the identity in H
(resp. H'). Now H and H' are evidently isomorphic as Lie groups so that
it follows that dim U=dimN. Note that the graph in Nx.V of any
extension a of p will be a connected closed subgroup containing H' and
of dimension equal to N, hence must coincide with the group U. On the
other hand since n^y is l-l, U is indeed the graph of a map p viz. n2 °{ni\v)~\
Clearly p is the unique extension of p. Hence the theorem.
Corollary 1. Let N be a connected simply connected nilpotent Lie group
and H a uniform subgroup. Then any automorphism of H extends to a
unique automorphism of N.
Corollary 2. Let Nt and N2 be simply connected nilpotent groups and
Hlt H2 uniform subgroups of Nt and N2 respectively. Any isomorphism of
Ht on H2 extends to an isomorphism of Nt on N2. In particular Nj/Z/j and
N2/H2 are homeomorphic. Conversely if H^ and H2 are uniform lattices and
NjHi and N2/H2 are homeomorphic, then Nt and N2 are isomorphic.
Proof. Observe that homeomorphic manifolds have isomorphic
fundamental groups.
We will now give a criterion to decide when a simply connected
nilpotent Lie group admits a lattice.
2.12. Theorem. Let N be a simply connected nilpotent Lie group and
let n be its Lie algebra. Then N admits a lattice if and only if n admits a
basis with respect to which the constants of structure are rational.
More precisely we have: Let n be a Lie algebra with a basis with
respect to which the structural constants are rational. Let n0 be the
vector space over Q spanned by this basis; then if JS? is any lattice of
maximal rank in n contained in n0, and exp: n—>N is the exponential
map, the group generated by exp JSf is a lattice in N. Conversely if r is a
lattice in N, then the Z-span of exp_1r is a lattice (of maximal rank)
JS? in the vector space n such that the structural constants of n with
respect to any basis contained in JS? belong to Q.
Proof Assume first that n admits a basis with rational structure of
constants. Let n0 be the Q-span of this basis. Clearly n0 is a Lie algebra and
n^n0®QR.

11. Lattices in Nilpotent Lie Groups
35
Let p0 be a faithful representation of n0 in M(n, Q), the Lie algebra of
nxn matrices with entries in Q. We assume further p0 chosen such
that p0(X) is upper triangular nilpotent for all Xen0. Such a choice of p0
is possible in view of Ado's theorem (Preliminaries § 1.3). Let p be the
canonical extension
n~n0®QR-^^M(n,Q)®QR = M(n,R).
Integrating p we obtain a faithful unipotent representation
p: N->GL(n,R).
Let U (resp. u) be the group (resp. Lie algebra) of upper triangular
unipotent (resp. nilpotent) matrices in GL(n, R) (resp. M(n, R)). Then the
diagram
exp
where exp is the exponential map, is commutative. Since p restricted
to n0, maps n0 into M(n, Q), the lattice JS? is mapped by p into the set
/ ' • M(n, Z) where /. is a suitable rational integer. Since p is a homeo-
morphism of N onto a closed subset of U, to show that the group
generated by exp JSf is discrete it suffices to show that the group generated
by exp(u n A-1 • M(n, Z)) is discrete. To see this, let £y denote the matrix
all of whose entries other than the ij-th entry are zero and the ij-lh
entry is 1. Now any matrix Xinur\M(n,Z) A-1 can be written in the form
x=x-xY,atJEij
where ai} are integers. Now let B be the diagonal matrix whose i-th
diagonal entry is (n! Xf~l. Then we have
BXB-i=k-lYJ^\>.)>-ial)Elj
so that BXB~1en\ M(n, Z). Since X is nilpotent, we have then
Bexp^B-1=expB^B-1 = l + BXB-1+- +
n!
;md since BXB"1enl M{n,Z\ it follows that
BexpXB-'eSL(n,Z)
for all Xeu n M(n, Z) A-1. Thus the group generated by
{expXlXeunMfn.Z).}-1}

36
II. Lattices in Nilpotent Lie Groups
is discrete. It follows then that exp JSf generates a discrete subgroup T
of N. Iff is the Zariski closure of T in N, exp_1f is a Lie subalgebra
of n containing jSP; since jSf is a lattice in n, exp_1f=it Thus f = N and
hence N/r is compact. This proves the first part of the theorem.
We now prove the converse assertion. Let r<=N be a lattice. Let n0
be the Q-span of exp_1(r). We will show that n0 is a Q-Lie subalgebra
of n and that n ^ n0 ®Q R. We will argue by induction on dim N. According
to Corollary 1 to Theorem 2.3, Tn [N, AT] is a lattice in [N, ATj. It follows
from Theorem 1.13 that T. [N, N] is closed in N. Hence if n: N -»• N/[N, N]
is the natural map n(F) is discrete in N/\_N, N]. The group N/\_N, N]
being a real vector space, n(F) is the Z-span of a basis el5..., ek of this
vector space. Let V be the R-span of elt ...,ek_t and let N' = n~1(V).
Let n' be the Lie algebra of N'. Then n' may be identified with a subalgebra
of n of codimension 1. Now by induction hypothesis the Q-span of
exp_1(rn N') is a Q-Lie subalgebra n'0 of n' and the natural map
n^®QR->
iris an isomorphism. Now r/F is infinite cyclic since it is a lattice in
N/N'(s.R); lety0er be any element such that its image in r/F generates
r/F. Clearly y0$N'. Let ven be the unique element such that exp v=y0.
Then we have for wen'0,
[r,w] = [exp-1y0,w]
= E((-ir1/«)(Ady0-l)-(w)
where Ad: N^Aut„(n) is the adjoint representation. ((Ady0—1) is
nilpotent so that the summation is finite.) Moreover, since N' is normal
in N, N'nTis normal in T; hence N'nTis stable under inner conjugation
by y0; it follows that exp_1(rnN1) and hence n'0 is stable under Ad y0.
Thus for wen'0 we have _ n ,
[i;,w]en'0.
Thus Q v+n'0 = n'0' is a Q-Lie subalgebra of n and since v4 n0, the natural
n0 ®Q R —> n
is an isomorphism. We claim that n'o=n0, the Q-span of exp_1(r).
To prove this let p0: n'0'—> M(n, Q) be a faithful representation such that
p0(X) is upper triangular nilpotent for all Jfen'o. Let p be the natural
extension . „_,.w/M m
p. n—>M(n, R)
and let p be the integrated (faithful) representation of N
p: #->GL(n,R).

II. Lattices in Nilpotent Lie Groups
37
Let T (resp. t) denote the group (resp. Lie algebra) of upper triangular
unipotent (resp. nilpotent) matrices. Since the diagram
GL(n,R)
is commutative and for Xen,
exp*=l-r*+—+...+^-I5r,
it follows that p(y)eGL(n,Q)-for yeN'nT and p{y0)eGL(n,Q). Hence
the group generated by p{y0) and p(N'nr) is contained in GL(n,Q).
Now if a: N-* N/N' is the natural map, a(F) is infinite cyclic. It follows
that the sequence
is split and a(y0) is a generator of a(r). Thus T is generated by rnN'
and y0. It follows that p(r)<=GL(n,Q). Hence
t n exp- V (T)=p (exp-1 (T)) <= M (n, Q).
Thus exp~1(r)c:p~1(M(n,Q)) and since p is the extension of p0, it is
easily seen that p~1(M(n, Q))=n'o. Thus we have exp_1(r)c:no. Hence
it'o=n0, the Q-span of exp"1 (F). This completes the proof of Theorem 2.12.
2.13. Remark. We give a slight variation of the proof of the first part.
Suppose then that n ~ n0 ®Q R where n0 is a nilpotent Lie algebra over R.
Let p0: n0—>M(n, Q) be a faithful representation of n0 such that for
every Xen0, p0(X) is nilpotent. Let p: n—>M(n, R) be the natural
extension of p0 to n. Let p be the representation of N obtained from p:
p: N-*GL(n,R).
Let r=p~l(SL(n, Z)). Then we claim that N/r is compact. To see this
let N be the image of p, and let n=p(n). Then ft is a vector subspace
of M(n, R) defined over Q. Let ft0=Po(no)- We will now show that N is
the smallest connected subgroup of N containing t=p(r). To prove
this it suffices to show that exp~l(t) spans ft as a real vector space or
equivalently ft0 as a Q-vector space. Let Xen0 be any element. Then
lor an integer A, > ^ ^2 ^^
exp AX = 1+AX+—-—+•••+-
2!

38
II. Lattices in Nilpotent Lie Groups
Since XeM(n, Q\ for a suitably large integer k, exp /.XeSL(n, Z). Thus
AXeexp-1f for a suitable AeZ. Hence n0 is contained in the Q-span of
exp-'(f). Now if JS? is a lattice of maximal rank in ft contained in ft0,
then for a suitably large integer X, exp X£? a SL(n, Z). It follows that iff'
is the group generated by expJSf and f" that generated by expAjSf,
then N/t' and N/t" are compact. Clearly, on the other hand F'ct.
It follows that f and f' are commensurable i.e. tnt' is of finite index
in f as well as f'. This shows that if p is any representation of AT induced
by a representation p0 of n0 in M(n, Q) such that p0(^Q *s nilpotent for
all Xen0, then though p_1(SL(n, Z)) is dependent on p, its "commen-
surability class" is not.
2.14. Remark. We will now prove the existence of a nilpotent Lie
algebra over R which does not admit a basis for which the structural
constants are rational. For this purpose we start with a vector space E of
dimension at least 6 and let V be a vector space of dimension 4. We will
then look for Lie algebra structures on E® V such that V is central and
[_E® V, E® K] = V. Such a Lie algebra structure is evidently given by a
surjective homomorphism
q>: A2E-^V.
Now if dim E=n, the set of all homomorphisms of A2 £—► Kis a vector
space of dim — 4=2 n2 — 2 n and the set of homomorphisms which
are surjective is open and non-empty (note that n (n —1)/2 > 10 since n > 5).
Thus the set of q> which are surjective is an open subset in a vector space
of dimension 2n2 —2n. Now if the Lie algebra structure on E@V given
by q> has rational constants of structure, we can choose a basis XU...,X„
of E and 7,,..., Y4 of V such that with respect to the basis {Xt a Xj}^ j
of A2 E and {r/}ig1i4 of V, the matrix of q> has all entries rational. We
now fix once for all a basis X°,..., X° of E and a basis y,°,..., Y£ of V.
Suppose we are given bases Xl,...,X„ of E, Yj,..., Y4 of F and a linear map
(/>: A2 E-^V
which with respect to the bases {Xt a ^}i<; and {y,}1gig4 is represented
by a rational matrix. Let u (resp. r) be the automorphism of E (resp. K)
which carries the basis Xl,...,Xn into X,0, ...,X°(resp. ¥,,..., YA into
Y10,...,y4°).Then
v°q>°u '
isamapof/l2£in K which is a rational matrix with respect to {X?AXi-}i<j
and {Yi°}i^i^4. Thus in the set Q of surjective linear maps of A2E
on K those which admit rational structure constants are of the form
vocpou~l where i;eAutK ueAutE and q>: A2E^>V is rational with

11. Lattices in Nilpotent Lie Groups
39
reference to {X? a X?};</ and {Yi°}lgig4. If A denotes the set of linear
maps <p\ A2 E-+V which are rational matrices with reference to
{X° a Xf}iKj and {Y,0}l£igA, then the set of i// for which the Lie algebra
admits rational structure constants is {v°q>°u~l\veAut V, q>eA and
ueAut E} i.e. the union of the orbits of the elements cpeA under the
group Aut FxAut£. Now A is countable and dim(Aut Vx Aut£) =
n2 + 16. Thus the set of i// for which the corresponding Lie algebra admits
a Q-structure is a countable union of submanifolds of dimension
,i2 + 16. The dimension of Q on the other hand is 2n2 — n. Since n>5,
2«2-2n-n2-16 = n2-2n-16^36-12-16>0. Thus the set of \ji
which give rise to Lie algebra structures which admit structural constants
in Q has a nowhere dense intersection with Q. This proves that there
exist nilpotent Lie algebras which do not admit any basis with respect
lo which the constants of structure are rational.
2.15. Remark. There is no uniqueness of the Q-structure (if one exists)
on a nilpotent Lie algebra. Let K = Q(y/2) = Q[X]/(X2-2) (j/f is the
image of X) and let rrj denote the Lie algebra over K of all upper triangular
(3x3) nilpotent matrices with entries in K. Evidently rtj is a Lie algebra
over K of dim 3. rt', may be considered as a Lie algebra over Q and when
considered as such will be denoted rt!. Let u denote the Lie algebra of
upper triangular nilpotent (3 x 3) matrices over Q and, let rt2 be the
direct product u x u. We claim that rtj ®QR and rt2 ®QR are isomorphic
while rt! and tt2 are not. In fact we will show that tti <g,Q K and tt2 ®Q K
are isomorphic. Let a: Q("|/2)—>Q("|/2) be the unique non-trivial Galois
automorphism of K. Evidently u®Q K~rti; we fix one such isomorphism
ip: u<8)q-K —>«!. Since rt2<g,QK can be canonically identified with
(ii®q K)x(u®Q K), the pair (<p-1, <p-1°(l(8)o-)) together give an injection
•t,-»ii28Qf..
This extends evidently to an isomorphism of rtjfg.QK on n2<S>QK.
Finally a simple computation shows that for .Yen, the centraliser of X
is either n, or is of codimension 2 in rt!; on the other hand there exist
elements in rt2 whose centralisers are of codimension 1. This shows
lhat rt! and tt2 are not isomorphic.
2.16. Remark. Let N be a nilpotent Lie group rt its Lie algebra. Let F
be a lattice in N and tt0 the Q-span of exp_1(r). Let tt' be a subalgebra
of rt such that tt' is spanned by tt0 n tt'. Let N' be the subgroup of N
corresponding to tt'. Then N'nT is a lattice in N'. In fact if JS? is any
..mice of tt contained in tt0, then expi>? generates a lattice H in N such
I hilt HnT has finite index in both H and r. Thus it suffices to show that
llnN' is a lattice in N'.To see this first observe that exp~' //nn'^J5?nn'
iiud the latter is a lattice in rt' since rt' is the linear span of rt'nn0. It

40
II. Lattices in Nilpotent Lie Groups
follows that the discrete group H n N' is such that it is not contained in
any proper connected closed subgroup of AT. Hence H n N' is a lattice AT.
In particular we obtain
2.17. Proposition. Let r be a lattice in a simply connected nilpotent
Lie group N. Let {C'k(N)}k^0 be the ascending central series of N. Then
C'k{N)nr is a lattice in C'k(N). In particular the intersection of r with
the centre of N is a lattice in the centre. (C'k(N) is defined inductively as
follows: C'0{N)={e); assume Ck_i{N) is defined and normal in N and let
rc: N —► N/Ck_i(N) be the natural map. Let Z be the centre of N/C'k_,(N).
Then Ck(N) = TTl(Z).)
2.18. Theorem. A group r is isomorphic to a lattice in a simply connected
nilpotent Lie group if and only if
1) r is finitely generated
2) r is nilpotent and
3) r has no torsion.
Proof. If r is isomorphic to a lattice in a simply connected nilpotent
Lie group N, evidently T is nilpotent. According to Theorem 2.10 /' is
finitely generated. Since N is simply connected, N admits a faithful
unipotent representation and a unipotent matrix of finite order is
necessarily the identity; N and hence r has no element of finite order
other than the identity.
Now assume that T has the properties 1), 2) and 3). We will argue by
induction on rank (r). To do this however, we need
2.19. Lemma. // T is a nilpotent torsion free group /"*/[/% r] is infinite.
Proof. Let r=r0 =>/^ =>•••=>/]■._, =>Tk={e} be the descending central
series for T. We assume as we may that T is non-abelian (in case T is
abelian, the lemma is trivial) and that rk_, =# {e}. Let y0 eTk_ 2 be an element
such that {[y0, r]} =# {e} and let H be the subgroup generated by y0 and
/;_,. Since [T, rt_J = {e} and {[T, y0j} ^[r, ^.Jcr,.,,/. is a normal
subgroup of r. Moreover since y0 commutes with /""._,, H is abelian.
Also, \H, [r, r\] = {e} since H<=rk_2. It follows that the action of r
on H by inner conjugation factors through T/[r, r]. We thus obtain an
action of T/[r, T] on H. Writing the group law in H as addition, we
have for aeH and yeT,
lnt(y)(ix)-<xerk_l
and lnt(y)(a)-a=0 if ae/^. Thus (lnty-l)2(H)=0. It follows that
Int y when extended to H ®z Q is a unipotent automorphism. If T/[r, r]
is finite, Int y would be of finite order as well. Thus the extension of
Inty to H®z Q is trivial. Since H is torsion free this implies that Inty

II. Lattices in Nilpotent Lie Groups
41
acts trivially on H for all yeT. But y0eH was chosen such that {[r,yo]} 4= {«}>
a contradiction. This proves the lemma.
Proof of the theorem. According to the lemma r/[T, r\ is infinite.
This group is in addition finitely generated and abelian. It follows that
we can find a normal subgroup /"J, T => /} => \T, r] such that T//J is infinite
cyclic. Then T is the semidirect product of a cyclic subgroup J of r and rt.
Now by induction hypothesis there is a simply connected nilpotent
Lie group N, and an injection i,: r1e-*Ni such that i,(/J) is discrete and
Ni/im) is compact. We identify /"" as a subgroup of N, via the map i,.
Now the group J acts as a group of automorphism of /"J and hence we
obtain from Theorem 2.11 a homomorphism
a: J-^Aut(Nj)
extending the action of J on /"J. Now let p: N,—>GL(.?j, R) be a faithful
unipotent representation. We define the groups /"J'*' inductively as
follows: /70)=r„ /I(t, = [r, J?*-1']. Let N/*' be the Zariski closure of I™
in N,. Then p{Nlk)) and p{T^k)) have the same Zariski closure in GL(m, C)
(Theorem 2.1). Moreover if Ulk) is the Zariski closure of p{N[k)) in
GL(m,Q, U<k)nGL{m,R)=p(N\k)). Now for 0eJ, yel?-l\
(I) eye-iy^eft",
The automorphism ct(0): Nl—*N1 extends canonically to an algebraic
automorphism
a(ey. l/<°>->l/(0>
so that from (I) we conclude that
5(9)(y)y-leU<k)
lor all deJ and ye t/(*_1). Taking intersections with GL(n, R) we conclude
that a(e)(y)y-leN[k) for Be J and yeN[k~l). If we let a{6) denote the
automorphism induced by d{9) on the Lie algebra n, of N,, we deduce
that for xen'j*-1', the Lie subalgebra of n corresponding to N[k~l), we have
(<r(0)-l)(x)enf>.
It follows that (j(0) is unipotent for OeJ. Now (T is a homomorphism
of J in Aut(n,) the group of Lie algebra automorphisms of n,. Since this
group is the set of real points of an algebraic group defined over R and
ci (J) consists of unipotent elements, a (J) is contained in a connected
unipotent subgroup V of Aut(nj). V is then simply connected and
nilpotent. Now by Theorem 2.11. we can find a unique homomorphism
</>: R-^KcAutftt!)

42
II. Lattices in Nilpotent Lie Groups
Z=" >R
J—2—► Autn1=Aut(JV1)
making the above diagram commutative. Clearly then we have an
injection i of T, the semidirect product of J and /"J into the semidirect
product of R and JV,. (Note that since JV, is simply connected, Aut n, =
Aut JV,, the group of all continuous automorphisms of JV,.) Clearly the
diagram ^___ .
1—ir^Nt
is commutative. It follows easily from this that i is injective, that i(r)
is discrete and that N/i(iT) is compact. This completes the proof of
Theorem 2.18.
2.20. Theorem. Let N be a nilpotent Lie group and N° its identity
component. Let r = N/N° be finitely generated and torsion-free. Assume
moreover that N° is simply connected. Then there exists a connected simply
connected nilpotent Lie group U and a continuous injection i: N^-*U such
that i(N) is closed and U/i(N) is compact.
For proving this one argues once again by induction on rank N/N°.
The details follow closely the proof of Theorem 2.18 and are left to the
reader.
We close this chapter with the following
2.21. Theorem. Let r be a lattice in JV, a simply connected nilpotent
group. Ihen if dim JV=n, r is generated by a set of n elements.
Proof. We argue by induction on dim JV. Let n: JV--> JV/[iV, JV] be
the natural map. Then n(r) is a lattice in N/[N,N~]. (Corollary 1 to
Theorem 2.1 and Theorem 1.13) since N/[N, N~] is isomorphic to a
vector space, n(r) is generated by dim N/[JV, JV] elements. On the
other hand Tn [JV, JV] is by induction hypothesis (note that dim [JV, JV] <
dim JV) generated by dim [JV, JV] elements. Since the sequence
{e} -> rn [JV, JV] -► r -► 7i(T)->. {e}
is exact, the theorem follows from the fact
dim JV=dim [JV, N] -(- dim JV/[N, JV].

Chapter III
Lattices in Solvable Lie Groups
Lattices in solvable Lie groups are much more difficult to handle than
those in nilpotent Lie groups. This and the next chapter are devoted to
studying them. Most of the results in this chapter are due to Mostow
([4] and [5]). The proofs given below are however often different from
those of Mostow.
The first main result is
3.1. Theorem. Let G be a solvable Lie group with countably many
connected components. Let H be a closed subgroup then G/H carries a
G-invariant finite measure if and only if G/H is compact.
Proof. We argue by induction on dim G. Let G° be the e-component
of G. Then H normalises G° and HG° is an open and hence closed
subgroup of G. If G/H is either compact or carries an invariant finite
measure then the same is true of the discrete space G/HG° so that this
set is finite. On the other hand HG°/H is compact (resp. carries an
invariant finite measure) if and only if G°/G° n H is compact (resp. carries
a finite invariant measure) (Lemma 1.7). Thus it suffices to prove the
result for G° to obtain it for G. We assume thus that G is connected.
By passing to the universal covering we may assume G to be simply
connected as well.
The start of the induction when dim G = 1 is obvious. Let N be the
maximal normal nilpotent subgroup of G and C0 its centre. Then the
Lie subalgebra c0 of the Lie algebra g of G corresponding to C0 is a
G-module. If G =t= e, N + e and hence C0 + {e}. Let c=#0 be a G-irreducible
submodule of c0 and C the corresponding analytic subgroup of G.
Since G is simply connected and C is normal in G it is closed in G. Let Gt
be the closure of HC. Assume that G, =#G. Then since G is connected,
dim G, <dim G. If G/H is compact, so are G/Gt and GJH. By induction
hypothesis, since dimG^dimG, GJH carries a finite Gj-invariant
measure; also G/G1={G/Q/{GJC) and dim(G/C)<dim G since C + e;
hence by the induction hypothesis, G/Gt carries a G-invariant finite
measure; it follows that G/H. carries a G-invariant finite measure
(Lemma 1.6). Conversely if G/H carries a finite G-invariant measure,

44
III. Lattices in Solvable Lie Groups
G/Gj carries a finite G-invariant measure and GJH a finite G-invariant
measure (Lemma 1.6); again since G/Gl =(G/Q/{Gl/Q and dim G/C<
dim G, G/G, is compact; similarly dim Gx < dim G so that GJH is compact
Thus G/H is compact.
We can therefore assume that Gj =G, i.e. HC=G. Now let U be the
smallest connected Lie subgroup of N containing HnN (see Lemma 2.1).
Now U/HnU is compact (see Theorem 2.1). It follows that G2=HU is
closed in G. Since H r\N is normal in H, H normalizes U. By Theorem 2.1,
U/H n U carries a [/-invariant finite measure as well. Thus by Lemma 1.7,
HU/U carries an H[/-invariant finite measure. Also HU/H being
homeomorphic to U/H n U, is compact. Now C normalizes U. Since
HC=G, U is normal in G. Thus G/G2=(G/U)/(G2/U). By induction
hypothesis, it then follows that G/G2 is compact if and only if it carries
a G-invariant finite measure, provided that dim t/=#0. We are thus led
to consider the following case:
HC = G; HnN = {e}.
Now [fl,ff|ciVn[G,G]clfnJV = {c}. It follows that H is abelian.
Since HC = G, c is //-irreducible as well. Moreover [//, C] <= C, [//, H~\ =
[C, C] = {e} so that [HC,HC]<=C and C being closed, [G,G]<=C.
If [G, G] = {e} G is abelian and the result follows from Theorem 2.1.
If c is a trivial //-module, then C is central in G so that G is nilpotent
and Theorem 2.1 applies again. Thus we can assume that c is an irreducible
non-trivial G-module. H being abelian, dim C = 1 or 2 We will now show
that this case cannot occur: i.e. under these assumptions G/H cannot be
compact nor carry a finite invariant measure. In fact G/C contains the
image of H as a dense subgroup and H is abelian. Thus G/C is abelian
and Ad H acts trivially on g/c. Since H is abelian, g decomposes into a
direct sum
g=g°©c
where g° = {t;eg|(Ad h — 1) v=0 for heH}. (Note that c is an irreducible
and nontrivial //-module and H acts trivially on G/C.) Let G° be the
Lie subgroup corresponding to g°: G° is the identity component of the
centraliser of H in G and is hence closed. Moreover since exp: c—> C is a
homeomorphism no element of C can commute with all of //. It follows
that if Z(H) is the centraliser of H in G, Z(H)n C=(e). On the other
hand G°<=Z{H) and G° C=G. It follows that Z{H)=G° so that H<=G°.
Clearly now G is the semidirect product of G° and C. If G/H is compact
or carries a finite G-invariant measure so does G/G°. But then since
the map C-^G-^G/G° of C on G/G° is a homeomorphism, C would
carry a finite C-invariant measure. But C being a noncompact (abelian)
Lie group, this is a contradiction. This proves the theorem.

III. Lattices in Solvable Lie Groups
45
The next result is an important step towards the proof of Theorem 3.3
which seems to play an important role in the study of lattices in solvable
Lie groups.
32. Theorem. Let G be a solvable connected Lie group. Let H be a
closed subgroup of G such that G/H is compact. Let pbea finite dimensional
representation of G in GL(n, R). Let G* be the Zariski closure of p(G)
in GL(w, Q and let N be the (normal) subgroup of all unipotent elements
in G*. Then p(H) and p(H)-N have the same Zariski closure in GL(n, C).
Proof. Let G1=p(H)N. Let GJ (resp.H*) be the Zariski closure of
p(H)N (resp. p(H)) in GL(n, C). Let a be a representation of GL(n, C)
in a vector space V admitting a vector v0 such that
H* = {h\heGL(n, Q, a(h) • C v0=Cv0}
(cf. Preliminaries § 2.1). Evidently it suffices to show that N<=H*. Now G
being solvable we can find a filtration
V=V0=>V^.-^Vk^Vk+1 = {0}
and characters {xt: G*-^C*}0^t^k such that
(i) for Ogigfc, Vt is a(p{G)) (and hence a{G*)) stable and
(ii) for geG* and veVt,
a{g){v)=Xl(g)vmodVi+1.
Since a is a representation of GL(n, C), o(g) is unipotent for all geN.
It follows that xt(N) = l for all i with 0^/_fc. Thus &|„(h,=JOUh, ""^d
only if Xi\p{H)N=Xj\fi(H)N for O^ijgfc. We now make the following
Claim. If veV is a common eigen-vector for all of p(H), it is also a
common eigen-vector for all p(H) N.
Proof of Claim. When ve Vk, v is an eigen-vector for all of G*. Assume
inductively that if veVr, r^r0 +1, is an eigen-vector for p(H) then it is
an eigen-vector for p(H)N. Let veVro be an eigen-vector for p(H) and
assume that v$Vro+1. Let arg denote the representation of G* on
VrJVro+1 obtained from a and let n: KPo-> VrJVro+l be the natural map.
Then we have for geG
°r0(p(g))(x)=Xr0(p(g))-x for xeVJVro+1.
Now for heH,v/e have
a(p{h))v=X(h)v
for some X(h)eC*. On the other hand
Hh) n(v) = n{a(p (h))(v))=Xro(p (h)) • n(v)

46
III. Lattices in Solvable Lie Groups
so that xrQ(p(h))=X{h) for all heH. Hence function
g,->Zr„(p(g))~1ff(p(g))i'
is invariant under the right action of H. Let $ be the function on G/H
obtained from the above. Clearly $ is continuous. Let
Vi= \&(x)dn
G/H
where d\i is the (finite) invariant measure on G/H of total volume 1
(cf. Theorem 3.1). Now
G/H
= in(Xr0(pM)-1-o(p{x!))v)dn
G/H
where xfeG is a representative for xe G//f. Thus
«("i)= J xX*'))-1 ^„(pM) «(p) dp=K(v).
G/H
It follows that Vi — veVrg+l. Now t^ is an eigen-vector for all of G as is
seen from the definition of t>,; for ge G, the eigen value of g corresponding
to Vi is clearly x,0 P(g)- Thus v and t^ are both eigen-vectors for all H,
the eigen values of any heH corresponding to these two vectors being
the same. It follows that vx — v is an eigen-vector for all of H. Since
vt — v e Vr +1, in view of the induction hypothesis, vt — v is an eigen-vector
for all of p(H) N. Thus we have necessarily for hep(H) N,
{a(h))(vl-v)=Xj(hHvl-v)
for some j with r0<j^k. Now tfj is an eigen-vector for all of p(G) and
hence G* and a{g)(v1)=x,a(g)(v1) for all geG*; moreover; for heH,
Xro{p(h)) = xMh)) ie. x^pm=xhw But tfaen Xr0\p(H)N=Xj\PlH)N- Thus
fj—r and fj are eigen-vectors for all of p{H)N corresponding to the
same eigen-values. Hence v=v, — (vt — v) is a common eigen-vector for all
of p{H) N. Hence the claim.
Taking v=v0, the theorem follows from the claim.
We now come to a cental result about lattices in solvable Lie groups.
The theorem is due to Mostow [4]. The proof below however is different
from that of Mostow.
3.3. Theorem. Let G be a connected solvable Lie group and N its
maximum connected (closed) normal nilpotent Lie subgroup. Let H be
a closed subgroup of G such that G/H is compact. Assume that HnN
contains no connected (closed) Lie subgroup normal in G. Then N/H n N
is compact.

III. Lattices in Solvable Lie Groups
47
For the proof, we need
3.4. Lemma. Let Gbea connected simply connected solvable Lie group
and N its maximum connected nilpotent normal subgroup. Let g be the
Lie algebra of G and ttcg the subalgebra corresponding to N. Let U be a
closed connected subgroup of N and ucg the corresponding subalgebra.
Assume that U contains no proper connected normal subgroup of G and
let Gl be the normaliser of U. Let xeGt. Then Adx is unipotent if and
only if the automorphism of g/u induced by Ad x is unipotent.
Proof. Let gc be the vector space g®„ C and let
9c=LIgA
AeE
be the decomposition of gc into generalised eigen-spaces with respect to
Ad x: E is the set of eigen-values of Ad x and for Xe E,
gA = {vegc|(Ad x—kI)' v=0 for some r}.
Then we have [g*\ a*1] <= g^ (setting gv=0 for v$E). Now suppose Adx
acts unipotently on g/u. Then for X 4= 1, the natural map
n '• 9c-^9c/«c
where uc is the subspace of gc spanned by u®lc:gc=g®ltC, clearly
maps g; into zero i.e. for A+l, g*"-c:uc. Let Vc= \J q\ Then it is
AeE.-l+l
easily seen that Vc n g spans Vc as a C-vector space and that
g = g'ng©Kcng.
Now let h be the subalgebra of g generated by V= Vc n g. Evidently this
subalgebra is normalised by any XeV—in fact it contains V. On the
other hand [g1, V^\ <= Vc: V is stable under ad X for Xeg1 n g. It follows
that h is an ideal in g. Since Vc is contained in u^., V is contained in u
and hence hcu. Since U contains no connected normal subgroup of G
different from (e), h=(0). Thus gjl=0 for A=#l. We conclude therefore
that all the eigen-values of Ad x (acting on g) are equal to 1 i.e. Ad x
is unipotent. The lemma is proved.
We now take up the proof of Theorem 3.3. Let p: G-^G be the
universal covering group of G. Let N be the maximum connected normal
nilpotent subgroup of G and H=p~1 (H). Evidently, then G/H is compact
and p(fi)=N. It is also clear that Br\fj contains no connected normal
nilpotent subgroup of G. Now if N/N n H is compact, so is NH/H. On
the other hand this last homogeneous space is p~l{NH)/p~l{H) which is

48
III. Lattices in Solvable Lie Groups
homeomorphic to N/H n Af. Thus we may without loss of generality
assume that G is simply connected. In this case we note that N is also
simply connected as also any closed connected subgroup of N.
Let U be the identity component of H n N and let Gj be the normaliser
of U in G. Then Gt => H so that GJH is compact. Let n: Gt-> GJU be the
natural projection. Since H=>U, n(H) is a closed subgroup of Gl/U = G2,
say. Evidently G2/n{H) is compact. Let G' be the identity component
of G2 and let H' = n (ff) n G'.Then G'/H' is compact. We will now determine
the maximum connected normal nilpotent subgroup N' of G'. Let N"
be the identity component of 7t-1(.V')- We assert that N"=N. The group U
is normalised by H. Hence the Lie subalgebra u of g corresponding to U
is stabilised by Ad(H). Theorem 3.2 applied to the adjoint representation
shows that Ad(JV) stabilises u as well. Hence Gt =>N. Suppose now xeN".
Then the action of Ad n(x) on the Lie algebra g2 of G2 is unipotent. We
have a natural inclusion n/ur-»g2 compatible with the natural action
Ad x on n/u and that of Ad 7t(x) on g2. We conclude therefore that Ad x
acts unipotently on n/u. On the other hand for any yeG, the action of
Ad y on g/n is unipotent. It follows that Ad x acts unipotently on g/u.
From Lemma 3.4, we conclude that Ad x is unipotent on g. Since N"
is connected we conclude that N"<=N. On the other hand since n(N) is
nilpotent and connected, N<=N". We have thus proved that n(N)=N'.
Consider now the group n(H). The intersection n(H)nN', we claim, is
discrete. In fact n(H)nN'=n(H)nn{N)=n(n-1n{H)nN)=n{HnN).
It is therefore isomorphic to (HnN)/U and is hence discrete. Clearly
N/HnN is homeomorphic to N'/H'nN'. We see thus that we are
reduced to proving the theorem under the following additional
hypothesis:
(*) G is simply connected and H n N is discrete.
We assume, as we may, that G<=GL(n, R) and that N is contained in
the group of upper triangular unipotent matrices. Let H be the Zariski
closure of H in GL(n, C) and let H* = GnK. Assume the theorem proved
under the additional hypothesis (*) for all solvable Lie groups of dimension
less than that of G. The start of the induction when dim G=l is trivial.
Now, according to Theorem 3.2, H*=>N. If H*±G, dim/J*<dimG.
Now N is precisely the maximum connected nilpotent normal subgroup
of H*° the identity component of H*. Further since G is simply connected
so is H*°. Now H*/H is compact H*° H being an open subgroup of H*,
is closed in H* as well. It follows that H*°/H*°nH is compact. It follows
then from the induction hypothesis that {H*°nH)r\N(=Hr\N) is a
lattice in N. We may therefore assume that H* = G=H*° i.e. G is
contained in the Zariski closure H of H in GL(n, C). It follows that a
Lie subgroup of G is normalised by H if and only if it is normalised by G.

III. Lattices in Solvable Lie Groups
49
Now \H, H~\ is Zariski dense in [H, H]. Since GcH, we have
[ff,fl]<=[G,(F]<=[H,H].
Now [H, H] is defined over R and hence the group V of all unipotent
elements in [H, H] is also defined over R (cf. Preliminaries § 1.2 and 2.5).
From Lie's theorem (cf. Preliminaries § 1.2) we conclude that
[G,G]cVnGL(n,R).
Combining these facts with Theorem 2.1, we conclude that [G, G]/[H, //]
is compact. Let [G,G] = G' and g' <= g be the subalgebra of g corresponding
to G'. Since G'cJV, in view of (*) H'=Hr\ G is discrete. Evidently G'/H'
is compact. Let Jif be the Z-submodule of g' generated by exp_1(W).
According to Theorem 2.12, £C is a lattice in the vector space g'. Clearly,
SC is stable under AdH acting on g'. Now let a denote the adjoint
representation of G on g'. Then g' admits a filtration
9' = 9o ? 9i •?-•? Q'k i» 9i+1 =(0)
with the following properties:
(i) for 0gigfc+1, g', is stable under a(H);
(ii) Se'i=^r\Se is a lattice in g'f for 0gi^fc+1;
(iii) let £| be the Q-linear subspace of g| spanned by ^{; then £j+1
is a maximal proper <7(//)-invariant Q-subspace of E\.
Now since H=>G each of the g'( is a(G)-stable as well. It is moreover
easy to see that the representation at of G on g'i/gj+1 deduced from a is
completely reducible. Let C be the centre of G. Now for an element ge G,
(Adg-l)(g)c:g' so that geNC if and only a(g) is unipotent. On the
other hand o(g) is unipotent if and only if ot(g) are for 0^i<k. al being
completely reducible <7,(g) is unipotent if and only if <7j(g)=l. Thus CN
is the kernel of the map
k k
T=]>.:G-^nAut(g;/g'(+1).
1-0 i.O
Also N contains the connected component of e in C so that N is the
connected component of e in CN. Now it is easy to see that Jif n q'JSC ng'i+1
is indeed a lattice in gj/gj+i so that ot{H) maps into the group preserving
this lattice. In particular at{H) is a closed discrete subgroup of Aut(g'i/gS+1).
Hence t(ff) is closed in ]J Aut(g;/gJ+1). Now HJVcr1 t(ff). Since r{H)
is discrete, N is the connected component of e in t ~' t (if). Clearly t ~x t (if)
is a closed subgroup of G. Since jVcMcr't(fl)=JVCH, N// is a

50
III. Lattices in Solvable Lie Groups
closed subgroup of G. Hence N/NnH = HN/H is closed in G/H and is
therefore compact. Hence the theorem.
3.5. Corollary. If G is a connected solvable Lie group and N its
maximum connected normal (closed) nilpotent Lie subgroup then for any
lattice r in G, Tn N is a lattice in N.
3.6. Theorem (Mostow). Let G, and G2 be two simply connected
solvable Lie groups. Let /"J (i= 1,2) be a lattice in Gt. Then given any
isomorphism q>: /"J—> T2 there exists a dijfeomorphism (/>: Gt—> G2 such that
(i) <£!/;=(?
(ii) <p(xy)=<p(x)q>(y)for all ye/;, xeGt.
In fact if G[ is the smallest connected subgroup of Nt (the maximum
normal nilpotent connected subgroup of GJ containing [/],/]] we can
choose q> such that (i) and (ii'), (iii') below hold
(ii') q>{xy)=q>{x)q>{y) for all ye/} • G\ and xeG
(iii') iplgj is a homomorphism.
Proof. Let N2 be the maximum normal nilpotent subgroup of G2 and
G'2 the smallest connected subgroup of N2 containing [/^,/^]. Now
•pK.^cG',. [/J,/3 (and hence rtr\G^ is a lattice in G'( for i=l, 2.
Hence the group /] n G'J\J[, /]] is finite. On the other hand if H is the
normaliser of G'2 in G2, H/G'2 is torsion-free. This follows from the fact
that G2 is simply connected. For yeGJn/], q>(y) normalises [7^,/^]
( = (/>[7i, /]]) and hence G2 as well. It follows that q>(y)sH for yeT^nGi.
Since a power of y belongs to [/"J, /]] and H/G2 is torsion free <p(y)sG'2.
Now according to Theorem 2.11, (/>|Ginr. extends to an isomorphism
(Pi: G[ -> G2. Define a map <p2: /] G\ - ► T2 G2 by setting
for ye/]andgeGj.This map is well defined: in fact if yg=y'g',y'~"1y=g'g~1
so that we have
<p(Y~l)<p(y)=(p(y'~l y)=<pdg' g~l)=<Pi(g')(Pi(g~l)
leading to <p2 (g y)=$2 (g* /)• $2 is evidently an extension of q>. Moreover
it is easily seen that
<p2: ^GJ-^GIj
is a homomorphism; it is continuous since the restriction to G\, the
connected component of e in /"i G\ is continuous. Since we could have
started with q>~x instead q> we see that <p2 has an inverse so that <p2 is an
isomorphism. Now G„ i= 1, 2, being simply connected is diffeomorphic
to a Euclidean space. Thus G,//} G\ is a classifying space for the group

III. Lattices in Solvable Lie Groups
51
rt G't, i= 1,2. Thus it follows that there exists a homotopy equivalence
V: G^G'^G^G,
and a map
<P: Gi^G2
such that the diagram
G,-^^G2
GjrlG\-~^->G2ir2G'2
where nt, i= 1,2, are the natural projections, is commutative (cf.
Preliminaries §4.1). Further we have
(i) ^\rlGi = 92
(ii) <F(gx)=<P(g)(j>2(x) for geGu xe^G^.
This means that the (r2 • G'2)-principal bundle on Gjrx G\ induced by *F
is bundle-equivalent to G, -»Gjr^ G\. The same is true if we replace V
by any homotopic map (cf. Preliminaries § 4.1). In particular if we replace
T by a diffeomorphism «P0 and denote a covering bundle map by 0 we
obtain the desired result. (Note that if $(e)=t=e to start with, we may
replace <p by the map g\-*q>{e)~l q>{g) so that (i) can always be secured,
once (ii) is secured.) Thus it suffices to show that «P is homotopically
equivalent to a diffeomorphism. We will do this in two steps:
Lemma A. Gl/rl G\ is diffeomorphic to a torus.
Lemma B. Any homotopy equivalence between two tori is homotopic
to a diffeomorphism.
Proof of Lemma A. By Theorem 3.2 G\ is normal in A/,. Consider the
fibration
n: Gjr.G'^Gjr.N,.
Since G\ is normal in A/, and moreover A/,/Gi is abelian (note that
[N,, N,]cG',: Theorem 3.2) this is a principal fibration with N,/(r, ■
Gi n N), a torus, as fibre; the base G,//^ N, is also evidently a torus. To
show that Gj/7] G[ is a torus, it suffices then to show that n is a trivial
fibration i.e. admits a section. Consider the surjective map
/IGi/Gi-^Ai/ty.
Now both groups being free abelian, this map admits a splitting a:
Tx Nj/Nj -»• Tx G'JG\. Now the spaces G,//] G\ and GJN^ rt are respectively
classifying spaces for /"J G'JG\ and .TJ N,/N, so that we obtain a map
a: G./JV, >GJG\

52
III. Lattices in Solvable Lie Groups
compatible with the action of J] NJNt on the two sides (on the right it
acts via a). Now the induced map o0: Gj/A/, /"j —> GXIG\ /] is such that the
composite map u
GJN, r, ^U GJG' r, -s- Gt/Nt rt
we find induces an isomorphism in the fundamental groups. The two
spaces being tori this is a homotopy equivalence. Hence u = noa0 is
homotopic to a diffeomorphism ul (Lemma B). On the other hand since u
admits a lift a0, u, admits a lift as well; u, being a diffeomorphism this
implies that n admits a section.
Proof of Lemma B. Let u: Tn—> T" be a homotopy equivalence. By
altering u by composition with a translation (which is homotopic to the
identity) we may assume that u{e)=e. Setting Tn=R"/Zn (Z"=set of
points with integral coordinates), we can lift u to a map u: Rn—>R" such
that u(0)=0 and u(x+a)=u(x)+u(a) for all xeR" and aeZ". u\z„ is then
a homomorphism of Z" on Z". This homomorphism is an isomorphism
since this map after suitable identifications is precisely the map induced
by u (a homotopy equivalence) on the fundamental group of T". Let ii0
be the linear extension of the map fi|z„: Zn—>Z". Then t\->tu + (l — i)u0
is a homotopy between u and u0 which goes down to T" as a homotopy
between u and u0, the group automorphism induced by u0. Hence the
lemma.
Our next aim is to generalize Theorem 2.19. Unlike the proof of that
theorem the generalization is a lot more difficult to prove. Also we need
the right generalization of the properties of a lattice in a solvable group
as an abstract group. Toward this end we first note that we have the
following
3.7. Proposition. Let r be a lattice in a solvable simply connected Lie
group G. Then T admits a sequence r=ro=>ri=>---=>rp={e} of subgroups
such that ri+l is normal in rt and r(/rl+l is infinite cyclic. Moreover
dim G = p.
Proof We first note that if N is the maximum normal connected
nilpotent subgroup of G, then r/N n T is isomorphic to a lattice in G/N;
the latter being isomorphic to R' (for some I) r/N n T is free abelian. On
the other hand if JV"0 is defined inductively by setting JV<0) = N and
tf<» = |W,#»-«], N(k)nT is a lattice in N(t) so that Nik)nr/Nik+"nr
is isomorphic to a lattice in Nik)/Nii+1); the Lie group Nik)/Nik+l) is
isomorphic to a euclidean vector space and for large k, Nik) = e. It follows
that r admits a sequence r=r0"=>/]"=>••■=>rs={e} of subgroups such
that rk is normal in /""._, and rk_i/rk is free abelian. Evidently this sequence
of subgroups can be refined to obtain a sequence as in the proposition.
That dimG=p is easily seen.

III. Lattices in Solvable Lie Groups
53
Corollary. A lattice in a connected solvable Lie group is finitely
generated.
We now generalise the corollary and next obtain a generalisation of
Proposition 3.7 itself.
3.8. Proposition. Let G be a connected solvable Lie group and H a
closed subgroup. Let H0 be the connected component of the identity in H.
Then H/H0 is finitely generated.
Proof. Let G' be the universal covering of G and p: G'^G the covering
map. Let H' = p~l(H) and H"=p~l(H0). Let H'0 be the identity
component of H'. Then p{H'0)<=H so that H/H0 is a quotient of H'/H'0. Thus it
suffices to prove Proposition 3.8 under the assumption that G is simply
connected. In particular we may assume that G<=GL(n,R). Let H* be
the Zariski closure of H in GL(n,Q and let Gt = H* n GL(n, R). Then
Gt is solvable, has only finitely many connected components and
normalises H0. Consider now the map
n: G1^G1/H0.
Under this map n(H0) is a discrete subgroup of the solvable Lie group
Gl/H0. We see thus that we are reduced to proving the following: let
r<=G be a discrete subgroup of a connected solvable Lie group G; then r
is finitely generated. (Note that G1 has only finitely many connected
components.) Once again we may by passing to the simply connected
covering (if necessary) assume that r<=Gc:GL(n,R). Let r* be the
Zariski closure of T in GL(«,C) and let 7i* = r*nGL(n,R). Now /J*
is a solvable Lie group with finitely many connected components. Let
r<f be the identity component of /J*. Then since rf/rf is finite, T is
finitely generated if and only r0 = rnr^ is. We are thus reduced to
proving that r0 is finitely generated. Let V= [/^*, 7^*]. Then V is a uni-
potent real linear group. Since evidently V and [/J,, /^] have the same
Zariski closure in GL(n, C) it follows from Theorem 2.1 of Chapter II
that V/\T0, /J,] (and hence K/Kn/J) is compact Let n: r^^r0*/V be the
natural map. Since V/VnT0 is compact, one deduces easily that n(r0)
is a discrete subgroup of the abelian Lie group .7^*/^ A discrete subgroup
of an abelian Lie group is finitely generated. On the other hand, Vn r0
is a discrete subgroup of a nilpotent Lie group V and is hence finitely
generated (Theorem 2.20).
Since the sequence
{e}->Kr./5-»r0-»B(ro)-»{e}
is exact we conclude that /J, is finitely generated. Hence the
proposition.

54
III. Lattices in Solvable Lie Groups
3.9. Corollary. Let G be a connected solvable Lie group H a closed
subgroup and H0 be the identity component of H. Then any subgroup of
H/H0 is finitely generated.
Proof. Let 7^c:////jf0 be any subgroup. Let n: H^H/H0 be the
natural map and let n~1{r1) = H1. Then we obtain the desired result by
taking for H in the proposition, the group H1.
3.10. Proposition. Let G be a connected simply connected solvable Lie
group and HcGa closed subgroup. Let H0 be the identity component ofH
and let r=H/H0. Then r admits a filtration
r=r0=>r1=>-=>7i=>7;+1-(e)
where each L+l is a normal subgroup ofTt and ri+l/ri is infinite cyclic.
Proof. We argue by induction on dimension of G. Evidently, the
proposition holds in the case dim G = 1. Let U=[G, G] and let
n: G^G/U
be the natural map. Then diml/<dimG and G/U is isomorphic to a
finite dimensional vector space over R. Let n(H) = H' and n{H0)=H'0.
Then H'0 is a connected closed subgroup of G/U and can be identified
with a subspace of the vector space G/U. It follows that H'/H'0 is a
subgroup of a vector space and is hence abelian and torsion-free. Since
H'/H'0 is a quotient of H/H0, it is finitely generated. Thus H'/H'0 is a free
abelian group. Now let Hl=Hr\n~l(H'0). We then find that H/Hl is a
finitely generated free abelian group. Clearly H0 is the identity component
of Hl as well. Now Hl is a subgroup of n~l(H'0). Clearly n~l{H'0) is a
connected closed subgroup of G and is hence simply connected. Thus if
n-1 (//{,) #G, then HJH0 = F has a filtration (induction hypothesis)
r=>/r =>•••=>/; =>c+i=(e)
such that J,'+1 is normal in r{ and r//I]'+1 is infinite cyclic. Since T/T1 is
free abelian (and finitely generated), the proposition follows in this case.
Thus we may assume that n~1(H')=G. i.e. n(H0)=G/U. In this case, let
Hl=HnU, and let Hl0 be the identity component of H1. Consider the
natural map i: Hl—>H. Evidently i(Ho)<=H0, so that we obtain a map
j: Hl/Hl0-+H/H0.
We claim that; is an isomorphism. Suppose xeH and x* in H/H0 is its
image under the natural map. Since n(H0)= G/U we can find x0eH0 such
that n (x0) = n (x). Clearly x Xq ' € H' and j(xx„')=x*. Thus j is surjective.
Next let xeH be such that i(x)eH0, then xeH1 r\H0. Now H0r\Hl is
the same as H0nU. Since n(H0) = G/U, we see that H0/H0nU is iso-

III. Lattices in Solvable Lie Groups
55
morphic to a vector space. The fibre space H0^> H0/H0 n U is necessarily
a product since the latter is homeomorphic to a euclidean space. Thus
H0nU is necessarily connected. It follows that W0n[/cflJ. Thus; is
an injection and hence an isomorphism. Now H1 is a closed subgroup of
U and //£ is the identity component of//1. Hence by induction hypothesis
Hl/Ho admits a filtration of the required kind.
3.11. Proposition. Let H be a closed subgroup of a connected solvable
Lie group G. Then [//, H] is nilpotent.
Proof. Note that [G, G] is nilpotent hence so is \H, H] <= [G, G].
Hence the proposition.

Chapter IV
Polycyclic Groups and Arithmeticity
of Lattices in Solvable Lie Groups
We continue our study of lattices in solvable Lie groups in this chapter.
Our starting point is Proposition 3.7 of Chapter III which gives
considerable information on the structure of lattices in simply connected solvable
Lie groups. We prove a converse to that proposition in this chapter. This
is a fairly easy task once one has a theorem due to L. Auslander [2]. Our
proof of Auslander's theorem is essentially due to Mostow [6]. Mostow's
technique leads also to a theorem (proved by him) which asserts that a
solvable Lie group G with a lattice T admits a faithful representation in
GL(n, R) which takes r into GL(n, Z). The proof given here is again a
slight variant of Mostow's proof— the essential ideas however are due to
him.
4.1. Definition. A group T is polycyclic if it admits a sequence
r=r0=> jj =>•••=> rk={e}
of subgroups such that each /] is normal in /]_, and /]_,//] is cyclic.
4.2. Remark. From the definition it is evident that T is finitely
generated. Moreover one sees immediately that every subgroup of T is
again polycyclic and is hence finitely generated. It follows that T satisfies
the ascending chain condition for subgroups; or equivalently every
family of subgroups of T has a maximal element.
43. Remark. Evidently a polycyclic group T is solvable. Now if G
is a solvable group and G = G0=>Gl=>--=>Gr={e} is any sequence of
subgroups such that Gs is normal in Gt_i and Gi_JGi is abelian, we set
r(G)=rank(G)= £ rank(G./G(+1).
From a standard argument it follows that r(G) depends only on G
(thereby justifying the notation) and not on the particular sequence
{GJo^.ir chosen. (We set r(G)=oo if rank (G,/G,+1) is infinite for
some i.)

IV. Polycyclic Groups and Arithmeticity of Lattices
57
4.4. Lemma. Let Tbe a polycyclic group and n a positive integer. Then
the subgroup r(n) ofT generated by
{^\xer}
has finite index in T.
Proof. We argue by induction on r{r). Let r = T0zDrl zd• ■ • zdrk= {e}
be a sequence of subgroups such that Tt is normal in /]_, and /]_i//] is
cyclic for all i with O^i— l^fe— 1. Let r^k be the first integer such that
r/rr is infinite. If such an integer does not exists, r is finite and there is
nothing to prove. Clearly then /""_, has finite index in r. Let /""_, (n) be the
subgroup generated by
{x-lxeT;.,}.
Evidently r(n)=>^_j(n) so that it suffices to show that Tr_i{n) has finite
index in /""_,. Now consider the exact sequence
{e}^r;_^r;_1_^r;_1/rr-,{e}.
rr_i/rr is infinite cyclic so that it is evident that n(rr_l{n)) has finite index
in rr_i/rr. Thus we see that /""_, (h)-./"" has finite index in /""_,. On the other
hand r{rr)<r{rr_i) so that by induction hypothesis we see that
rr{n)=group generated by {xn\xeri.},
has finite index in Tr. It follows that rr_l{n)rr{ri) has finite index in /""_,
(note that rr(ri) is a characteristic subgroup of Tr and is hence normal in
rr_i). But evidently rr{n)<=rr_i{n). It follows that /;_,(«) has finite index
in i|!_i. Hence the lemma.
4.5. Definition. A group r is strongly polycyclic if it admits a sequence
r=rQ=>rl=> •••zirk={e} of subgroups such that r, is normal in J] _, and
ri_l/ri is infinite cyclic for l^igfe.
A strongly polycyclic group is polycyclic. Also a subgroup of a
strongly polycyclic group is strongly polycyclic.
4.6. Lemma. Let r be a polycyclic group. Then r admits a subgroup F
of finite index which is strongly polycyclic.
Proof. We argue by induction on rank (T). Let
r=r0zDrlzD---zDrk={e}
be a sequence of subgroups such that each Tt is normal in /]_, and
rl_l/rl is cyclic for 0< i^k. Let m <k be the first integer such that r/Tm +1
is infinite. We can then evidently replace r by Tm. In other words we may
assume that T/L] is infinite. Now rank(/^)<rank(r). Hence by induction
hypothesis we may assume that L] admits a subgroup /]" of finite index
which is strongly polycyclic. By passing to a smaller subgroup if necessary

58
IV. Polycyclic Groups and Arithmeticity of Lattices
we can assume without loss of generality that /i" is normal in /"J. Let n be
the index of /j" in /J and let /j(n) be the subgroup generated by
{X"|X€/^}.
Then /"j(n) is of finite index in /"J and is evidently strongly polycyclic and
in addition normal in /". Now T is a semidirect product J x /] of an
infinite cyclic subgroup J with /"J. The group J stabilizes /^(n) and
r{ = J. r^ri) is clearly a strongly polycyclic group which has finite index
in r. Hence the lemma.
The following result and its corollaries are easily proved by induction
on the length of the descending central series of the nilpotent groups
involved. Their proofs are left to the reader.
4.7. Lemma. Let Gbe a group and H and K two nilpotent normal
subgroups. Then HK is nilpotent.
Corollary 1. If{H(}, §i^n are nilpotent normal subgroups of a group G
then H = HxH2,...,Hnis nilpotent.
Corollary 2. A polycyclic group admits a unique maximal (nontrivial)
normal nilpotent subgroup.
4.8. Definition. For a polycyclic group T the unique maximal nilpotent
normal subgroup is called the nil-radical of r (and denoted T in the
sequel).
4.9. Remark. For a polycyclic group r, T is evidently a characteristic
subgroup of r. Suppose P is a normal subgroup of r such that r => P => T,
then T is the nilradical of P as well. In fact clearly "P => T; on the other
hand since P is normal in r and "P characteristic in P, "P is normal in r
as well; hence by maximality T=nP.
4.10. Proposition. Let r be a strongly polycyclic group and T its nil-
radical. Then T is isomorphic to a lattice in a simply connected nilpotent
Lie group N. Moreover the action ofTonT by inner conjugation can be
extended to an action a: .T—> Aut(AT) ofTas a group of automorphisms of
the Lie group N. Let n be the Lie algebra of N and GL(n) the group of
R-linear automorphisms of the vector space n. The group Aut(N) may then
be identified with a subgroup o/GL(n) and
T= {x | x € r, a (x) is unipotent}.
Proof. T is finitely generated, nilpotent and torsion-free. According
to Theorem 2.18, T is isomorphic to a lattice in a simply-connected
nilpotent Lie group N. Moreover in view of Corollary 1 to Theorem 2.11
again, the action of yer on T by inner conjugation extends to an action
a{y): N^N. Thus we obtain a homomorphism a: f—> Aut(N). An auto-

IV. Polycyclic Groups and Arithmeticity of Lattices
59
morphism u: N—>N defines an automorphism of the Lie algebra n which
enables us to identify Aut(N) as a subgroup of GL(n).
Let A be the kernel of a and "A the nil-radical of A. Then "A is a normal
nilpotent subgroup of T (cf. Remark 4.9). It follows that "AcT. Thus "A
is central in A. Let B=A/"A and n: A—>B be the natural map. Let B° be
any abelian normal subgroup of B. Then n~l (B°) is nilpotent and normal
in A so that 7t"1(B°)=BA Since B is solvable, it follows thatB={e} i.e.
A="A. It follows that A is the centre of T (and is in particular abelian).
Let H be the Zariski closure of a{r) in GL(nc) where nc = rt®RC
and GL(nc) is the group of C-linear automorphisms of the vector
space nc. Then H is a solvable algebraic group. Let H° be the identity
component of H. Then the unipotent elements of H belong to H° and
since H° is a connected solvable algebraic group the set U of unipotent
elements of H° is a nilpotent normal subgroup H° (Lie's theorem, cf.
Preliminaries § 1.2). Evidently U is a normal subgroup of H. Let
F = {x\xeT, a{x) is unipotent}.
Since <7(T")c:U, a{F) is nilpotent Define inductively r'ik) by setting
rm=r and r'ik)= [r, r (*-I)]. Then for large k we have
r<k)<=A.
Now let i: Tc->jV be the homomorphism which identifies T with a
lattice in N. Let C be the centre of N. In view of Theorem 2.1, i{A)<= C.
Moreover we have for yeT and as A,
i{yay-1 a-l)=a{y){i{a)).i{a)-1.
Let ccnbe the subalgebra of n corresponding to C and exp: n—>N the
exponential map. Let Xec be any element such that expX = a. Then we
n nvc
i(yay-la-l)=exp(o(a)(X)).exp(-X)
= exp(a(a)-l)(X),
where / is the identity endomorphism of n. Now since o(F)<=.U, we can
find an integer r such that for any 0U ...,dreF and any Xec,
(<7(01)-;)(<7(02)-/)...(<7(0r)-;)(x)=o.
(Engel's Theorem, cf. Preliminaries § 1.2.) In view of (I) and the fact that A
is abelian, it follows that F is nilpotent. Thus fcT. On the other hand
for xeT, <7(x)=Adi(x) where Ad is the adjoint representation of N
in n. N being nilpotent it follows that a(x) is unipotent. Hence T<=F.
Thus /"" = T. Hence the proposition.
4.11. Corollary. Let The a polycyclic group and T its nil-radical. Then
I '/T admits an abelian subgroup of finite index.

60
IV. Polycyclic Groups and Arithmeticity of Lattices
Proof. In view of Lemma 4.6 and Remark 4.9 we can assume that T
is strongly polycyclic. It suffices then to show that a{r)/a{T) has an
abelian subgroup of finite index. Evidently o{r)/o(T) is a subgroup of
H/U. Now H is an algebraic group so that H/H° is finite. On the other
hand H° is a solvable linear algebraic group and U its subgroup of
unipotent elements. It follows from Lie's theorem that H°/U is abelian.
Thus H°/U is an abelian subgroup of H/U of finite index. Hence the
corollary.
After these preliminaries we now state the central result about
polycyclic groups.
4.12. Theorem. Let T' bea polycyclic group and F a strongly polycyclic
normal subgroup of finite index. Let T' be the nil-radical of /"". Then T
admits a faithful representation p: r—>GL(n, Z) such that ("/"") consists of
unipotent matrices.
The rest of this chapter is devoted essentially to proving this theorem.
We need some special machinery.
For a group G let F(G; Z) (resp. F{G; Q)) denote the algebra of
Z-valued (resp. Q-valued) functions on G. Evidently we have an inclusion
i: F(G;Z)^F(G;Q)
which gives an isomorphism
F(G;Z)®ZQ^F(G;Q).
We identify F(G; Z) with its image by i in the sequel. We denote by L
and R the left and right regular representation of G in F(G; Z) or F(G; Q):
if A = Q or Z,/eF(G; A) and x, ueG, we have
L(x)f(u)=f(x-lu)
and
R(x)f(u)=f(ux).
Clearly, R (x) R(y)=R (x y) and L(x) L(y)=L{x y) for x, y e G. In the sequel
A will stand for either Q or Z or R. Let p be a representation of G on an A-
module M. We then define a map Yp: M ®AM*^>F{G,A) as follows:
(here M* = KomA{M, A)): for msM, oleM* and xeG, we set
%(m,a)(x)=(p{x)m, a>
= <m,p*(x)-1a>
(where p* is contragredient to p). It is then evident that IfJ, is 4-bilinear
and hence defines a linear map
<P„: M®AM*^F(G,A).
We have moreover setting Rx=R(x) and Lx=L(x),

IV. Polycyclic Groups and Arithmeticity of Lattices
61
4.13. Lemma. For meM, aeM* and xeG,
%{p{x)m,ai) = Rx%{m,0i)
and
%(m,p*(x)ot)=Lx%(m,ot).
We omit the proof which is straight-forward.
4.14. Remark. From Lemma 4.13 it is evident that the image of $p is
stable under Lg and Rg for all geG. G x G acts in a natural manner on
M®AM* and F{G;A) and it is clear that «PP is compatible with the
action of G x G on the two ^-modules. We denote the image of <Pp by [p]
in the sequel. Assume now that M is finitely generated and free over A.
Let ex ek (resp. a, <xk) be a basis of M (resp. M*) over A. Let
/: M-> LJ [p] (resp. i#: M*-> 11 [p]) be the following map: for
k copies k copies
meM set i(m) = {$p(m<g)aj)|l:£i:£Jc} (resp. for aeM* set i#(a)=
{$p(ej®a)} l^i^k). Then i (resp. /„.) is a G-module injection if we make
[p] a G-module via the right translation action R (resp. the left translation
action L). Also we note that if px and p2 are two representations of G
over A which are equivalent then [p,] = [p2]. Thus [p] depends only on
the equivalence class of p.
4.15. Remark. We note that if M is finitely generated over A so is M*
and hence so is M ®A M*. Thus if M is finitely generated over A so
is [p].
4.16. Remark. An 4-submodule E of FG; A) of finite rank over A
is finitely generated and free. When A = Q this is obvious. When A = Z
we argue by induction on the rank of E: let f0eE be a non-zero element
and xeGan element in G such that/o(x)=|=0. Consider the linear map
/: E->Z defined by setting
for all /eE. The kernel of / is of rank less than rankE. It follows by
induction hypothesis that the kernel of / is finitely generated. Since the
image of / is finitely generated the assertion follows. At the start of the
induction rank of E is zero; but then since F{G; Z) is torsion free, E=0.
We now prove
4.17. Lemma. The following conditions on an element feF{G;A) are
equivalent
(i) There exists a representation p oj G in a finitely generated A-
module M such that /e [p].
(ii) The A-span of {Lxf\xeG} (resp. {Rxf\xeG}) is of finite rank.

62
IV. Polycyclic Groups and Arithmeticity of Lattices
Proof. That (i) implies (ii) is clear from Remarks 4.14 and 4.15. Let M
be the ,4-span of {LxfxsG}. Assume that M is an ^-module of finite
rank. Then M is finitely generated (Remark 4.16). Evidently M is
Instable for all xeG. Let t be the representation of G on M thus obtained.
Let e be the identity element of G and e*\ M—>A the ^-linear map
e*{f)=f{e). Then clearly we have
f(x)=Lxf(e)=(x(x)f,e*y
i.e. fs\x"\. Hence the lemma.
4.18. Definition. A representation p of a subgroup H of a group G in
an ^-module M is said to be extendable to G if there exists
representation t of G in an ^-module N and an injection i: M—>7V such that
i p(x) »i=t(x) i(»i) for all xefJ and msM. x is called an extension of p.
4.19. Remark. If M is finitely generated as an ^-module, it follows
from Remark 4.14 that p is extendable to G if and only if [p] regarded
as a G-module through the right regular representation is extendable.
Suppose now that H is any group and B is a group of automorphisms
of H. Then B acts on F{H;A) as follows: for beB and fsF(H;A) we set
K{b){f){x)=f(b-1 x b\ With this notation we have
4.20. Lemma. Let G be a semidirect product of a subgroup B and a
normal subgroup H. Let p be a representation of H in a finitely generated
A-module M. Then p admits an extension to G in a finitely generated
A-module N if and only if the A-span of
{K{b)f\bsB,f€lp-]}
is finitely generated. Here K(b)=K(Intb).
Proof It is easy to see that if p admits an extension t of the desired
kind, then K(b)f'belongs to the image of [t] under the restriction map
F(G;A)^F(H;A)
for all b eB and /"e [p]. Hence the implication in one direction.
Conversely assume that E, the 4-span of {K(b)f\beB,fe[p']} is
finitely generated over A. We define an injection
i: F(H;A)^F(G;A)
as follows: any element xeG can be written uniquely in the form bh
where beB and heH; we set
if(x)=if(b-h)=f(h).
It is evident that i is a homomorphism of //-modules for the H-module
structures obtained from the right regular representation. Now any

IV. Polycyclic Groups and Arithmeticity of Lattices
63
element x0eG can be expressed as a product h0b0 with h0eH and
b0eB and we have for/
(RX0(if))(x)=if(xx0)
= if(bhh0b0)
= if(bb0bolhh0b0)
=f(bElhh0b0)
= (K(b0)(R„J))(h)
= i(K(b0)(Rhof))(x).
Thus RXo(i(f))ei(E). It follows that the A-span E of
{Kv(/)|xoeG./€LP]}
is a finitely generated ^-module. This is evidently stable under Rx for
all xeG. We denote the representation of G in E by t. Evidently t is an
extension of the right regular representation of H in [p]. The lemma now
follows from Remark 4.16.
Finally the following general lemma (which seems to be of some
interest in itself) is needed.
4.21. Lemma. Let G be a finitely generated group. For a
representation pofG on a finitely generated free A-module let %p denote the character
of p and Mp the representation space for p (%p: G—>.4 is the function
gi-»tr pig))- Let p0 be a fixed representation of G on a finitely generated
free A-module Mpo and let S be any set of representations of G on finitely
generated free A-modules such that for all reS, xPQ=xz. Then the A-linear
span of
{f\fel?l*eS}
in F(G;A) is finitely generated over A.
Proof. We prove the result in several steps.
In view of Remark 4.16, the result for the case A — Z follows from the
case A = Q.
4.22. Assume the result proved for A=R. Since the map
i: F(G;Q)®QR^F(G;R)
given by i(f®X)=l. f is evidently an isomorphism, a set of elements
/j, ...,/.€F(G; Q) are linearly independent over Q if and only if
{/'(Jk)\ 1 rSfc^r} are linearly independent over R. It follows therefore that
if suffices to consider the case when A=R. In the sequel thus we will
consider only the case A = R.

64 IV. Polycyclic Groups and Arithmeticity of Lattices
4.23. If 2. is any set of finite dimensional representations of G, then the
following two conditions on 2. are equivalent.
(a) There exists a (finite dimensional) representation a of G such that
for any re2., there exists an injection
iza: Mz-^Ma
of R [G]-modules.
(b) There exists an integer n such that dim Mz^n for all re2. and the
R-span F0 of {f\ fs\x], xeE} in F(G, R) is finite dimensional.
To see that (a) implies (b) one need only notice that for all re2.,
[t]c[<t]. Conversely we see that if F0 is considered as a G-module
through the right regular representation, then for xeS, Mt admits a
R [G]-module injection in F0 © • ■ • © F„ (cf. Remark 4.9).
n copies
4.24. F or each x € S, let Mt=Mt° 3 M' 3 • • • 3 M*' = {0} be a composition
series for Mt as an R [G]-module. We fix such a series for each x once
for all. Let E0{x) denote the R[G]-module U M\-l/M\. Then we have
ZEo (t) Xt Xpo ^Eo (Co) '
Since E0{x) and E0(p0) are completely reducible, they are equivalent
(see van der Waerden [1], Exercises 1 and 2, § 125, p. 175). In particular
kz=kpo=k is independent of xeS. Let au l^i^k be the irreducible
representations associated to the composition series forp0;
Let
S,= {t|teS,JW?/MtI^M(r/}.
Evidently it suffices to show that for each i, l^i^fc,
{/1/eW.teS,}
spans a finite dimensional subspace of F(G; R). In the sequel we may
assume S=St for some i, l^i^fc. By renumbering the {<7.}iii<t we may
thus assume that for all xeS,
We make this simplifying assumption in the sequel. We may still continue
to assume that p0eS by replacing p0 by a suitable element if necessary.
4.25. For each xeS, we now have an exact sequence of R [G]-modules.

IV. Polycyclic Groups and Arithmeticity of Lattices
65
We denote the representation of G on Ml by q>(z) so that the exact
sequence may be put in the form
4.26. We now start on the proof of the lemma. We argue by induction
on the length of p0. If the length of p0 = l, p0 is irreducible so that
[t] = [p0] for all zeS; thus in this case the lemma is true. Now let length
(p0)=k. Assume then that for all p of length <fc, {fs[x]\zel,} has a
finite dimensional span where 2. is any set of representations having the
same character as p. In particular, we can take
Z = {<p{T)\xeS}
(in the notation of 4.25) and p = q>(p0) (note that we have assumed
p0eS). Now in view of 4.23 and the induction hypothesis we can find a
finite dimensional R [G]-module a such that for every 6si, there exists
an injection
ae: Me^Ma.
From this injection we obtain a commutative diagram
0 >Me=„(t)—*—» Mz >Mai >0
<*0 fit .1
0 > Ma >M,(t) >Mai-. >0
(M^(t) can in fact be defined as the quotient of Mz@Ma by the image
of the map ae@i: M^Mz@Ma) with the rows exact. Clearly, then
Wc["I*(■"")]• We are thus reduced to proving the following:
4.27. Let M„, Mz be two finite dimensional R[G]-modules a, z being
the corresponding representations. Consider any set S of extensions of t
by a i.e. representations p such that we have an exact sequence of
G-modules:
0-> Ma-*-+ M„-^y Mt-> 0.
The the R-span of
{/l/e[p],peS}
in F(G, R) is finite dimensional. For each peS fix an exact sequence
as above once for all. Choose also once for all an R-splitting r: Mt—> Mp
of the above sequence. We then obtain R-isomorphisms
Ma@Mz-^Mp

66 IV. Polycyclic Groups and Arithmeticity of Lattices
and dualising
M*®M*^-*M*.
We identify Mp with MZ@M„ through fp as above. Now for geG and
veMz, wEMa, we have
p(g)w = o(g)w
P(g)v=?(g)v + cp(g)v
where cp(g): Mz—>Ma is a R-linear map. One checks easily that for
veMz, gt, g2eG we have
Cp{gigi)v=cp(gl)z(g2)v + a{gl)cp{g2)v.
Now since
Mp®M*^Ma®M*®Ma®M*®Mz®M*®Mz®M*
and «P„Im,.®ms = #<. while «Pp|Mt®Mt = ^t, « suffices to show that the
R-span of
{$p{Ma®M?®MT®M;)\peS}
is finite dimensional. One sees moreover easily that $p(Ma®Mz*)=0.
Now suppose veMz and aeM*. then we have
#pfoa)(g)=<p(g)*-,a>
= <j(g)v + cp(g)v,ai>
= {cp{g)v,a.y.
Now let Zza be the vector space of all Hom„(Mt, MJ-valued functions c
on G such that
(I) c(g1g2)=c(gi)oT(g2) + a{g1)oc{g2).
Then it is easily seen that an element c of Zaz is determined entirely by
its value on a finite set I of generators for G. Hence Zaz may be identified
with a vector subspace of the vector space of all maps of I in HomR(Mt, Ma).
We see that Zaz is thus finite dimensional and for peS, cpsZaz. Now we
have a multi-linear map
B: Z(TtxMtxM*->F(G;R)
if we set
B(c,i>,a)(g)=<c(g)u,a>.
Clearly, since cpeZaz for pES, the .R-span of {$p(v,a)\vsMz,aeM*} is
contained in the R-span of the image of B. Since Zaz Mz and M* are
finite dimensional the image of B spans a finite dimensional subspace of
F{G, R). This proves the lemma.

IV. Polycyclic Groups and Arithmeticity of Lattices
67
We now proceed to prove Theorem 4.12.
Proof of Theorem 4.12. Let T be a polycyclic group and let F be a
strongly polycyclic normal subgroup. We have seen that the group
F/"F admits an abelian normal subgroup of finite index (Corollary 4.11).
Replacing F by a subgroup of finite index if necessary we may assume
that r'/T' is free abelian. We will construct an integral representation
of F in this situation. We argue by induction on rank of F/T'. First, T'
admits a faithful unipotent integral representation (Theorem 2.18). Hence
we can start the induction when F/"F has rank 0. Let F' be a normal
subgroup of r such that F'=>nF and F/F'^Z. Then clearly F is a
semidirect product of the form J x /"*" where Js.Z is a cyclic subgroup
of F. Now let p be a faithful integral representation of /"*" on a free abelian
group V which is unipotent on T"(T' is the nilradical of /"*" so that by
the induction hypothesis such a representation exists). For 6eJ let dp
be the representation
y^piOyd'1)
of /"'. Then for OeJ and fe[p]
KWfe[8J.
On the other hand, we claim that xP=Xep ■ To see this let W= V®z R. Let
Wk = {v\vEW,(p(y)-If v=0 for yeT'}.
By Engel's theorem we see that if fe0 is the largest integer such that
Wio-i =t= W, then we have a strictly increasing sequence
0=W0<=W1<=W2,...,Wko_1<=Wko = W
of subspaces of W. Evidently for 0^i^fco, Wt is stable under p{T").
Moreover on WJWi_ t, T" acts trivially. The same remarks apply to 0p(F').
Since we have dyd~ly~lsT' for ye/""' and 6eJ it follows moreover
that on WJW^ t the representations obtained from p and 6P are the same.
Thus xP=Xep- We conclude now from Lemma 4.21 that
{K(0)f\OeJ,fe[p]}
spans a finitely generated Z-module in F(F, Z). Applying now Lemma 4.20
we see that the integral representation p extends to a representation p
of F over Z. Let M be the representation space for p~ and M that for p.
We identify M as a /""'-submodule of M. Let
M' = {v\veM,(p(x)-lfv=0 for large k and xeT'}.
Then M' is stable under F and contains M and is hence nonzero. Let p*
denote the representation of F on M'. We claim that p* is faithful.

68
IV. Polycyclic Groups and Arithmeticity of Lattices
Since M<=M' as a /""'-submodule, the kernel of p* intersects T" in {e}.
Thus this kernel is isomorphic to a subgroup of P/F'^Z. Hence this
kernel is abelian and being normal must be contained in T'. But T'<= /""'.
We conclude thus that kernel p* = {e}. To obtain an integral
representation for r we take the induced representation obtained from p*. Since
r/P is finite, this induced representation is on a finitely generated free
abelian group. This proves the theorem.
We will now deduce the following corollary from Theorem 4.12.
4.28. Theorem. A lattice in a simply connected solvable Lie group is
strongly polycyclic. Conversely every polycyclic group r admits a normal
subgroup r of finite index in r which is isomorphic to a lattice in a solvable
simply connected Lie group.
Proof. The first part is a restatement of Proposition 3.7. For the second
assertion we may clearly assume that r is strongly polycyclic and
that r/T is free abelian. Let p be a faithful integral representation,
p: r->GL(n, Z), which is unipotent on T. Let G be the Zariski closure
of p(f) in GL(n, C); let Gj = G n GL(n, R) and G be the connected
component of the identity in Gj. We can assume that G=>p(T) by replacing T
by a subgroup of finite index containing T. Now if U is the unipotent
radical of G then p(T)<=U. Let N be the smallest connected closed
subgroup of U containing p("/'). Since p(F) normalizes both U and p(T)
it normalizes N as well. Moreover one sees easily that G/N is abelian
(note that [T,/"]<= "D. Now N/Nnp{r) is compact (Theorem 2.1) so
that under the natural map n: G—> G/N, p{f) projects into a discrete
subgroup of G/N. G/N is an abelian Lie group. It follows then that
n(p(r)) has a subgroup H which is free abelian and has finite index in
np(r). Now one sees easily that H is contained in a closed connected
subgroup L of G/N as a lattice. Moreover L can be assumed to be
isomorphic to R* where k is the rank of H. This follows for instance
from the following argument. Let yu ...,yk be a set of free generators
for H and let Xu ..., Xk be chosen in the Lie algebra of G/N such that
expX,=yi. Then the Lie subgroup corresponding to the subalgebra
(= subspace) spanned by Xt,..., Xk is of the required kind. Then n~l{L)
is a simply connected solvable group containing p(r)nn~l(H) as a
lattice. Evidently p{r)c\n~l{H) is isomorphic to a subgroup of finite
index in r. Hence the theorem.
An argument closely resembling the above can be given to conclude
the following. Given a polycyclic group r we can find a solvable Lie
group G with finitely many connected components all of which are simply-
connected such that r is isomorphic to a lattice in G.
4.29. Counter examples. A strongly polycyclic group cannot in general
be realised as a subgroup of a connected solvable group. In fact suppose

IV. Polycyclic Groups and Arithmeticity of Lattices
69
that r is polycyclic and is contained in a connected solvable group G;
let N be the maximum connected nilpotent normal subgroup of G;
then rnJVa nilpotent normal subgroup of T; hence rnNcT so that
since G/N is abelian r/T is abelian. Thus a necessary condition is that
r/T be abelian. We will give an example of a polycyclic group which
does not satisfy this condition. Let a: J1—>AutJ2 be the action of the
infinite cyclic group Jj on the infinite cyclic group J2 defined by setting
<7(x)(a)=a_1 for a generator x ofJt and any element aeJ2. Let F be the
semidirect product Jt ■ J2 obtained from the action a, J2 being normal
in F. The group F'=J\ ■ J2 is normal in F and F/F' = F" isomorphic to
the semidirect product of Z/(2) and Z/(3) for the unique nontrivial action
of Z/(2) on Z/(3). Let t" denote the regular representation of F" on its
group-ring Z(F") and t the representation of F obtained from t". Let T
be the semidirect product of F and Z(F") obtained from this action.
It is then easily seen that T is strongly polycyclic and T=F'- Z(F") and
r/T^F". Since F" is not abelian, T cannot be isomorphic to a subgroup
of a connected solvable group.
4.30. We will next show that if r/T is abelian r can indeed be realised
as a discrete subgroup of a connected solvable group. To see this let
p: r—> GL(n, Z)<=GL(n, C) be a faithful representation unipotent on T.
For each integer i let
Vi={v€C\(p{6)-l)iv=0 for a\\ 6eT}.
Since p(T) consists of unipotents, for some m £n, Vm = C while V„_l + V.
We thus have filtration of C:
c»=Vm^K„-i-='--='i/D=o.
Each of the Vh 0<ig.m is p(/>stable since T is normal in T. Now we
obtain for each i a representation pt on £,= Vi/Vi_l. Evidently p,(T) is
trivial. Since r/T is abelian p,(r) is abelian; we can decompose £,- into
generalised eigen-spaces for r/T. It follows that each E{ admits a
composition series whose composition factors (as a T-module) are all
1-dimensional. We conclude then that after a suitable conjugation we
may assume that p(F) is contained in the connected solvable group S
of all upper triangular matrices in GL(n, C).
4.31. Once again the condition that r/T is abelian is not enough to
ensure that T be realised as a discrete subgroup of a simply connected
solvable group. In fact if r<=G, G solvable and simply connected and N
is the maximum connected nilpotent normal subgroup, G/N is isomorphic
to a euclidean space as group so that r/rn N is in addition to being
abelian also torsion free. Moreover [T,r~\<=rnN<=T. Thus we have a

70
IV. Polycyclic Groups and Aritbmeticity of Lattices
necessary condition in order that T be a discrete subgroup in a connected
simply-connected solvable group:
(C): There is a subgroup F of T such that [T,F]<=F<=T such that
r/F is free abelian. It turns out that (Q is also sufficient. To see this let S
be the group of upper triangular matrices in GL(n, Q and assume
that r<=S as a discrete subgroup (This assumption is evidently admis-
i>
sible.) Let F'=T/F and let F'-^Z". Let q>: r-»RB be the composite
homomorphism
*
Let St=Sx R", the direct product and let «P: -T-> St be the map
#(y)=(%),</>(y))> yer.
Evidently $ imbeds T as a discrete subgroup of St. Now let N be the
maximum connected nilpotent normal subgroup of S and let Nt = Nx (e).
Then for yeT, 0(y)€Nt if and only if yeF. The group S1/N1 = A is a
connected abelian group; if n: S1—>A is the natural map, one sees easily
that 7t(«P(r)) is discrete in A and isomorphic to r/F. It follows that
n(${F)) is torsion free. Since A is abelian and connected, we can find a
closed connected simply-connected subgroup A'cA such that n $ (r) <= A'.
It follows that ^{r)<=S' = n~1 {A'). Since Nt is simply-connected it follows
that S' is simply-connected.
4.32. We will next give an example of a polycyclic group T with r/T
abelian but not satisfying (C). Let Ju J2 be infinite cyclic groups. Let
6sJl be a generator. We make Jt act on J2 by making 6 act on Jt as the
automorphism xi->x_1 and form the semidirect product F=Jl-J2. Let
F be the normal subgroup Jf • J\ of F. Then F/F' is finite and isomorphic
to Z/(2)xZ/(2). Let a be the regular representation of F" = F/F' on its
group algebra Z(F"). Let T be the semidirect product of F and Z(F")
where we make F act on Z{F") through the regular representation of F";
r=F-Z(F"). Let H = {o(x)v-v\xeF", veZ(F")}. Then one sees easily
that the nilradical T of r is precisely J\ ■ J\ ■ Z(F") and that [/", T] = J\ H.
Now if d0eJ2 is a generator, d0$T while 0o<=[^. r]; thus if F is any
subgroup of r such that T=> F => [r, T], the image of 60 in r/F is a torsion
element.
4.33. Finally we note that condition (C) is not adequate to guarantee
that a strongly polycyclic group F be realised as a lattice in a connected
solvable group. This is shown by the following example.
A matrix AeSL(2, R) belongs to a connected abelian subgroup of
SL(2, R) if and only if it is in the image of the exponential map. Now if
XeM(2, R) has trace 0 then its semisimple part Xs is conjugate either to

IV. Polycyclic Groups and Arithmeticity of Lattices
71
a diagonal matrix or to a skew symmetric matrix. Now the trace of exp X
is the same as that of expXs. If Xs is diagonal
then, trace expX = eXl + e~"""'^l; if X is skew symmetric, g=expX is
orthogonal and the eigenvalues of g are eie, e~ie for a suitable 0eR; thus
in this case trg=2Cos0_^ -2. We see therefore that if AeSL(2,R)
belongs to a connected abelian group, then trace(^)"g; —2. Let
Then B acts as an automorphism of Z2<=R2. Consider the semidirect
product r of the cyclic group J and Z2 where we make J act on Z2 as
follows: fix a generator 6eJ and we let 6 act on Z2 as the automorphism B.
Evidently Z2 <=T so that T/T is abelian. Moreover T/Z2 is infinite cyclic.
Thus r is strongly polycyclic and satisfies condition (C) as well. We
claim that T cannot be isomorphic to a lattice in a connected solvable
group. Suppose r is isomorphic to a lattice in a connected solvable
group G. Let N be the maximum connected normal nilpotent subgroup
of G. Let K be the maximal compact subgroup of N. K is connected and
central in N and normal in G (cf. Preliminaries § 1.8). It follows that N/K
is the maximum connected normal subgroup in G/K. Since T has no
torsion and K is compact r n K=(e) and the image of T under the natural
map p: G—>G/K is a discrete subgroup. We may therefore assume that
K=(e) so that N is simply connected. Now in view of Theorem 3.3,
rn N is a lattice in N. Now [T, r]crnJVcT. Moreover it is easily seen
that T=Z2 and [r, /"] is a subgroup of finite index in Z2. Thus rn N
is a free abelian group of rank 2. We conclude now from Corollary 2 to
Theorem 2.9, that N^R2. The group G normalises N while N is abelian.
We thus obtain an action a of G on N which factors through the connected
abelian group G/N. On the other hand this would imply that B belong to
the connected abelian group a{G), a contradiction.
We have seen that a lattice in a simply-connected solvable Lie group G
is polycyclic and hence admits a faithful integral representation. We will
now show the existence of such an integral representation which extends
to all of G. We will prove the following theorem due to Mostow.
4.34. Theorem. Let G be a connected simply-connected solvable Lie
group and r<=-G a lattice. Then G admits a faithful representation p:
G->GL(rc, R) such that p(r)cGL(n, Z).
The proof presented below is essentially due to Mostow. We need a
lew preliminary lemmas.

72
IV. Polycyclic Groups and Arithmeticity of Lattices
435. Notation. In the sequel we will use the following notation which
will enable us to deal with polycyclic groups and solvable groups
simultaneously. A (P/S)-group is a solvable Lie group which is either connected,
simply-connected and solvable, or, discrete and polycyclic. Let H be a
(P/S)-group. If H is polycyclic, as hitherto nH will denote its nilradical;
if H is connected, "H will denote its maximum connected normal
subgroup. If H is polycyclic, rank(ff) denoted r(H) has the same meaning
as hitherto. If H is connected and solvable, by rank (if), in the sequel we
mean the dimension of H as a real Lie group. If GcGL(n,C) is an
algebraic subgroup dim G will denote its dimension as an algebraic group
(it is the same as the dimension of G as a complex Lie group).
4.36. Lemma. Let H be a (P/S)-group and p: H—>GL(n, R) a
representation. Let H denote the Zariski closure of p(H) in GL(n, Q and U be
the unipotent radical ofH. Then dim U^ rank H.
Proof. We argue by induction on rank(ff). The start of the induction
when rank (if)=0 is trivial. Assume that the lemma is proved for all
(P/S)-groups of rank less than r=rank (if). Suppose now that H' is a
closed subgroup of finite index in H then one sees easily that the Zariski
closure H' of p{H') has finite index in H and that U is also the unipotent
radical of H\ Thus for proving the lemma for H it suffices to prove it for
any subgroup of finite index. Now from the definition of a (P/S)-group
one sees that if rank (H) > 0, we can find a subgroup H' <= H of finite index
and a normal subgroup H" of H' such that H'/H" is infinite cyclic or
isomorphic to R. It follows that by replacing H by the subgroup H' we
can assume the following H is a semidirect product J ■ Ht where J is an
infinite cyclic group or J^R and Ht is a normal subgroup with
rank(/fj)=r—1. Let J (resp. Ht) be the Zariski closure of p(J) (resp.
p(Hi)) in GL(n, Q. Then J normalises Ht and H=J • Ht. It follows that
H is a quotient of the semidirect product of J and Ht. Let V (resp. Ut) be
the unipotent radical of J (resp. Ht). Then Ut is normalised by V and U
is a quotient of the semidirect product of V and Ut. By the induction
hypothesis we have dimUj^r— 1. It follows that it suffices to show that
dimV^l. J being abelian it decomposes into a direct product SxV
where S is the group of all semisimple elements in J. Let p: 3—>\ be the
natural projection of J on V. Then p°p{J) is Zariski dense in V. Now if
xeJ one sees immediately that p°p(x) is the unipotent part of p{x).
Now, let o be the Lie algebra of V. We choose an element xeJ as follows:
if J is infinite cyclic x is a generator; if J^R, x is any non-zero element
of R. Let Ye» be chosen such that exp Y=p<> p(x). Consider the group
V' = {expz- Y\zeC}. V is an algebraic subgroup of GL(n, Q of
dimension 1 containing p»p(J). Thus V=V and hence dimV=l. This proves
the lemma.

IV. Polycyclic Groups and Arithmeticity of Lattices
73
4.37. Definition. Let H be a (P/S)-group G an algebraic group and
p: if—>G a representation. Let H be the Zariski closure of p{H) in G
and U the unipotent radical of H. The representation p is full if dimU =
rank(ff).
4.38. Lemma. A (P/S)-group H admits a full representation p: H—>
GL(N, Q such that p(H)c:GL(N, R) and for xeH, p{x) is semisimple if
and only if it is of finite order. Moreover ifH is polycyclic p can be chosen
such that p{H)cGL{N, Z).
Proof. Let a: if—>GL(.n, R) be a faithful representation such that
o{"H) consists of unipotents. Let A<=H/"H=A' be an abelian normal
subgroup of finite index and n: H^A' the natural map. Let t: H—>
GL(n, R) be a representation with kernel the group "H, and such that
x(n~1(A)) consists of unipotents. Moreover when H is polycyclic we
assume in addition that o(H)<=GL(m,Z) and r(/f)c:GL(n,Z). Let
N=m + n and let p = o@x: fl->GL(N,R). If H is polycyclic p{H)cz
GL(N, Z). (Note that a, t can be chosen as described above: when H is
polycyclic this follows from Theorem4.12; when H is connected this is a
consequence of Ado's theorem.) Let p be the Cartesian projection
GL(m, C) x GL(n, Q-> GL(n, C). Since H <= GL(m, C) x GL(n, C), we
obtain an exact sequence
c^Hn GLM x (e)-> H-^> p (H)-> e.
We will now show that the unipotent radical B (resp. B') of H n GL(»i) x (e)
(resp. p(H)) has dimension greater than or equal to rank {"H) (resp.
rank (A)). To see this we observe that B=BnGL(m,R)x(c) contains
oCH); now o{"H) is either a real nilpotent Lie group of dimension rank
(nH) (as a real Lie group) or a discrete subgroup of the unipotent group B
of rank=rank(nH). It follows from this that dim B"g rank(nH). Similarly
from the fact that B'=B'nGL(n, R) contains the group t(ti~1(A))(^A)
as a closed subgroup, we conclude that dim B' = rank (A). Since the
sequence
e->B->U->B->e
is exact, dim U=dim B + dim B'"S rank ("//)+rank (A)=rank {H). On the
other hand according to Lemma 4.36, dim U^ rank (H). Thus rank(i/)=
dim U. We have only to prove the assertion that p{x) is semisimple if
and only if x is of finite order. This is seen as follows. If p(x) is semisimple
so is p(x") for every integer q. On the other hand xqen~l(A) for a suitable
i nteger q (=t= 0). If x" =t= e, then one sees easily from our definition of a and t
that either a(x) or t(x) is a nontrivial unipotent so that p(x) is not semi-
simple. We conclude that x"=e if x is semisimple i.e. that x is of finite
order. This completes the proof of the lemma.

74
IV. Polycyclic Groups and Arithmeticity of Lattices
4.39. Definition. Let fee C be a subfield and H a (P/S)-group. A fc-
algebraic hull of H is a linear fe-algebraic group H* together with a
continuous injective homomorphism i: H—>H* such that the following
conditions hold
(i) i is full.
(ii) i(H)<=H$ and is Zariski dense in H*.
(iii) if U* is the unipotent radical of H*, the centraliser of U* in H*
is the centre of U*.
If k is a numberfield with O as the ring of integers we demand in
addition to (i)—(iii) also
(iv) for any representation p: H*—>GL(n, C) defined over k,
i'1 p-l(GL(n,C))
has finite index in H.
(Our interest is limited to the case fc = R and fe=Q.)
4.40. Proposition. A torsion-free (P/S)-group H admits an R-algebraic
hull. IfH is polycyclic, H admits aQ-algebraic hull.
Proof. Let p: tf->GL(n,C) be a full representation of H satisfying
the conditions of Lemma 4.38. Let H be the Zariski closure of p (H) and U
the unipotent radical of H. Let Z(U) be the centraliser of U in H. Let
S={xeZ(U)|x semisimple}. Now the unipotent radical Z(U)nU of
Z(U) is central in Z(U). It follows that S = {xeZ(U)|x semisimple} is a
subgroup of Z(U) (cf. Preliminaries § 2.5). Since Z(U) is normal in H.
S is normal in H as well. Note that all the algebraic groups above are
defined over R and that when H is polycyclic by our choice of p, p(H)cz
GL(n, Z) so that all these algebraic groups are seen to be defined over Q.
Let H*=H/S and i: tf->H* be the composite map
tf_i_H-*->H*=H/S.
Once again H* is a linear algebraic group defined over R (Q, if H is
polycyclic). It is clear that i (H) <= Hjf where fc=R when H is connected
and fe=Q when H is polycyclic. That i(H) is Zariski dense in H* is
evident. Since kernel of n=S and SnU=(e), U^U*, the unipotent
radical of H*. Since p is full so is i. Since any semisimple xeH, x$S acts
nontrivially on U, the third condition of Definition 4.29 is fulfilled. When
H is polycyclic, p(H)c:GL(n, Z); one deduces from this that condition
(iv) of Definition 4.39 holds for (H*, i: tf->H*) chosen as above. Since H
is torsion-free, p(H)nS = e. This shows that i is injective. This completes
the proof of the proposition.
4.41. Lemma. Let H be a torsion-free (P/S)-group. Let (H*, /': H^H*)
(resp. (H'*, i'\ H—>H'*)) be a k-algebraic (resp. k'-algehraic) hull for H

IV. Polycyclic Groups and Arithmeticity of Lattices
75
where k and k' are subfields of a field KcC Then there exists an isomorphism
q>: H*—>H'* such that i'=<p o j and q> is defined over K.
Proof. Let A be the subgroup (i(x), i'(x))eH* x H' *, xeH of H* x H' *
and A the Zariski closure of A. Let p (resp. p') be the Cartesian projection
of H*xH'* on H* (resp. H'*). Let a=p\A and a!=p'\A. Let U be the
unipotent radical of A. a. is surjective so that a(U)=U* (=unipotent
radical of H*). According to Lemma 4.35 dim U^ dim U*. It follows that
a maps U isomorphically onto U*. Since U={xeA\x unipotent} one
concludes that the kernel of a consists entirely of semisimple elements
(note that every element x of A is uniquely a product of its semisimple
part xseA and its unipotent part x„eU). Now for xekernel a and yeU,
a(xyx~ 1)=a(x)ct{y)a(x)~1 =a(y) and a maps U isomorphically onto U*.
It follows that xyx-1=y for xekernel and yeU. Now once again
arguing as we did for a, we see that a' maps U isomorphically onto U' *
so that we have for xekernel and yeU,
a'(x)a'(y)a'(x-1)=a'(xyx-1)=a'(y).
Thus a'(x) centralises U'* and hence belongs to U'*. Since x is semi-
simple so is a'(x). Thus a'(x) = e. We find thus that if a(x)=e a'(x)=e as
well. Since <x=p\A, a.'=p'\A, we conclude that x=(e,e) i.e. a is injective.
Hence a is an isomorphism. Similarly a! is an isomorphism. We can set
(/)=a'oa-1. That cp is defined over K follows from Condition (ii) in the
definition of algebraic hulls. This proves the lemma.
4.42. Proof of Theorem4.34. Let G be a simply-connected solvable
Lie group and T a lattice in G. Then T is polycyclic and both T and G are
torsion-free. Let (G*, i: G—> G*) be an R-algebraic hull for G and
(r*,j: r^>T*) a Q-algebraic hull for T. Let T be the Zariski closure of
i{r) in G*. We then claim that (J\ i\r: r^T) is also an R-algebraic hull
for T. Since i(r)<=i(G)<=G$, T is defined over R and i|r(r)c:rR. Evidently
i\r is injective. Let U* be the unipotent radical of G*. Then according to
Theorem 3.2 U*c:r. It follows that U* is the unipotent radical of T as
well. Since dim U* = rank(G)=rank(O, i\r is full. The centraliser of U*
in r is contained in the centraliser of U* in G* and is hence the centre
of U*. We see thus that (T, i\r: r^T) is an R-algebraic hull for /".
Now let q>: r*^T be an R-isomorphism such that <p°j=i\r- The
group r* is defined over Q and hence admits a semidirect product
decomposition r*=M*-V* where V* is the unipotent radical of T*
and M* is a reductive abelian Q-subgroup of T* (V* is defined over Q
as well). Now q> (M*)=M is a reductive R-subgroup of G*. It follows that
we can find a torus T* defined over R such that T*=>M and G* is the
semidirect product T* • U*. The map q>: r*^>r induces an isomorphism

76
IV. Polycyclic Groups and Arithmeticity of Lattices
of the unipotent radicals of these groups:
->=(/>!v.- v*-»i/*.
We then have for xeM*, ve V*
i/,{a(x)){v)=il,{xvx-1)=(p{x)ik{v)(p{x-1)=<j{(p{x))(ik{v))
where a denotes the adjoint action of M* and T* on U* and V* respec-
tively-Let <p: AutV*-AutU*
be the isomorphism induced by \f/. Since V* is a unipotent group defined
over Q, Aut V* has a natural structure of an algebraic group defined
over Q: note that if »* is the Lie algebra of V*, Aut V* can be identified
with the group Aut »* of Lie algebra automorphisms of »*. Also the
map a: M*—>Aut V* is defined over Q. Similarly AutU* is in a natural
fashion and R-algebraic group and a: T*—>AutU* is defined over R.
We now have a commutative diagram
M*-V*=r*—^—>T = ► G*=T*U*
A*=AutV*-V*
where A* is the Q-group obtained by forming the semidirect product of
Aut V* and V*, a (resp. /}) is an injection defined over Q (resp. R) such
that a(M*) (resp. /S(T*)) is contained in AutV*c:A*. Clearly a(.T)c:A$
and further if p: A*—>GL(«,C) is any representation defined over Q,
a-' o p~1 (GL(n, Z)) has finite index in T and p o a (r) GL(n, Q). Choose
now p to be a faithful representation of A* defined over Q. Then pop is
a faithful representation of G such that pofj(G)<=GL(n, R) and clearly
pop\r=a. It follows that p o j?(r)<= GL(n, Q) and that for a subgroup V
of finite index in /", poj? (/"")<= GL(n,Z). Let jSf=Z" the standard lattice
in R"( c C). Then 2"=f)po j?(y)(jS?) is again a lattice in R" stabilising 2".
It follows that after a conjugation by an element geGL(n, Q) we can
assume that p o /S (r) <= GL(n, Z). This completes the proof of Theorem 4.34.
4.43. Arithmeticity: a counter example. Let G be a connected solvable
Lie group and r <= G a lattice. (G, F) is said to be arithmetic if there
exists an algebraic subgroup G* <= GL(n, C) and an injective homo-
morphism i: G->GJ (= G* n GL(n, R)) with the following properties:
(i) i(G) is closed in GJ and G£/G is compact.
(ii) If G£=GJ n GL(n, Z), TnGJ has finite index in T and Gj£. (We
have here identified f with i(r).)

IV. Polycyclic Groups and Arithmeticity of Lattices
77
Theorem 4.34 suggests the possibility that for any lattice T in a
solvable simply connected Lie group G the pair (G, f) is arithmetic. We
will now give however an example of a lattice T in a simply connected
solvable Lie group G such that the pair (G, f) is not arithmetic. (Note
that Theorem 2.12 asserts in particular that if N is a simply connected
Lie group and rcJVisa lattice, then (N,r) is indeed arithmetic). Let
Be SL(4, Z) be the matrix
0 -1
1 4
0 -1 '
1 6
Then B is diagonalisable over R and has eigen-values 2+"|/3, 3+2"|/2.
These eigen-values being positive we can find XeM{4, R) such that
expX=B. Let a: R—>GL(4, R) be the representation t\-*exptX. Form
the semidirect product G=R x R4 for this action of R on the abelian Lie
group R4. Now the subgroup Z4<=R4 is stable under a(Z) so that
r = Z x Z4 is a subgroup of G. Evidently, T is a lattice in G. We claim
that the pair (G, T) is not arithmetic. Let G* <= GL(n, C) be an algebraic
group defined over Q and G<=GJ = G*nGL(n, R). We can replace G*
by the Zariski closure 'G*, of G in GL(n, C); if (G*, G, -T) has the
properties (i) and (ii) so does ('G*, G, T). We assume therefore in the sequel
that G is Zariski dense in G*. Now R4 <= G is a normal subgroup of G. It
follows that R4 is also stable under G&. We obtain thus a representation
of GJ on R4 and hence a representation a* of G* on C4: a: G*-> GL(4, Q.
Now o{r)c GL(4, Z) is a discrete subgroup of <7*(GJ|) and since Gr/T
is compact so is o*(G£)/o*(r). On the other hand ct*(GJ) has for its
identity component, the identity component H, of <7*(G*),. We claim
that H/(H r\ a*{r)) is compact. Now H has finite index in <7*(GJ|) so that
H r\ a*(r) is a subgroup of finite index in <7*(r). a*(r) being infinite cyclic,
Hno*(r) is also infinite cyclic. We will now show that H contains a
closed subgroup isomorphic to R+ x R+. This leads to a contradiction
since in this case H/a*{r) cannot be compact. Now <7*(G*) contains a(Z)
and hence the Zariski closure of a{Z) in GL(n, C). Now a{Z) can be
diagonalised over R so that the Zariski closure T of a{Z) in GL(n, Q
has all characters defined over R. Now let X(J) be the group of all
characters on T. One sees then that it suffices to show that X(T) is of
rank 2, and this follows from the fact that the units 2 + j/3 and 3+2|/2
are multiplicatively independent elements of C*.
Thus the pair (G, f) is not arithmetic.
4.44. Remarks. The ideas of the proof of Theorem 4.4 can be applied
to prove Ado's theorem on locally faithful representations of Lie groups.

Chapter V
Lattices in Semisimple Lie Groups: The Density Theorem
of Borel
Lattices in semisimple Lie groups are of course the most interesting and
difficult to study. The results obtained so far for these lattices have not
yet reached the stage of completeness achieved in the solvable and
nilpotent cases.
This is only the first of many chapters dealing with semisimple Lie
groups. Our main aim here will be to establish a theorem due to A. Borel
[3]. Unlike most of the other results which we will establish later, this
theorem requires a minimum of preparation. The proof given below is
the same as that given by Borel but for some minor changes.
Let G be a connected Lie group and H a subgroup of G.
5.1. Definition. H is said to have property (S) in G if the following
holds. Given any neighbourhood Q of e in G and any element geG, there
exists an integer n=n(g, Q) greater than zero, elements a>1,a>2EQ and
hsH such that g"=a>1ha>2.
We make a few simple observations.
5.2. Remarks. (1) If G is compact any subgroup H in G has the
property (S).
(2) If a compact subgroup H of a connected Lie group G has property
(S) in G, then G is compact.
Proof of (2). If G is not compact we can find a continuous homo-
morphism /": R—> G of the real line R in G such that / is injective, /"(R)
is closed and /: R—>/(R) is a homeomorphism. Now let i.0eR be any
element not equal to 0. Let Q be a compact neighbourhood of the identity.
We can then find an integer At such that
f{t0)*eQHQ.
Assume then that we have found integers At,..., Xr such that Xi>Xi_l
for l^i-l^r-1 and f(t0)x'sQHQ. Consider the element f{t0Yr+1;
according to the definitions of property (S) we can find an integer fc>0
such that f{tf(*r+1)<=S2HS2. Setting (Ar +1) k= yL+1, we obtain an inductive

V. The Density Theorem of Borel
79
construction for a strictly monotone increasing sequence of integers
R}iS,< oo -Clearly
/(t^eQHQ.
QHQ is a compact subset of G. It follows that {Xr-t0ef~1(QHQ)},
a compact subset of R, a contradiction. Hence the assertion.
(3) If H has property (S) in G and n: G —► Gt is a surjective continuous
homomorphism then re (if) has property (S) in G.
(4) If n: G—>Gj is a surjective homomorphism and HjcGj has
property (S) then re" '(ifi) has property (S) in G.
Proo/" o/" (4). We need only remark that a surjective homomorphism
of Lie groups is a necessarily an open map.
5.3. Remark. Contrary to what one might expect in view of (1) above,
it will be seen later that if H is a closed subgroup of G such that G/H is
compact, then H need not necessarily have the property (S). However
we do have
5.4. Lemma (Selberg). If H is a closed subgroup of G such that G/H
has a finite invariant measure then H has property (S) in G.
Proof. We will prove a slightly stronger statement. We will show,
that given a neighbourhood Q of G, we can find an integer N = N(Q)
depending only on Q and H such that for any geG, there exists n = n (g, Q)
with l^ngN with the property gne£iH£i. Fix an invariant measure n
on G/H such that p.(G/H)=l. Let it: G-^G/H be the natural map and
let a=n(n(£i)). Clearly a>0. Let N1=N1(ii) be the smallest integer such
that Nja^l. Consider now the sets {n(gkii)}likiNl. Now n(n(gkS2)) =
n(ji{S2))=a. for all lgfcgNj since \i is G-invariant. It follows that the
sets {n{gkQ)}1<k^Ni cannot all be mutually disjoint. Thus we can find
integers k,I with l^k<l^Nt such that g*f2Hng!fi=#0 i.e., we can
find (Wj, (w2 in Q such that gk a>lh=gl a>2 for some heH; in other words,
we have
gl~k = wlhoi2l
and there is no loss in generality in assuming that Q=Q~l so that we
may assume that (wj'eft. This proves the lemma.
We will now state the main result of this chapter.
5.5. Theorem (Borel). Let G be a connected semisimple Lie group
without compact factors. Let H<=.G be a subgroup with property (S).
Let pbea linear representation of G in a finite dimensional vector space V
over the field k( = R or Q: p: G—►Aut^F). Then the Linear span s/
of p(H) in Endjt(F) is the same as the linear span 9& of p(G) in Endk(V).

80
V. The Density Theorem of Borel
It is easy to see that the case k=R can be deduced from the case k = C.
We will therefore assume that k = C in the sequel. We will first prove
that Theorem 5.5 is equivalent to
5.6. Theorem. Let G be a connected semisimple Lie group without
compact factors and H<=.G a closed subgroup with property (S). Let p
be a finite dimensional representation of G in a complex vector space V.
Then if p is irreducible so is p restricted to H.
To see the equivalence of Theorems 5.5 and 5.6 we argue as follows: if
Theorem 5.5 is true, then since for an irreducible representation p,
^=Endc(K) (Burnside's theorem: see for instance Bourbaki [2], §3,
No. 3) we have si = Endc(F) as well. Thus V is an irreducible H-module.
Conversely, assume Theorem 5.6. Let p = LJ pt be the decomposition of p
into irreducible representations with respect to G. Then pt restricted
to H is irreducible. Thus V is a semisimple H-module as well. Now
consider V® V*. This again breaks up into a direct sum LI Et of irreducible
16/
G-modules. F® F*i:Hom(F, V) so that the commutant C{&) of 31 is
precisely the set of G-invariants in V® V*. Let J = {i\iel, Et is 1-dimen-
sional}. Clearly, C{Sf) = LI Et. Now each Et is an irreducible H-module
as well. Also, dim Et= 1, if and only if Et is a trivial G-module (since G
is semisimple). Thus Et is a trivial //-module if and only if ie J. It follows
that the commutant C(s/)ofs/is LI Er Thus we see that C(sf) = C(3f).
Now V is a semisimple j^-module as well as a semisimple ^-module.
Hence by Burnside's theorem (Bourbaki [2], loc. cit.) si and 9& are equal
to their respective bicommutants. It follows that s&=gt. This proves our
claim.
For the proof of Theorem 5.6, we need a few lemmas.
5.7. Lemma. Let Gbea semisimple Lie group and pt, p2 be two
irreducible representations of G over C. Assume that pt is non-trivial Then pt®p2
is irreducible if and only if p2 is irreducible and trivial on every simple
component on which pt is non-trivial.
Proof. Let p=Pi®p2- Then p is irreducible if and only if p®p*
contains the trivial representation exactly once where p* is the
representation contragredient to p. Now
p®p*=(pl®p2)®(pl®p2)*^{pl®pt)®(p2®pl)-
Let Gj be a simple component of G on which both pt and p2 are non-
trivial. Let gt be the Lie algebra of Gt and t the adjoint representation
of G in Qt. Then pt®p* and p2®p* both contain the direct sum t©1
where 1 is the trivial representation. It follows that p®p* contains

V. The Density Theorem of Borel
81
(t©1)®(t®1). Now t is self-dual (the Killing form gives an isomorphism
oft with its dual). Hence t ®t contains a copy to the trivial representation.
Thus p®p* contains the trivial representation atleast twice and hence
it cannot be irreducible. The converse is easy to prove.
5.8. Lemma. Let G <= GL(n, Q be a linear semisimple Lie group and
H<=.G a subgroup with property (S). Let s/ be the linear span of H in
M(n, C). Let g be the Lie algebra of G. We identify g canonically with a
Lie subalgebra of M(n, C). For a nilpotent matrix Xe$ let n(X) be the
greatest integer such that X"m=#0. Then Xn(X)es/.
Proof Let g = exp X. Since H has property (S) in G, we can find a
strictly monotone sequence {Am}1Sm<00 of positive integers, a sequence
hmeH and elements £,m,nm in G converging to e in G such that g*m=
gk=l + kX+k2X2/2\ + ---+kn<X)-Xn<X)/n(X)\
and hence Lim{gXk/XnkiX)} = X"m/n(X)\es/. This proves the lemma.
5.9. Lemma. Let h be the Lie subalgebra of M(n, C) generated by
{XniX)\XeQ,X nilpotent Xn{X)+0, XniX)+1=0}. Then g is contained in
the normalizer of\).
Proof. This follows from the fact that the set of generators
{Xn(X)\XeQ, X"m+0, XB(*)+1=0 for some integer n(X)}
is stable under inner conjugation by elements in G.
5.10. Lemma. Let G <=■ GL(n, Q be an irreducible connected semisimple
Lie subgroup. Let g, h be as in Lemmas 5.8 and 5.9. Then the Lie algebras h
and Qj = g + h are semisimple. The Lie algebra Qtisa direct sum of h and a
(unique) supplementary ideal ht.
Proof. C is an irreducible G-module, hence an irreducible g-module.
It follows that C is an irreducible g-module as well; in other words g:
admits a faithful irreducible representation. It follows that gt is reductive.
Now gt is generated by the semisimple subalgebra g together with some
nilpotent elements. Thus gt is semisimple. h being an ideal in gt is semi-
simple as well. The last assertion is immediate from the (unique)
decomposition of gt into its simple ideals.
5.11. Lemma. We assume the hypothesis of Lemma 5.10 to hold. In
addition we suppose that G has no connected compact normal subgroups.
Let it: gj—>h be the natural projection o/gt on the ideal h which respects
the direct sum decomposition gt = E)©!),. Then n\g is injective.

82
V. The Density Theorem of Borel
Proof. Let g0 be any simple component of g. It suffices to show that
7t|go is non-zero. Since g0 is noncompact, we can find an element Xeq0
such that Xn(X)+1 = 0, Xn{X)+0 for some integer n(X)>l-in particular
X =#0. Moreover from the structure theory of real semisimple Lie algebra
we can find also an element Teg such that (cf. Preliminaries § 1.7)
Now lT,XniX)] = n(X)'XniX); on the other hand since X"weh and h
is an ideal,
[ti(T), tcOT'*')] = [ti(T), AT"'*'] = [T, X"m],
we see that te(T)=#0. This proves our contention.
5.12. Lemma. Let g be a semisimple Lie algebra and g=^®^i a
decomposition of g into a direct sum of two ideals fj and fjj. Let n (resp. nt)
be the projection of g on (resp. g J. Let p be an irreducible representation
of g on a complex vector space. Then p is equivalent to the tensor product
ctotcOctjOTCj where a (resp.at) is an irreducible representation of h
(resp. h J.
Proof. Let V be the representation space for p. Let M (resp. Afj) be a
minimal p(h)-stable (resp. p (hj)-stable) subspace of V. Now p(X) and
p{Y) commute with each other if Xeh and Yehj. It follows that as a
module over fj (resp. i)t),p(Y). M (resp. p(X)- Mt) i.e. either zero or
isomorphic to M (resp. Mt). From this, using the irreducibility of p one
sees that considered as an h-module (resp. hj-module) V is a sum of
simple h-modules (resp. hj-modules) all isomorphic to M (resp. Mt). Now
let B (resp. Bt) be the associative subalgebra of Endc V generated by p(h)
(resp. p(hj)). Then from the preceding arguments one sees that B (resp. Bt)
admits a faithful simple module viz. M (resp. Mt). Hence B (resp. Bt)
is simple (cf. Bourbaki [2], Proposition 9, No. 2, § 5). Now B and Bt
being algebras over C, we conclude that they are isomorphic Endc Mt
and Endc M respectively (Bourbaki [2], Corollaire 3, No. 4, § 5). Now
let a (resp. <7t) denote the map of h (resp. ht) in B (resp. Bt) obtained by
restricting to p(h) (resp. ht). Now since elements of B commute with those
of Bj we have a natural homomorphisms (of associative algebras)
/: BfglcBj-^EndF.
The map / is surjective since the image by / contains p(g). On the other
hand B®cBj is isomorphic to Endc(M (g^ Mt) and is hence simple.
Thus / is an isomorphism. Endc V being simple all simple modules over
Endc Vare isomorphic. Now M(g)cMt is a simple module over B^B^^.
It follows that there exists an isomorphism (of vector spaces)
F: M^-Mt-^V

V. The Density Theorem of Borel
83
such that
F(ax)=f(a)-F(x)
for all aeB ®c Bt and xeM<S>cMt. Clearly then F defines an equivalence
of p with ffoTtgjffjOTtj. This proves the lemma.
Proof of Theorem 5.5. We replace, as we may, G by p{G). Thus we
have G <= GL(n, Q (for a suitable n). Let g <= M(n, Q be the Lie subalgebra
of M(n, Q corresponding to G. To show that H is irreducible, it suffices
to show that C" is irreducible as an h-module where h is defined as in
Lemma 5.9. Let g^g+h and g1=h©h1 be the decomposition of gt
into a direct sum of ideals. Let i: g^M(n, C) be the natural inclusion.
According to Lemma 5.12, i=(xon)^)(x1onl) where t (resp. Tt) is a
representation of h (resp. hj) and n: gt—► h (resp. ii{. gt—► ht is the canonical
projection. Now 7t|8 is injective (Lemma 5.11) and since i={z°n)®{xlon^)
is faithful, t is a faithful representation of h. Thus t o n is a faithful
representation when restricted to g. Now i restricted to g is irreducible so that
it follows from Lemma 5.7 that z^^ is necessarily trivial on g. Since i
is faithful rt restricted to ht is faithful so that 7ij is trivial i.e. g£h. But
then any h-stable subspace of C is g-stable. Thus C is h-irreducible,
hence H-irreducible. This proves the theorem.
5.13. Corollary. Let G be a connected semisimple Lie group without
compact factors and H a subgroup with property (S). Let p be any linear
representation of G. Then the space of invariants for p(G) and p(H) are
identical.
Proof. Let V be the representation space for p and let V=LI Vt be the
16/
decomposition of G into irreducible G-modules. Now each Vt is an
irreducible //-module as well. Finally dim V~l if and only if Vt is a
trivial G-module. It follows that Vt is a trivial H-module if and only if it
is a trivial G-module. This proves the corollary.
5.14. Corollary. Let p be a finite dimensional representation of G
in a vector space V. Let H be a subgroup of G with property (S) in G.
Then an H-submodule of V is l-dimensional if and only if it is trivial.
Proof. Let F=LIv;- be the decomposition of V into irreducible G-
ieJ
modules. Let £ be an H-submodule of V of dimension 1. Now each Vt is
H-irreducible. It follows that E is isomorphic to V{ for some iel, say i0.
Since dim£=l. diml<0=l; but then Vio is a trivial G-module (G is
assumed semisimple). Thus £ is a trivial H-module. Hence £ is contained
in the space of H invariants. It now follows from Corollary 5.13 that £
is G-stable.

84
V. The Density Theorem of Borel
We have therefore
5.15. Corollary. Any H-invariant 1-dimensional subspace is G invariant
as well.
Passing to a suitable exterior power we deduce from Corollary 5.15
the first assertion of
5.16. Corollary, (i) Let p be a finite dimensional representation ofG on a
(real or complex) vector space V. A subspace W<=. V is G-stable if and only
if it is H-stable.
(ii) Let p: G—>GL(n, Q be any finite dimensional representation ofG.
Then p(G) and p(H) have the same Zariski closure in GL(n, C).
(iii) If G is linear, any algebraic subgroup of G normalised by H is
normalised by G.
Proof, (i) has already been proved. To prove (ii) let G' (resp. G) denote
the Zariski closure of p(H) (resp. p(G)) in GL(n, C), clearly G' <= G. Then
we can find a representation a of GL(n, C) on a vector space E and an
element v (#0) in E such that
G'={geGL(n,C)|<7(g)t>eCt>}.
Applying (i) to the representation ffopofGwe find G' = G. This proves
(ii). The third assertion is evidently an immediate consequence of the
second.
5.17. Corollary. Let G be a connected semisimple Lie group and G' be
the minimal connected normal subgroup of G such that G/G' is compact.
Let n: G—>G/G' be the natural map. Let tcG be a lattice such that n(r)
is dense in G/G'. Then the normaliser N(F) of r in G is discrete. Also, if
Z<=G is the centre of G, ZT is discrete. (Note that if G has no compact
factors, G = G' so that these conclusions hold for any lattice in G.)
Proof. Since Zr<=N(r), it is evidently sufficient to prove that N(F) is
discrete. Let G" be the maximal compact connected normal subgroup
of G and n': G-^G/G" the natural projection. Let N(r) be the normalizer
of r. Then N(r) is closed in G. Let N0 be the identity component of N(r).
Since N0 normalizes T and T is discrete N0 centralizes f. It follows that
n(N0) centralizes n{r) hence G/G'. Thus n{N0)=e. Now ri{r) is a lattice
in G/G"; the last group has no proper compact connected normal
subgroup. The Lie algebra of ti'{N0) being stable under Ad ri{r), it reduces
to zero (in view of Lemma 5.4 and Corollary 5.16). Hence n'(N0)=e.
It follows that N0 = e. Hence the corollary.
5.18. Corollary. Let Gbea semisimple Lie group without proper compact
connected normal subgroups and H a subgroup ofG with property (S). Then
the centralizer of H in G is the centre of G.

V. The Density Theorem of Borel
85
This is an immediate consequence of Corollary 5.17.
5.19. Corollary. Let G be a connected semisimple Lie group without
compact factors. Let H, H' be connected closed proper normal subgroups
ofG such that HH' = G and HnH' is discrete in G. Let n (resp. n1) be the
natural map of G on G/H' (resp. G/H). Let r<=G be a lattice. Then the
following conditions on r are equivalent:
1) 7t(r) is a discrete subgroup of G/H'.
2) re'(r) is a discrete subgroup of G/H.
3) rr\H is a lattice in H.
4) fnff is a lattice in H'.
5) r contains (rnH)(r<-\H') as a subgroup of finite index.
If in addition G is linear conditions l)-5) are equivalent to
6) fnfl is Zariski dense in H.
7) rr\H' is Zariski dense in H'.
Proof. The implications 1) o 4) and 3) <*• 2) are simply restatements
of Theorem 1.13 (Chapter I). Let rH=Tr\H. Then rH is a discrete
subgroup of H normalised by L It follows that 7t(7^) is normal in n(T). IfrH is
a lattice in H, n(rH) is a lattice in G/H'. It follows from Corollary 5.17 that
7t(r) is discrete. We have thus shown that 3) implies 1). An exactly
analogous argument shows that 4) implies 2). Thus we have the cycle of
implications
1) => 4) => 2) => 3) => 1).
Under this set of equivalent conditions evidently (rnH)(rnHr) is a
lattice in G and hence is of finite index in f. Thus these conditions imply 5).
Conversely we will now show that 5) implies 1). For this we observe first
that in view of Corollary 5.17 we may assume that T contains the centre
of G and further that r={rr\H)(r'r\H). Now if BeH is any closed
subgroup containing the centre of H, BH' is easily seen to be a closed
subgroup of G. It follows that since TnH contains the centre of H,
{rr>H)-H'={rnH){rr\H')-H' = r-H' is closed in G. Thus n{r) is
discrete in G. Thus (5) implies (1).
Evidently (3) (resp. (4)) implies (6) (resp. (7)). Now if rH = Tr\H is
Zariski dense in H, n{rH) is Zariski dense in G/H' and normal in re(F).
Now the normalizer N of n(rH) is closed in G and its identity component
N° (being connected) normalises n(rH) if and only if it centralises n(rH).
Since n{rH) is Zariski dense in the semisimple group G/H', N°={e). Thus
N and hence n(r) is discrete in G. It follows that 6) implies 1). Similarly
one concludes that 7) implies 2). This completes the proof of the lemma.
Now let r cz G be a lattice, G being connected semisimple and without
compact factors. Let H be a closed connected normal subgroup of G and
p: G-*G/H be the natural map. Suppose p(f) is not dense in G/H. Let

86
V. The Density Theorem of Borel
G' be the identity component of the closure of p(F) and let H' = p~1(fil).
If 7i. G-^G/H' is the natural map, evidently n{r) is discrete and /"nff
is a lattice in H'. If H" is the unique connected normal subgroup such
that H' H" = G and H'n H" is finite, then T contains (To H') • (Fn H") as
a subgroup of finite index.
We summarise this discussion in Corollary 5.21 below. For the
convenient formulation of the corollary we make the following
5.20. Definition. A lattice Tc: G in a connected semisimple group
without compact factors is reducible if G admits connected normal
subgroups H,H' such that HH' = G, HnH' is discrete and r/(rr\H)-
(Fn H') is finite. A lattice is irreducible if it is not reducible.
With this definition we have
5.21. Corollary. The following conditions on a lattice r in a connected
semisimple group G without compact factors are equivalent.
1) r is irreducible.
2) IfH is any proper connected normal subgroup ofG,Hr\T is not a
lattu e in H.
3) If H is any proper connected normal subgroup of G, Hr\T is
central in H.
4) If H<=-G is any connected normal subgroup and p: G—>G/H is the
natural map, p(r) is not discrete.
5) IfH and p are defined as in (4), p(F) is dense in G/H.
Only the equivalence of (3) to the rest needs some clarification. One
easily reduces the proof to the case when G is linear. If HnT is not
central for some connected normal proper subgroup H, its Zariski
closure B is a non-central normal subgroup of G (cf. Corollary 5.16); in
particular B is not discrete. The identity component B° of B is thus a
connected proper normal subgroup of G such that BnT is Zariski dense
in B. And in view of Corollary 5.19, BnT is a lattice in B. Thus T is
reducible.
A simple induction argument leads now to the following
5.22. Theorem. Let Gbea connected semisimple group without compact
factors. Let r<=.G be a lattice. Then we can find a (finite) family {HJie/ of
connected normal subgroups ofG with the following properties:
(i) if HI = n HP Ht n Hi is discrete,
(ii) g=y\h!,
(iii) r^HinT is an irreducible lattice in Ht and
(iv) Yl^i 's a (normal) subgroup of finite index in r.

V. The Density Theorem of Borel
87
For convenient future reference we state a consequence of
Corollary 5.21.
5.23. Corollary. Let G be a connected semisimple Lie group without
compact factors and T<=.G an irreducible lattice. Let OeT be any element
not in the centre ofG. Let {G,}ie/ be the simple quotients ofG and p{: G—► Gt
the natural map. Then p{(0)^e for any iel. In particular if G is linear a
unipotent element OeT does not belong to any proper connected normal
subgroup of G.
Most of the results of this chapter admit suitable generalisations when
G is allowed to have compact factors. The proofs given above for
Theorems 5.5, 5.6 and the Corollaries need only be slightly modified to yield
5.24. Theorem. Let G be a connected semisimple group and H a
subgroup with property (S). Let Gt be the minimal connected normal subgroup
ofG such that G/Gt is compact. Let p be a finite dimensional linear
representation ofG in GL(n, k) (k=R or C). Then we have
(i) The linear span ofp(H) in M(n, k) contains that of p(Gl).
(ii) A vector vek" is invariant under p(Gt) if it is p{H)-invariant.
(iii) A p{H)-stable subspace ofkf is also piG^stable.
(iv) The Zariski closure of p(H) in GL(n, Q contains that of p(Gt).
The centralizer ofH in G centralises Gt as well.
Note that Corollary 5.17 has been stated for all semisimple Lie
groups. To enable us to obtain exact anologues of Theorems 5.5, 5.6 and
the Corollaries 5.13-5.18 we need to strengthen slightly the property (S)
of H. We introduce towards this end
5.25. Definition. A subgroup H of a semisimple Lie group G has
property (SS) if H has property (S) and Gt H is dense in G where Gt is
the minimal connected closed normal subgroup such that G/Gl is
compact. (Clearly if G has no compact factors H has property (S) if and only
if it has (SS).)
We then deduce from the proofs of Theorems 5.5 and 5.6 and their
Corollaries the following
5.26. Theorem. Let Gbe a semisimple Lie group and H a subgroup with
property (SS). let p be a finite dimensional representation ofG in GL(n, k)
(fe = R or C). Then we have:
(i) The linear span ofp(H) in M(n, k) equals that ofp(G).
(ii) p is irreducible if and only if p restricted to H is irreducible.
(iii) A vector vek" is H-invariant if and only if it is G-invariant.
(iv) A subspace E <=.!<" is H-stable ij and only if it is G-stable.

88
V. The Density Theorem of Borel
(v) IfH is discrete the normalizer ofH in G is discrete; in particular
ifZ is the centre ofG,ZH is discrete.
(vi) p(G) and p(H) have the same Zariski closure in GL(n, C). The
centralizer ofH in G is the centre ofG.
(vii) If G=GtxG2 and pu p2 are the Cartesian projections of G
on G1; G2 then if H is a lattice such that pt(H) is discrete so is p2{H) and
H has finite index in p1(H)xp2(H).
5.27. Remark. Let G be the complex semisimple Lie group SL(n, Q
and B the subgroup of upper triangular matrices in SL(n, C). Let p be
the natural representation of G. Then G/B is compact. The Zariski
closure of p{B) on the other hand is itself. It follows from Corollary 5.16
that B does not have property (S) in G. However we have the following
5.28. Theorem. Let G be a connected semisimple Lie group without
compact factors. Let H be a closed subgroup such that G/H is compact.
Let p be a finite dimensional linear representation of G in a vector space V
over k (= C or R). Then the space of G-invariants is the same as the space of
H-invariants.
Proof. Let E be the space of //-invariants. Let
El = {v\veV> p(G)v is relatively compact in V}.
Clearly Et is a subspace of V. It is moreover G-invariant and contains E
since G/H is compact. Let pl denote the representation of G in Ev Let
eu ...,ek be a basis of Ev Then since {pi{G)ei}l^i^k are all relatively
compact, Pi{G) is a relatively compact subset of Autc(£j). In other words
pt is equivalent to a unitary representation. But G has no compact
factors. It follows that pt is trivial i.e. £t consists of G-invariants. Hence
£ = £j = G-invariants. This proves the theorem.
529. Remark. The proof given above is due to A. Weil [2]. Weil
states the theorem for discrete subgroups H of a connected semisimple
group G but his proof requires no modification. In the case when H is
discrete, however, Theorem 5.28 is a particular case of Corollary 5.13 in
view of Lemma 5.4.
5JO. Remark. S.P. Wang[l] has shown that in a solvable Lie
group G, if H is a closed subgroup with property (S), then G/H is compact.

Chapter VI
Deformations
Let r be a finitely generated group and G a Lie group. Let R(F, G) be
the set of all homomorphisms of T in G topologised by the topology of
pointwise convergence. In this chapter we obtain some results which
show that homomorphisms ueR(F, G) near a given one u0, are controlled
considerably by u0 if u0 has certain special properties.
6.1. We begin with some rather general remarks on the space R(r, G).
We observe first that the group G acts on R(r, G) in a natural fashion:
for geG and u: T—► G, we define u*: r—>G by setting
u*(x)=gu(x)g-1,
the map (g, u)\-*ug is a continuous action of G on R{T, G) as is easily
checked.
6.2. Next we remark that a finite set S of generators for T enables us
to identify R(r, G) as a closed subset of f| G. To see this we argue as
s
follows: Let F be the free group on the set S and p: F—>T the canonical
map induced by the inclusion S <"->/". By the definition of the free group
on S, R{F, G) can be identified with the space f] G. Let H be the kernel
s
of p. Any element xeH can be written in the form
x = sf'-...-s£"
with steS, ££=±1. We fix one such expression once and for all and
denote by fx: ]"] G—► G the map
s
JxyXosfseSJ os i '0S2 ''"' osm'
Evidently fx is a continuous map of ]~] G in G consider now the restric-
s
tion map r: R(r, G)-^R(F, G): r(ti)=j.op. The identification of R{F, G)
with Y[ G, one sees yields an identification cps of R(r, G) with the closed
s
set f] fx~1(e). In the sequel when we fix a set of generators S for f, we

90
VI. Deformations
will often consider R(r, G) as a subspace of ]~] G through this
identification cps without explicit mention. s
6.3. We note further the following. Suppose T<=H is a subset
generating H as a normal subgroup,
n/x-iw=n/x-i(e).
This shows that if T is finitely presentable (so that H has a finite system
of generators as a normal subgroup) R(r, G) can in fact be identified
with a finite intersection f] f~ 1(e)(T a finite set). Since each/x is analytic
xeT
it follows that R(r, G) can be identified with an analytic subset of ]~] G.
s
6.4. Suppose now G<=GL(n, Q is an algebraic group defined over
a field ItcfJ. Then [~[ G is defined over k as well; the maps/x: ]"] G-^G,
s s
xeH are evidently morphisms defined over k. It follows that R(r,G)
has a natural structure of an affine variety defined over the field k.
We now give a definition the significance of which is illustrated by
the following proposition.
6.5. Definition. A homomorphism ueR(r, G) is locally rigid if the
orbit of u in R(r, G) under G is open in R(r, G).
6.6. Proposition. Let G be an algebraic group defined over a number
field k and r a finitely generated group. Let K=R or C and assume that
kcK. Suppose ueR(r,GK) is locally rigid. Then we can find a number
field L,k<=L<=K and geGK such that gu(y)g~1<=GLfor allyeT. (Here
GL is the group of L-rational points of G.)
Proof. We set GC = G. Evidently R(r, G) is a sub variety in C" (for
some m) defined over k. That is, the ideal / of polynomials on Cm vanishing
on R(r, G) is spanned by polynomials with coefficients in k. The orbit R'
through u is a homogeneous space for G and hence an analytic sub-
manifold of Cm, of codimension p, say. It follows that we can find
polynomials f,...,fpel with coefficients in k such that in a neighbourhood
U of u in Cm, we have
UnR{r,G) = UnR' = {xeU\fi{x)=0for l^i^p}
and the rank of the Jacobian matrix {8fi\dxJ}l&i^ is exactly p in the
open set U. We assume as we may without loss in generality that
(dfi/dxJ)lgi ygp is non-singular. Let n: Cm-^Cm~,' denote the projection
C" onto the last (m—p) coordinates. If U'<=U is chosen suitably we find
that according to the implicit function theorem for R or C as the case

VI. Deformations
91
may be the map n\v.nKm: R' n U' n Km —>Km~p is an open map. Let k~
denote the algebraic closure of k in C. Since R' nU = R(r,G)nU, if
xeU'r\Km is such that n(x)ekm~p, xe(k~)r^R'. Thus we can find a
neighbourhood V of n(u) in /cm~p such that for all yeV, we can find
xeU'n{k~ynKmnR' such that n{x) = y. In other words we can find
a point g(w) = Int g ° w in the orbit of u which is contained in the set of
(k~ n ^-rational points of R(r, G). Reinterpreting this in terms of the
homomorphism, we see that for all yef, gu(y)g~1eG(fe~ n K). Now let
S be a finite set of generators. Evidently we can find a subfield L c k~ n K
which is finitely generated over fe and such that gu(y)g~ieG(L) for all
yeS. Since S generates T, g«(y)g~1eG(L) for all yef. Since k~ =>L, L
is a numberfield.
A useful criterion for local rigidity of u in R(r, G) is our first main
result.
6.7. Theorem (A.Weil [4]). Let ueR(r,G) be a point such that
Hl(r, Ad o u) (thefirst cohomology group ofT with coefficients in Adou:
cf. Preliminaries §3.2) vanishes. Then u is locally rigid.
For the proof of the theorem, we need the following simple
consequence of the implicit function theorem.
6.8. Lemma. Let X,Y, {Zt}i€l be Ck-manifolds (lgfegoo). Let h:
X-*Y and f: Y-*Zi be Ck-maps. For a point p on one of these
manifolds we denote by Tp the tangent space at p. Let x€X be a point such
that the sequence
t d/i 7- udft ^ rr -j-
iel
is exact. Suppose further that f h (X) = p{ for all i. Then the image h (X)
of X in Y is a neighbourhood of y in the space P)/i1 (/>.)•
.6/
Proof. Since the question is local we may replace the spaces X and
Y by open domains in euclidean space; the Z, may be replaced by eucli-
dean spaces and again (by enlarging the indexing set /) by the real line
itself. Since dim T is finite, we can find a finite subset Jc/ such that
U df,
j -""ut —— ► T
x 'y L-L Pi
lej
is exact. We can in fact assume that J contains exactly k elements where
k is the rank of \Jdf. Assume for the moment that we have shown that
iel
h(X) contains an open neighbourhood of x in f]frl(Pi). Then since
ieJ
C\fi~l(pl) is contained in the set f]fr1(Pi) and contains f(X), the
(€/ lej

92
VI. Deformations
assertion follows. If in addition to our choice of J we replace X by a
submanifold X' containing x and whose tangent space at x is
supplementary to the kernel of dh at x, we can actually assume that the sequence
Udf, T
o->r^r^—► LItw-»o
is exact. Now since dim. Image LI dft=dim ]~] Zt (note that dim Zt=l)
at y it follows from the implicit function theorem that Off1 (pi) is a
submanifold of Y (near x) of dimension equal to rank dh (at x). But then
h may be regarded as C*-map of X into this submanifold of maximal
rank at x. It follows once again from the implicit function theorem that
f(X) is a neighbourhood of y. This proves the lemma.
6.9. Proof of Theorem 6.7. As observed in 6.2, 6.3, R{r, G) can be
realised as the intersection f] fx~l(e) where {fx: f\ G—► G}xeH are ana-
xeH S
lytic maps. Consider now the map h: G-^>Y\G defined by h{a){y) =
s
au(y)a~l for all yef. In view of Lemma 6.8 it suffices to prove the
following. If B (resp. A) denotes the tangent space at u (resp. e) to \\ G
s
(resp. G) and M is the image of A in B under the tangent linear map
induced by h, then _
xeH
where Lx is the kernel of the tangent linear map B-^A induced by/x.
The tangent space B may be identified with the Lie algebra (of right-
invariant vector fields) LI 9 of FT G. Similarly A can be identified with
s s
the Lie algebra g of G An element of B is thus a map
q>: S->g.
Recall that if p is a representation of a group J on a vector space
V, a 1-cocycle on A with coefficients in p is a map q>: A —> V such that
for x,yeA,(p(xy) = (p(x)+p(x)(p(y); a 1-cocycle is a coboundary if there
exists veV such that for all xeA, we have (p(x)=p(x)v — v. A 1-cocycle
can be given another interpretation. Let At be the semidirect product
V xA: the multiplication in At is given by
{v,S){v\S')=(v+p{S)v',S8').
For a map if/: A —> V, let $: A —>At be the map ij/(x)=(il/(x), x); then i//
is a 1-cocycle with coefficients in p if and only if $ is a group homo-
morphism. From this interpretation we conclude immediately that an

VI. Deformations
93
element cp: S—>g in B extends uniquely to a 1-cocycle q> on F with
coefficients in a = Ad°u° p.
Next, \etfx: B—>q denote the tangent linear map induced by fx (at
u) xeH. We claim that fx{<p) = <p(x). To prove this we argue as follows.
Let x=s\l s£22 ...s^ where s^S and e, = ±l. For (peB, the curve q(t) =
{expr (p(s)-u(s)}seS has cp as its tangent vector at u (=q(0)). Its image
under fx is the curve
v(t)=(exp t p(s,) ■ «(s,))" • (exp t <p(s2) h(s2))£2... (exp t cp{sj ■ M(.s„,))rm.
Now for seS, cp(s~1)= —Ad m(s)_1 ■ cp(s). It follows that the tangent
vector to v(t) at i;(0) ( = e) is precisely
g»(*i)+ I Ad«(s1)£i«(s2r...«(srr-^(.sr+1)=^(x).
Evidently /^M is the tangent vector to v(t) at v(0). This proves our
claim. It follows that Lx consists of all 1-cocycles on F with values in
a which vanish at x. Hence f] Lx is precisely the set of 1-cocycles on
xeH
F which vanish on H. Now for xeH, zeF and (06 f] Lp we have
xell
q>(zx) = (p(z)+p(z)(p(x) = <p(z), so that <p is in fact of the form (pt°p
where q>l is a 1-cocycle on r with coefficients in Ad ° u. We see thus
M' = (~) Lx can be identified with the space of 1-cocycles on r with
xeH
values in Ad ° u.
The space M on the other hand consists of vectors of the form
{Adu(s) X — X}seS with Xeq: in fact the curve expiA" (which has
A"eg for its tangent vector at e in G) maps under h into the curve
{exp t X' u(s)exp — tX'}seS; and at u the tangent vector to this last curve
is evidently {X'-Adu(s) X'}seS, From the definition of a coboundary
we see immediately that M is precisely the space of all coboundaries
on F and hence on r. Thus M = M'= f] Lx. This proves Theorem 6.7.
xeH
We will next prove a result of a rather general nature and then study
the space R(r, G) when r can be imbedded in G as a uniform lattice.
We begin with
6.10. Definition. Let k be a field. An element geGL(n, k) is net if the
(multiplicative) group A(g) generated by the eigen-values of g does not
contain any nontrivial root of unity.
A subgroup rcGL(n, C) is net if every element yeT different from
the identity is net.
With this definition we will now prove
6.11. Theorem. Let rcGL(n, C) be a finitely generated subgroup.
Then r contains a normal subgroup of finite index which is net.

94
VI. Deformations
We consider first the case when Tc:GL(n, k), k a number-field. Let
J be the ring of integers in k. Since T is finitely generated we can find a
finite set of integers qu ..., qreZ such that Tc: GL(n, A) where A is the
ring J[l/qt,..., 1/qJ. Let N = n\ and ii0 the (finite) set of nontrivial
roots of unity which satisfy a polynomial over k of degree less than or
equal to N. For xsQ0, let FX(T) be the minimal polynomial of x over k.
Let S' = {Fx(l)\xeii0}, and S the multiplicatively closed set generated by
S'. Since S' is finite and does not contain 0 we can find a proper prime
(hence maximal) ideal a in the Dedekind domain A such that Sna=$.
Let B=A\_S20'] and b<=B a prime ideal such that a=bn 4. Let?;: B—>B/b
and »h: ,4—>,4/a be the natural maps. Consider now the natural map
n: GL(n,4)->GL(n,4/a).
Evidently A/a = k' is a finite field so that the kernel H of»/ is a subgroup
of finite index. We will now show that H is net. Let xei/ be any element
different from e. Let A(x) be the group generated the eigen values
Xlt...,Xn of x. Now the characteristic polynomial of x is a polynomial
of degree n belonging to k [T]. It follows that its splitting field L is an
extension of degree utmost equal to N( = n!). We conclude therefore that
A(x)<=L. In particular if coeA(x) is a non-trivial root of unity, coeQ0.
Suppose now coeA(x)nS20; then ri(co)EA(fj{x)). Since »/(Fro(l))=#0, while
Fro((u)=0, we conclude that »/(co)=M. It follows that ^(x)=#Idendity,
a contradiction. We conclude thus that A(x)r\Q0=$ and hence that
H is net. Clearly F=Hnr is net and is of finite index in T. This
completes the proof of Theorem 6.11 in the special case when r<=GL(n, k),
k a numberfield.
6.12. We now consider the general case. As before, for xeG=
GL(n, C), let A(x) denote the subgroup of C* generated by the
eigenvalues of x. Let p denote the natural representation of G and p* its dual.
For each non-negative integer p let pp (resp. p*) denote the p-th tensor
power of p (resp. p*). Let op q=pp®p*. One sees then that each element
of A(x) is an eigen-value of ap q(x) for a suitable pair of non-negative
integers (p,q). Let Q denote the set of roots of unity in C* and Q{x) =
QnA(x). For each xeT and coeS2(x) we fix a pair (p(x, co), q(x, co)) of
non-negative integers such that co is an eigen-value of ap(x>co)j,(Xj0)){x). In
the sequel we let a{x,to) denote op(xe>)q(xe>). Now G is an algebraic
group defined over Q. If we fix a finite set S of generators for T, as
remarked in 6.4, we can identify R(r, G) with a subvariety of J~[ G defined
over Q. Let s
R'{r,G)={ueR{r,G)\det(a{x,(o)u{x)-(o)=0 forxef, weQ(x)}.

VI. Deformations
95
If Q denotes the algebraic closure of Q in C, evidently R'(r, G) is a sub-
variety of Y[ G defined over Q. From our definition of R'(r, G) we see
s
immediately that if xsT is not net, u(x) is not net for any ueR'(T, G).
Now, according to the Hilbert nullstellensatz, we can find a Q-rational
point u in R'(r,G); this means that u(r)<=GL(n,Q) and, since r is
finitely generated that u(r)<=GL(n,k), k a numberfield. From what we
have already proved we can find a net subgroup A<=u(T) which is of
finite index in u(T). Let P=u~l(A). Then P is of finite index in /". We
claim that P is net. In fact if xsT (x=#identity) is not net, u(x) is not
net so that u(x)$A or equivalently, x$P = u~1(A). This completes the
proof of Theorem 6.11.
6.13. Corollary. Let G be a connected Lie group which admits a
representation p: G—>GL(V) such that the kernel of p is torsion-free. Let
r be a finitely generated subgroup of G. Then T admits a torsion-free
subgroup of finite index.
According to Theorem 6.11, p(P) admits a torsion-free (net) subgroup
A. Let H be the kernel of p. Then the sequence
e^>Hnp-\A)nr^>p-\A)nr^>A^>e
is exact. Since Hnp~1(A) and A are torsion-free P=p~1(A)nr is
torsion-free. Evidently, P is of finite index in T.
6.14. Corollary. Let T be a finitely generated subgroup of a connected
Lie group G. If every element of T is of finite order T is finite.
According to Corollary 6.13, AdT has a torsion free subgroup of
finite index. It follows that Ad T is finite. Let H be the kernel of Ad.
Then the sequence
e-> tfnr->r-> Ad T->e
is exact. Now H n T being of finite index in T, it finitely generated (see
for instance Kurosh [1], §36, p.36). The group HnT is abelian since
H is the centre of G. Since every element of T is of finite order, H n T
is finite. This establishes the corollary.
Our next result, among other things, guarantees the applicability of
the theorems proved hitherto to uniform lattices in Lie groups.
6.15. Theorem. Let G be a connected Lie group and T a discrete
subgroup such that G/r is compact. Then T is finitely presentable.
We will deduce Theorem 6.15 from the following
6.16. Theorem. Let Vbea compact connected manifold. Then the
fundamental group nx(V,x) of V at a point xsV is finitely presentable.

96
VI. Deformations
Proof. Let p: V—> V be a universal covering of V. Then we may regard
n1{V,x) = r as acting on Vas a group of homeomorphisms, the quotient
for the action being V. The map p then may be regarded as the natural
map of Konto its quotient. Now let {[/;} isl be & finite collection of open
subsets of V such that
(i) for each isl, U{ is connected;
(ii) for each isl, p when restricted to LJ is a homeomorphism of U{
onto its image p{U();
(iii) the union \J p(Ut)= Kand
iel
(iv) if p(t/f)np(Lf.)=#0, there is only one y = yy such that [/;>> n t/j=#0.
Condition (iii) is equivalent to saying that \J Ut T is the whole of V. (We
is/
assume that r acts on V on the right.) Since V is connected given any
pair ijsl, one can find elements i=j0,j1, ...,jk=j such that the
intersection p(Ujr)r\p(UJr+t)±$. We will call a sequence OoJi.--.Jj,) of
elements of / a chain if p(Ujr)np([/,r+1)=#0 for 0gr<fc. We fix once for
all an element i0sl. Also for each isl, we choose a chain (i0, iv ..., it = i)
(k depends on i, of course) and denote this chain by C£. We assume Cig
chosen such that Ch=(i0,i0). Now let
S = {{i,j)\p(tynp(UJ)±Q}.
For each pair (i,j)sS, there exists according to (iv) a unique element of
T, yi}, such that UiyijnUj+^. Now if C( is the chain {i0,iu...,ik = i),
then {ir,ir+1)sS. We set
Let OlJ=riyijTj1. Now if p(t/i)np(ll//)np(t/Jk) is nonempty, then it is
easily seen that
so that it follows that dijdjk = 8ik; in particular 0U is the identity element
e of r and diJ=dji1 for every pair {i,j)sS. Let F be the free group on
the set S. For each ssS we denote by s also the corresponding element
of F. Now for each isl, if Ci=(i0, iu...,ik=i\ let
C|=(»o.iiM<i.*2)----Gk-i.y
in the free group F. Also let
E = {{i,j,k)sIxIxI;p{Ui)np{UJ)np{Uk)±Q}
and for each (i,j, k)sE, let XiJk=(i,j)(J> k)(k, i). Finally for each {i,j)sS
let nij=(i,j)U>i)~1- Let R be the normal subgroup of F generated by

VI. Deformations
97
the Xtjk, nip c,. and (i, i). Let r* = F/R. Now the map (i,j)i-» 0y of S in T
extends canonically to a homomorphism ut: F^T. Clearly u1(XikJ) =
tii(cI)=u1((i, 0) = Ui^) = e- It follows that ut passes to a homomorphism
u: r*->r.
We claim that u is an isomorphism.
We first show that u is surjective. To prove this it is evidently
sufficient to show that {6u\(i,j)eS} generates T. We observe first that V is
connected and that (J p(t/()= V. It follows that given the pair (i0,y)el x F,
iel
we can find elements (i0, y0), (il5 yj,..., ft, yj in Z x r with it = i0, yk = y,
y0=e such that Utryrn [/,r+1yr+1=#0. Clearly we have then for 0^r<k,
Vr + l Jr = Vir + iiV
It follows then that
J~Jk~ Vffc.k-1 Vik- i ik- 2 - - • lii'o
= T"k ^■fcifc-iTifc-iT.fc-i^k-iifc-2Tifc-2---Tii ®hioXio
= "ifc.fc-i^.fc-i.fc-2---^"i,o
since Tio=Tifc = e. Hence u is surjective.
We have now to prove that u is injective. Let W be the union of the
mutually disjoint open subsets L^x(i, y) of Kx(Jxr*). We introduce
an equivalence relation R on Was follows: a point (xh i, y) is equivalent
to {xj,j,y'),i,jel,y,y'er* and x^U,, XjeUj, if p(x^=p(x^ and y=yfJy'
where for (i,j)eS, yf} is the image of (i,j) under the natural
homomorphism it: F^F/R = r*. Since(i,J)(J,i)^R> y*j=y*i~i s° that R is
symmetric; that R is reflexive follows from the fact n{i, i) is the identity
element of T*; finally if (x;, i, y) is equivalent to (xpj, y1) and (xy, j, y') to
(xk, k, y") then p(Xi)=p(xj)=p(xk) so that t/(n l/jn [£#0; it follows that
(iJ)U,k)(k,i)eR, i.e. y^y^y*.. and hence R is transitive as well. It is
easily checked that R is open so that the natural map
f: W^W/R=V*
is an open map. Since in addition V restricted to Ut x (i, y) is evidently
1-1, ¥ is a local homeomorphism of W on V*. Thus K* is a manifold.
We next define a map/t: W—> K by setting
/i(x..i.v)=x.*r1"(y)
where (x(, i, y)e W. Now if (Xy,;, y') is equivalent to (xt, i, y) we have

98
VI. Deformations
/i(x„i,y)=xitr1«(y)
= x,xf1u(YfJf)
= xirrieiJu(Y)
=xtyutjlu(y')
=xjrj1u(y')=f1(xj,j,y').
(Note that if yer is the element such that xiy = xJ, then p([r.)np([//)=#0
so that y=y,j.) Thus/t defines a map/: K*—> K Now the inverse image
under fx of the open set t/. in V consists of the set of points (Xj,j, y)
such that XjtJ1 u(y)eUl. For such an (Xj,j,y), p(xi)=p(Xj) hence
p(t/i)np(t/i)4=0. Thus (Xj,j,y) is equivalent to the point {x„i,yfjy) for
some X(€t/(. Thus the inverse image by f of U{ is the image under f of
the union of the sets
(U,xixy), u(y) = tt.
Moreover ¥ is a homeomorphism on this union onto its image while
fy is evidently a covering map of this union on Ut. Thus / is a covering
map. Clearly u is injective if and only if/ is 1-1. To prove that/ is 1-1,
it is sufficient to show that V* is connected (recall that V is simply
connected and hence has no non-trivial connected coverings).
Since the open sets Ut are assumed to be connected, in order to show
that V* is connected it suffices to show the following. Let (i, y)el x T*;
then we can find a sequence
(»o> «)> (»i> vi)> ••■»(*»»?»)=&y)
such that
nU,rxirxvJr\V(Utr.lxtr+1 xy,+1)*0 for 0£r<k.
Now
nUirxirxyr)nnUir+lxir+1xyr+1)
is nonempty if and only if
Uirxirxyr and Uir+ixir+1xyr+1
contain equivalent points; but in this case, p{Uir)np(Uir+l) is nonempty
and yr+1=yZ+liryr- But then we would have y=yjilt..1yi_lllt_,-yf1i0
with i = ik. Conversely if y has this form, setting yr=y*Ir.. y* _. ir. 2 •. • y* ,•„,
we see that the open sets {Uir xirx yr) are such that
¥{Uir x tr x yr)n V{Uir+1 * «V+i * 7,+i)*0.
Thus to prove that V* is connected it suffices to show that for every
is I any element y* can be written in the above form. Now since T* is
generated by the yfj we can write any element as a product of the yfh as

VI. Deformations
99
(j,h) varies over S. Now C( is the chain corresponding to i; if Cj=
(i0=JoJi>---Jp=J) and Ch = {i0 = h0, ...,hg = h) we have
y.*—y/0 jiyjih---yj,-ij y..* y ■■»■, -, • • • y^/., y*, .0
in view of the definition of fl. Thus writing y as a product of the yjh and
substituting for the yjh from the above formula we see that
y=y/c0/c1 y<ci)c2 •••y/cr/cr+1
where fe0 = kr+1=i0. Once again noting the fact that if C;=(i0, iv..., ip = i)
''p'p-i llp-l'p-2' ' " ' <1 -0 >
it follows that y = e • y is of the form
flklk-l 'Ik - 1 'k - 2 " "" ' 'l 'o
where lk = i, l0 = i0 and (/r/r+1)eS if 0^i<k. This proves that V* is
connected.
This completes the proof of Theorem 6.16: since r*^r, r is finitely
presentable.
6.17. Proof of Theorem 6.15. Let p: G—>G be the universal covering
group of G and let f = p~[(r). Then f is the fundamental group of the
compact manifold G/f and is hence finitely presented (Theorem 6.16).
Let n: F—>t be a surjective homomorphism of a finitely generated free
group onto f with kernel R, R being finitely generated as a normal
subgroup. Let p denote also the restriction of p to f considered as a
map off into r: p: f-*r. Then the kernel A of p is the fundamental
group of G. Moreover if K is a maximal compact subgroup of G, then
K and G have isomorphic fundamental groups. It follows that A is finitely
generated. Now r~tjA^Fjn~l{A). Moreover from the definitions of
R and A, the sequence
is exact. Since A is finitely generated and R is finitely generated as a
normal subgroup of F, n~1(A) is finitely generated as normal subgroup
of F. Thus r^F/n~1(A) is finitely presentable.
This proves Theorem 6.15.
6.18. Remarks. The question of finite generation of non-uniform
lattices is much more difficult to handle. In a later chapter of this book
we will prove that an irreducible non-uniform lattice (cf. Definition 5.20)
in a semisimple Lie group G with at least one rank-1 factor is finitely
generated. Kazdan [1] and S.P.Wang [2] have shown that a lattice in
a semisimple Lie group without rank-1 factors is finitely generated. (As

100
VI. Deformations
the methods employed for this are very different from the techniques
used in this book we will not be proving this result in this book.) As
will be seen later, these two results combined with Proposition 3.8 show
that any lattice in a connected Lie group is finitely generated.
We will now establish a theorem due to A. Weil [2] on uniform
lattices in a connected Lie group. The proof given below however is
different from that of Weil [2]. A third proof can be found in Koszul [1].
6.19. Theorem. Let G be a connected Lie group and r a finitely
presentable group. Let R{T, G) be the analytic set of all homomorphisms of
r in G and let
R0(r, G)={u\ueR(r, G); u: r^>G is injective,
u(f) is discrete and G/u{T) is compact).
Then for each u0€R0(T, G) there exists a neighbourhood Q ofu0 in R(r, G)
and a continuous map F: Gx.Q—>G such that
1) for each ueQ, the map Fu: G—>G defined by setting Fu(g) = F(g,u)
for geG is a diffeomorphism of G on G and
2) for ueQ, geG and yeT, we have
Fu(g-u0{y)) = Fu(g)u{y)
and Fu{e) = e.
In particular for ueQ, u is injective, u(r) is discrete and G/u{T) is
diffeomorphic to G/u0{T).
6.20. Corollary. R0(r, G) is an open subset of R(r, G).
6.21. We first establish Theorem 6.18 in the case when G is simply
connected and deduce the general case from this. Let p: G—>GL(n, R)
be a representation of G with p(G) closed in GL(n, R) and kernel
p discrete. We denote the Lie group p(G) by G'. We make the group r
act on the product P = GxG'xR{r,G) as follows: for geG, g'eG'
ueR(r, G) and yeT, setting u0(y)=y,
(g, g', u)y=(gy, g- • p{u(y)), u).
The action on the left of G' (through left translations on the second
factor) on this product space evidently commutes with the action of r
defined above. We thus obtain an action on the left of the group G' on
the quotient P of P by the above action of r. The Cartesian projection
G x G' x R(r, G)-> G x R(r, G) being compatible with the action of T on
the two spaces defines a map
p: P^G/rxR{r,G).

VI. Deformations
101
It is easily seen that p(g'Q=p(£) for g'eG', and that the above action
of G' makes P into a locally trivial principal fibre bundle on G/r x R(r, G).
Further the diagonal map A: G—> G x G' defined by A (g)=(g, p (g)), it is
easily seen defines a continuous section a0 of the principal fibration
p: P^G/rxR(r,G) over the closed subset G/rxu0. Such a section
can be extended to a section a over neighbourhood U of G/r x u0 in
G/rxR(r,G). Since G/r is compact t/ can be assumed to be of the
form G/rx V where V is a neighbourhood of u0 in R(r, G). Since G is
simply connected it is easily seen that if V is sufficiently small the map
a: G/rx K->(GxG'xR(r, G))/r
can be lifted to a continuous map
a: GxV-^GxG'xR{T,G),
such that for geG, ve V and yeT
ff(g,«o)=(&P(g)>"o)
and
a{gy,v)=a(g,v)y.
Since ct is a section we have
*(g, »)=(& *i(g>"), »)eG xG'x R(r, G)
where geG, i"€ V and
Ft: GxK-^G'
is a continuous function such that for geG, ve V and yeT,
Fl(gy,v) = Fl(g,v)-p(v(y))
and
Jri(g,«o)=Pte)-
6.22. Now, let / be a non-negative real valued C00-function with
compact support on G such that
J/fe)^fe)=l
G
where d\k is a bi-invariant Haar measure on G. (Since G admits a lattice
G carries a bi-invariant Haar measure.) Define F2: Gx V—> M(n, R) by
setting
F2(g,u)=^f(hg-i)p(gh-l)F1(h,u)dix(h)
G
for all geG and ueK Here we regard Ft as a function with values in
M(n, R) through the identification of G' as a closed subset of M(n, R).
The function F2(g, u) is evidently C00 in g for each fixed ueV and the

102
VI. Deformations
partial derivatives with respect to the variables along G are continuous
in both u and g.
Moreover for ge G, y eT and u € V, we have
F2(gy,u)= J/XfcyV1) pfeyfc-ViCMW*)
G
G
= J/Wpft-ViGg, «)(p(«(y)))<fo(*)
G
= f/(fcg-1)p(g^V1(M)p(«(y)Hp(fc)
G
=f2(ft«)'p(«(y)).
Also,
F2fe,"o)=J/(*g-1)pfe*-1)fi(*,«o)^(*)
G
= J/(*g-1)pfe)p(*~1)p(*)^(*)
G
= pfe).
6.23. Now, G' is a closed subgroup of GL(n, R). Let p=codim of G'
in GL(n, R). Then we can find an open neighbourhood D of 0 in Rp, an
open neighbourhood B of G' in GL(n, R) and a diffeomorphism
<P: DxG'^B
such that 4>(0,g)=g
4>(x,gW=#(*,g)-'»
for xeD and g, fceG'. Now let £ be any relatively compact open subset
of G such that Er=G. Since F2 is continuous in (g, u) we can find an
open neighbourhood S2ofu0,S2<=V such that
F2(ExQ)<=B.
Since F2(gy. ") = ^(g,") P • (u(y)) aQd ET=G and B is stable under the
right action of G', we see that
F2(GxG)<=B.
Let n: Dx G'->G' be the Cartesian projection and let F' = no4>~1oF2.
Then F' is continuous in (g, u)eG x K and for each fixed ue V, Cx in g.
Moreover, the partial derivatives of F with respect to variables along G
are continuous in both g and u. Also F(g, u0)=p(g) for geG and for
geG, yeT andueV
(*) F(gy,u)=F(g,u)p(u(y)).

VI. Deformations
103
6.24. The map gi->p(g) is a map of maximal rank. Hence by
replacing Q by a smaller open subset containing u0 we can assume that
for each fixed ueQ, the map
FJ: G^G
defined by F^(g) = F'(g, u) is of maximal rank the relatively compact
set E. Since Er = G, in view of (*) we see that FB' is of maximal rank
everywhere on G. We now claim that FJ is a covering projection for all
ueii. To see this we argue as follows: The set E is relatively compact in G
and FJ is of maximal rank everywhere on E. We can therefore find open
symmetric neighbourhoods W, W of the identities in G, G' respectively
such that
W F^\w^iWx^F^(W~1Wx) is a homeomorphism for all xeE and
(ii) for xeE, F^{Wx)=> W ■ FJfr).
Let geG" = Fl(G). In view of (ii) we see that Fl{G)=>W'F'u(G) and
hence G" = G' i.e. Fj, is surjective. Next if yeG' is any point consider the
set Fi~l{W'y). If zeF.-\W'y\ F^(z)eW'y or equivalent^ yeW'F^z).
Thus yeF^Wz). Now FB maps Wz homeomorphically onto its image.
We can therefore find an open subset Wy'a Wz which is mapped by FB'
homeomorphically onto W'y and contains z. Thus we see that F^l~1(W'y)
is the disjoint union of open subsets each of which is mapped
homeomorphically onto W'y. Hence FB' is a covering projection. Replacing
F'(g,u) by F'(e,u)_1-F'(g,u) we can moreover assume that F,,'(e) =
F'(e,u)=e for ueii.
6.25. Since G is simply-connected and the map F^p'.G^G'
evidently lifts to the identity map of G on G, it follows that each F'u: G->G'
lifts to a unique diffeomorphism Fu: G—>G such that p°Fu = F^ and
Fu(e) = e. Moreover F(g,u) = Fu(g) is a continuous map of Gxfi on G.
Now we have for yer
p F(gy,u) = p(F(g,u) ■ u{y))
so that (/)y(g,u)=F(gy,u){F(g,u)-u(y)}_1ekernel p. Now kernel p is
discrete. Since an analytic set is locally arcwise connected, we can
assume that Q is chosen to be connected. It follows that q>y(g, u) is
constant. Since q>y(g, u0)=e, we conclude that we have
F{gy,u) = F{g,u)-u{y).
This completes the proof of Theorem 6.19 in the case when G is simply
connected.
6.26. We now reduce the general case to the simply connected case.
We assume therefore that G is not simply connected. Let G be the
universal covering of G and p: G—>G the covering projection. Let

104
VI. Deformations
t=p~1{r) and Q0 be the inclusion f <"->& Let Scf be a finite system
of generators for t containing a subset S' which is contained in Z = kernel
p and generates Z. Let q> denote the inclusion S<=f and q> the map
po<J5: S—>r. Let F be the free group on S and S: F—>/* the map induced
by <J5. Let n=pofi. Let K be the kernel of S and T a finite set generating
K as a normal subgroup. Let U be an open neighbourhood of e in (5
such that the restriction p to t/ is a homeomorphism of t/ on the open
set p(U). We identify R(F, G) with J] & by means of the set S and R(f, G)
s
with the analytic subset f]fe~1(e) of R{F,G) as explained at the be-
BcT
gining of this chapter. fe: J~[ G = R(F, G)-> G we recall is the map
s
ui->u(0), ueR(F,G). Now let V<=R{F, &) = J~[ G, u0eV, be an open set
with the following properties s
(i) The map 9 = nP:n^'~>n^' restricted to V is a homeomorphism
s s s
onto the (open) image.
(ii) For t>eKand OeTvS', je(v)eu0(6)- U. Let q(V)=V and Q =
R (T, G) n V: once again we identify R (/", G) with a subset of R {F, G) = J] G
s
as explained earlier using the set S and the map q>: S—>T. Let i5=
{xsV\q{x)€Q}. Evidently q\a: fii—>ft is a homeomorphism. We assert
that Q<=R(r,G) and that we have in addition the following: (i) for
ueQ, u(x)=x for all x in the kernel Z of p and (ii). For usd and yet,
pu(y)=q(u)p(y). These properties (i) and (ii) show that it is enough to
prove the theorems for G in order to prove it for G: in fact by replacing
Q by a smaller neighbourhood if necessary one may assume the existence
of a continuous function F: (GxQ)-*G satisfying the requirements of
the theorem. Evidently F goes down to a map F: Gxii^G with the
requisite properties. This completes the proof of Theorem 6.19.
6.27. Remark. H.C.Wang [1] has shown that if G is a connected
Lie group such that the quotient G/R of G modulo its radical R has no
compact factors, R0{r, G) is (under the identification described in
6.3-6.4) an analytic submanifold []G (S a set of generators for T).
(See also H. Garland [1].) s

Chapter VII
Cohomology Computations
This is a relatively long chapter and is divided into five sections. § 1 and
§ 2 are in the nature of a quick survey developing the notation needed for
the formulation of the well known de Rham-Hodge theory for locally
constant sheaves. The reason for introducing this material here instead
of in the Preliminaries is the following: the notation used for these
results will be needed in subsequent sections; secondly, the de Rham-
Hodge theory is needed for this chapter and only this. It is however
impossible to prove these theorems here—evidently. Though these
results are deep, they are however to be regarded standard material.
§ 3 is just a reformulation of § 1 adapted specially for Lie groups and
their lattices. For § 4, the set up of § 3 is the starting point for obtaining
theorems on the cohomology groups of lattices in solvable Lie groups
due to Mostow [7].
The most difficult part of the chapter is §5. Here the Hodge theory is
specialised to the case of locally symmetric spaces. The exposition of the
differential geometry is essentially the same as that of Matsushima and
Murakami [1]. The machinery is then applied to obtain rigidity theorems
due to Weil [2] for uniform lattices in semisimple Lie groups.
The results proved in this chapter are not needed in the sequel.
1. de Rham's Theorem
Let M be a paracompact connected differentiable manifold and T the
fundamental group of M. Let M denote the universal covering of M
and p: M—>M the projection of M on M. The fundamental group r of M
operates on M and we denote by Ry the operation on M of an element
yeT (on the right). Let p be a representation of T on a finite dimensional
vector space.
p defines a local system (in the sense of Steenrod [1]) Lp (resp. a
vector bundle E(p)) with typical fibres as F with the discrete topology
(resp. F with the Euclidean topology). The local system Lp (resp. the
vector bundle E(p)) is defined as follows: we make r operate (on the

106
VII. Cohomology Computations
right) on M x F as follows:
(x,u)y={Ryx,p(y)-1(x))
for (x,u)eMxF, yeT where F is provided with the discrete (resp.
euclidean) topology; the quotient space Lp=(M x F)/r, F provided with
the discrete topology (resp. E{p)={M x F)/r, F provided with the
euclidean topology) is in a natural way a local system (resp. vector bundle)
over M. Let nL (resp. n) denote the canonical projection of M x F (F with
the discrete topology) on Lp (resp. of MxF {F with euclidean topology)
on E(p)). For each xeM, let cpLx (resp. cpx) denote the linear isomorphism
of F with the discrete (resp. euclidean) topology onto the fibre Lx (resp. Ex)
of Lp (resp. E(p)) over p(x)sM given by
<PLx(u) = nL{x, u) (resp. (px(u)=n(x, u)),
for all ueF. Now let A"(E(p)) be the vector space of all E(/>)-valued C°°
p-forms on M. An element a> in Ap(E{p)) is defined by associating
differentiably to each point xeM an alternating p-linear map cox of the
tangent space TX(M) at x into the fibre Ex at x of £(p).
Now, we can find a covering {Ut}ieI of M by open sets and homeo-
morphisms ^(: [/(xF->/*£ 1(^) (Ml'• Lp-* M) such that the diagram
is commutative and \jjt is an isomorphism on each fibre. (Ut x F is given
the product topology with F being taken to be discrete.) If \k: E(p)^M
is the natural projection, ^ define isomorphisms
UiXF-^n-'iU)
where F is provided with the euclidean topology and hence define a
continuous map gy: t/;n [/,•—>Aut(F) (provided with the discrete
topology). In other words, gtJ is a constant function on each connected
component of Ux n Uj and assuming as we may that t/; n Uj is connected
for all pairs i,;, gi} can be assumed to be constants. Through the
isomorphisms nh then a £(p) valued p-form can be regarded as a collection
{(Wj,- ie/} of F-valued p-forms on Ui such that on t/(n U,-,
(W — gytO;
for every pair (i,j)elxl with t/(nLr=#0. Since gy are constants, it is
clear that if {a>t; iEl} is such that on t/;n Uj
Oj = gij<»i>

1. de Rham's Theorem
107
then {ri^dcoi; iel} again satisfy on t/(n Uj, the condition
Here d denotes exterior differentiation. We thus obtain an operator
again denote d on LI^4''(£(p)):
d: A"{E(p))^A"+1{E(p)).
Let iip(E(p)) denote the sheaf of germs of E(p) valued p-forms on M and
let £?{p) denote the sheaf of germs of continuous sections of Lp. Then
we have
7.1. Lemma. JS?p is a locally constant sheaf and the sequence
0^2>ii^a>{E(p))-U...-U&{E{p))^-
is exact (d is a local operator and hence defined on the sheaves).
This is simply the Poincare lemma: since the question is local, we
can assume M to be simply connected and we are reduced to the Poincare
lemma since Lp in this case is trivial (for a proof of the Poincare lemma,
see for instance C.H. Dowker [1], Lecture 17).
The de Rham theorem for sheaves (see for instance C. H. Dowker [1],
Lectures 14-16) now gives
7.2. Lemma. The k-th cohomology group of the complex (\±Ap(E(p% d)
is isomorphic to the k-th cohomology group ofM with coefficients in S£p.
Next we identify the complex 0^4p(£(/>)) with a complex of C00
forms on M, Let a>eA"(E(p)) be any element; we define a F-valued form
co' on M as follows: for x e M and tangent vectors Lt,..., Lp to M at x,
o)'x{Lu ..., Lp) = q>;1 w^piLi),..., p(Lp)).
It is then easy to see that the map (wi-xw' defines an isomorphism of
Ap(E{p)) on the space A"{T,M,p) of F-valued C°° p-forms n on M
satisfying Ryr,=p{y)-1r,
for all yeT. Under this identification the boundary operator goes over
to the operation of exterior differentiation. It follows from Lemma 7.2
that we have
7.3. Lemma. The k-th cohomology of the complex (LI/lp(r, M, p), d)
described above is canonically isomorphic to Hk(M, £fp).
In the sequel the p-th cohomology of the complex LI/lp(r, M, p) is
denoted H"{r, M, p). Our interest will be mainly in the case when M is
contractible. In this situation a theorem due to S. Eilenberg [1] says the
following:

108
VII. Cohomology Computations
7.4. Lemma. IfM is contractible H"(r, M,p)^H"{M, &p) is naturally
isomorphic to HP{T, p) the p-th cohomology group of r with coefficients
in p. Moreover let TV be another Cx manifold with fundamental group F
and the universal covering TV contractible. Let q>:F—>rbea homomorphism
and <P: TV—>M a differentiable map such that <P(xy)=<P(x)q>(y)for xeN
and yeF. Let <P*: A (r, M, p)-> A (/"", N,poq>) be the map (of complexes)
induced by taking inverse image of forms by <P. Let <P* also denote the
induced map in the cohomology groups: <P*: HP{T, M, p)—> H"(F, TV, p o q>).
Let q>*: HP{T, p)—>HP{F, p °q>) denote the homomorphism induced by q>:
r^F. Then the diagram
Hp{F M, p) -^ H"(F, ft,po(p)
l\ l\
H"{r,p)^^H"{F,pocp)
is commutative.
2. Hodge's Theory for Local Systems
The results of § 2 are often not effective enough as means for computing
cohomology groups. For this reason we will here outline some of the
more sophisticated machinery needed—Hodge's theory of Harmonic
forms for locally constant sheaves.
We continue with the notation of § 1. We now assume that M is an
oriented riemannian manifold of dimension TV, say. Let p* be the
representation contragredient to p and E(p*) the associated bundle. Then
£(/>*) is canonically isomorphic to the dual £(/>)* of the bundle E(p); the
fibre Ex at xeM of E(p) is canonically dual to the fibre E* at x of £(/>*).
We denote
(u, u*)i-><u, u*>
this canonical pairing given by this duality. Suppose now that ^ (resp. r\)
is an infinitely differentiable E(p) (resp. E(p*)) valued p form (resp.
q-form). Then we define a scalar (p+q)-form £, ah as follows: let U be a
coordinate open set on M and let a1,...,ali be 1-forms on U which span
the dual of the tangent space to M at any point of U; then we can write
(7.5) £= E ukw
ii< — <lP
and
«j> 'i
1 Aa;
(7.6) n= E v
h<-<J,
h u«hA-AH

2. Hodge's Theory for Local Systems 109
where «;.j (resp. vh jq) are C00 sections of E(p) (resp. £(/>*)) over U;
then £, a r\ is the scalar form
(7-7) E <"., «,,»/, ^KfcA-Aa^Aaj.A-Aaj,
on the open set t/.
Now assume given a metric along the fibres of E(p) i.e. we are given a
positive definite scalar product ax on Ex for each xeM such that for a
C00 section a over an open set U, ux(<r(x), <t(x)) is a C00 function on [/.
Such a metric gives an isomorphism
#: E(p)^E(p*)
defined by setting
<mx, #vx>=ax(ux,vx)
for ux vxeEx, xeM. This isomorphism extends naturally to an isomorphism
again denoted # of A"(E(p)) on A"(E(p*)):
#: A"{E{p))^A"{E(p*)).
Finally, the riemannian metric and the orientation enable us to define
amap *: A>(E(p))^A"->(E(p))
as follows: if I;eAp(E{p)) is written in the form (7.6)
*f= E "i, f,*(«£,A —A«1f)
where on scalar forms the operator * is defined as follows: let {L.Jig.gjv
be an orthonormal basis to the tangent space at a point xeM; assume
that L1a---aLjv is positive with respect to the orientation on M;
then if Pi is defined by /?,(£,)=<5,y
where p+q = N and j\ jq is the set [1, ATJ —(it ip) arranged in
increasing order and g = +1 or — 1 according as
fr,A-•A&j)A^....0j<i = 01A---Aftv or -/?,a-a/?w.
From the definitions it is easy to see that for £eAp(E(p))
**Z = (-l)'W-p>Z.
For £, r\ sAp{E(p)) when one of them has compact support (or more
generally if the integral is finite) we set
K,!f)=J#(*flA!,.
M

110 VII. Cohomology Computations
If we denote by dm the volume form of the riemannian metric, we have
(Z,n)=f<Z,tl>dm
M
for a suitable C00 function <£,»/> on M. If £,,r\eAp(£(/>)) are in the
forms (7.6) and if in addition, {at, ...,ctN} is orthonormal, we see easily
that
(7.8) «,fj>(x)- E ax(uk ip,vu ip).
h<-<ip
We now define an operator 8: A"(E(p))—*Ap~1(E(p)) by setting
for £eA>(E[p)\
&Z={-lf'+"+\*)-l*-ld*(*)Z.
It is not difficult to see that if £eAp(E(p)) and rieAp+ ' (£(/>)) and one of
them has compact support, then
(7.9) (<*{,!»)=«;,*!»).
We define the Laplacian A as the operator
A = d5+5d.
A is a (degree preserving) differential operator on Ap(E(p)). A form
£eAp(E(p)) is harmonic if A £ = 0. Moreover if £, is a C00 form with compact
support we have
(AZ,ri)={dt,dn)+5Z,5Ti).
In view of (7.8) (d£, d£)and(dl;, ^are non-negative. Thus if M is compact,
£ is harmonic if and only if
<f£ = 0 and <5f=0.
A straight forward imitation of Hodge's theory for the usual cohomology
groups (see for instance de Rham [1]) leads to the following result.
7.10. Proposition. // M is compact, every closed form £ (i. e. a form
such that d£ = 0) in Ap(E(p)) is cohomologous to a unique harmonic form.
Consequently HP(M, &p) is canonically isomorphic to the space of harmonic
forms in A"(E(p)).
7.11. Corollary. Suppose M is compact and that for some p we have
a constant c>0 such that for every £eAp(E(p)),
{AZ,Q=(dZ,dQ+(8S,8Z)*c(Z,Q.
Then Hp(M,Lp)=0.

3. Discrete Subgroups in Lie Groups
111
Proof. It suffices to prove that any harmonic form £, in A"(E(p)) is
zero; but this is evident from the fact that A£=0.
The information that can be elicited from an inequality of the type
in the corollary when M is non-compact is limited. A weak vanishing
theorem that can be obtained is the following result due essentially to
Andreotti and Vesentini [1].
7.12. Proposition. Suppose that M is complete. Assume that there
exists c>0 such that for every £eAp(E(p)) with compact support, (A£, £)^
c{£,ti,). Then every closed form £eAp(E(j>)) such that {£, £)<oo is a
coboundary i.e. there exists n€A"~1(E(p)) with dn = i>.
7.13. Definition. A form £ eAp(E(j>)) is square summable if
«,0=J{A(*#{)<00.
Af
Proposition 7.12 can then be reformulated as follows:
Suppose that M is complete and that there is a constant c>0 such
that for all £ eAp(E(p)), with compact support, (J &£).£c (£,£). Then
every square summable closed form £eAp(E(p)) is a coboundary.
Remark. Evidently Corollary 7.11 is an immediate consequence of
Proposition 7.12.
3. Discrete Subgroups in Lie Groups
We will now specialise the results of § 1 and § 2. We will take for JWT a
space obtained as follows: Let G be a connected Lie group and K a
closed connected subgroup. Let T be a discrete subgroup of G such that
rngKg~1 = (e\ geG: r acts fixed-point free on the homogeneous space
K\G. Assume that this action is properly discontinuous. Let X=K^G
and the quotient X/T of X by T will play the role of M in § 1 and § 2. We
thus set X/r=M. We assume further that X is contractible. Evidently X
plays the role of JWT.
Let &0: G->K^G, n0: G-* G/r and n: X-+XJT be the natural maps.
Let g be the Lie algebra of right invariant vector fields on G. Then every
Ysq is projectable under n0 and the association to each Ysq of its
projection on G/r identifies g with a subalgebra of vector fields on G/r.
In the sequel we will thus treat elements of g as vector fields on G/r as
well as on G.
Finally let I be the Lie subalgebra of g corresponding to K. Assume
that the action of K on g is completely reducible. We can then find a
K-stable subspace p of g such that g=I©p. We fix this subspace once
and for all.

112
VII. Cohomology Computations
Next let p be a finite dimensional representation of G in a (real) vector
space F. We let p stand for the induced representation of g as well; for
Yeg, if q>t is the 1-parameter group corresponding to Y,
pO0«>=-"^-pfa>iHr-o-
Let E(p) be the vector bundle on M=XjT associated to p. (We let p
stand for p/r as well.) We have defined E(p) as the quotient of X x F
by an action of T. Now X itself is the quotient of G by K. Thus E(p) is the
quotient of G x F by the action of K x r as follows:
(&/)(*, vH*gy,p(y)-7)-
Let & denote the natural map (g,f)h->(&0{g), f) of GxF on XxF. For
geG,q>g is the isomorphism of F on the fibre over it o w0(g) defined by
(pg{f)=n°a> (g,f)
where n denotes also the natural map X x F-> E(p).
Now for an F-valued p-form £eAp(r,X,p), let £' be the F-valued
p-form on G defined by „ , w„ » ..
£ = p(s)({ oct),,),
for all seG. It is easy to see that £.-><!;' is an isomorphism of A"(r, X, p)
on the space of all F-valued p-forms r\ on G satisfying
(a) r\oRy=r\ for yeT
(7.14) (b) r\oLk = p{k)on for fceK
(c) i(Y)fr=0 for Yet.
Here for a p-form r\ (with values in F) i(r)»/ is the F-valued (p— l)-form
on G defined by i(Y)if(Z„ ....Z,.^^,, ...,Zp_,) for all Z„...,
Zp_t€g; also for yeT, /?v (resp. Lk) denote the right (left) translation
by y (resp. k). In view of (a), y\ may indeed be regarded as an F-valued
form r\' on G/r. We denote by A%(r,G,K,p) the space of F-valued
p-forms r\ on G/r such that
00 n°Lk=p{k)n for fc€K
(c') i(Y)if=0 for yel.
Here Lt is the left action of k on G/r and 7 is regarded as a vector field
on G/r. The map £.-><!;' defines thus an isomorphism of Ap(r, X, p) with
AP0(T, G, K,p) which we denote <P. We choose an orthonormal basis
Yj,..., YN of p. Now in view of (b') an element ifsA^T, G, K, p) is
completely determined if we know the functions (on G/r)
< i,=n\\,-,\)
for all iu ..., ip. We have then

3. Discrete Subgroups in Lie Groups 113
7.16. Lemma. Let dp be the operator defined by the commutativity of
A"(r,X,p)^^A"+1(r,X,p)
the diagram Aplr v ^ d
Al{r,G,K,p)—±-+Al+l{r,G,K,p).
Then for n° sAp0(r, G,K,p) and Zt Zp+1eg, we have
(dynz, zp+l)
,717, = E(-i)"+1(z..+p(W(Zi.--->z -zP+i)
+ E(-irV([z.,zj,z1,...,2. zv,...,zp+1).
In particular,
(d„n°) .,♦,
= i(-i)u+1(Yiu+p(Yij)nl ?u ip+1
p
(7.17)'
+ Yl{-lf+vn°{LYiu,YiJ,Yii,...,Yi)t,...,Yiv,...,YlpJ.
Proof. From our definitions, we have n°=p ■ (»/ o d>0) here we regard p
as a function with values in End F). It follows that we have
dn° = dpA{ri°a>0)+p- d{n o co0)
= dp a (n o dj0)+p (dn o (30).
Now for leg if q>, is the 1-parameter group of Y,
dp{Y)(g)=-^p((ptg)\t=0=-j^p(<pt)\t=0p(g)=-p(Y)-p(g).
It follows that we have
dpn°(Z1,...,Zp+1)=dri0(Z1,...,Zp+1)-dpA(rioa>0){Zu...,Zp+1)
= "i\-ir+1Zur,0{Z1,...,Z Zp+l)
+ 2j( — 1)"+"'/ ([Z,,, Z„], Zj, ...,Zb, ...jZ^, ...,Zp+1)
u<v
+ "J:\-irdp(Zu)p-1-r,0(Z1,...,Zu,...,Zp+1)
U-l

114
VII. Cohomology Computations
='i;1(-ir,z.ffo(z1 zu,...,zp+l)
+ Pi{-l)u+1p{Zu)r,°{Z1 Zu,...,Zp+1)
+ E(-i)B+,,'/0([zB,zj)z1,...)zI„...,z„)...,zp+1).
This proves (7.17). Evidently (7.17)' is an immediate consequence of (7.17).
Now let F* denote the space of F-valued C00 functions on G invariant
under the right action of r. We now define a representation p* of g on F*
using the representation p as follows: fovfeF* and Ysq,
p*(Y)f=Yf+p(Y)of
(here Yf is the derivative off with respect to the right-invariant vector-
field Y; because of the right-invariance of Y, Yf is again invariant under
the right action of T; also in writing p(y)o/we treat p(Y) as a function
from F* into itself: p {Y): F*—>F*). Evidently we can consider elements of
AP0(T, G, K,p) as alternating F*-valued p-forms on g in a natural way.
Moreover since K is assumed to be connected, this identification defines
in view of (7.15) an isomorphism of the complex (A0(r, G, K, p), dp) onto
the standard complex A (g, I, p*) of g relative to I with coefficients in p*.
Combining this with Lemma 7.4 (§ 1), we obtain
7.18. Proposition. Let G be a connected Lie group and K a connected
closed subgroup such that K\G is contractible. Let g be the Lie algebra
ofG and leg the subalgebra corresponding to K. Let r<=G be a discrete
subgroup of G such that r acts properly discontinuously and fixed-point
free on K\ G. Let p* be the representation o/g on the space F* of F-valued
Cx functions on G invariant under the right action ofT defined as follows:
for Xsq andfeF*, p*{Y)f= Yf+p{Y) of Then the complex A0{r, G, K, p)
is canonically isomorphic to the complex A(q, I, p*). In particular we have
H"{r,p)^H"{Q,t,p*).
7.19. Remark. The hypotheses of the proposition hold in the following
cases.
1. G any connected Lie group, K a maximal compact subgroup and T
a discrete subgroup which is torsion-free (cf. Preliminaries § 1.8).
2. G a simply-connected solvable group K=e and T any discrete
subgroup of G (this is a special case of 1) (cf. Preliminaries § 1.8).
3. G = K ■ U a semidirect product of a simply-connected closed nil-
potent normal subgroup U and a closed abelian subgroup K acting
reductively on U and fc U, a discrete subgroup.

4. Solvable Lie Groups
115
Thus Proposition 7.18 holds for all the triples {G,K,r) as above.
Moreover since the isomorphism described in the theorem is natural we
obtain in view of Lemma 7.4 (§ 1) once again the
7.20. Corollary. Assume that G = K-U as in (3) (7.19) above. Suppose
further that G is simply-connected. Then the diagram
H"(r,P)
H"(q,1,p*)—^^H"(q,P*)
where i„. is induced by the natural inclusion is commutative; in particular i^
is an isomorphism.
The case when G is semisimple and K is a maximal compact subgroup
is of special interest to us. Here we have
7.21. Proposition. Assume that G is connected linear and semisimple
and that K is a maximal compact subgroup of G. We take p to be the
orthogonal complement of I in g with respect to the Killing form. Then we
have for aeAl{T, G, K, p),
(7-22) (dpa)h ,,+I='i;1(-ir,(y<ll+p(iu)«i1 ?„ ,,+I.
i=I
Proof. We need only observe that since [Z,Z']€l for Z.Z'ep, the
second term on the right hand side of Eq. (7.17) is zero.
4. Solvable Lie Groups
In this section we sometimes use the following notation. If g is a Lie
algebra and p is a representation of g on a vectorspace F we let Hp(g, F)
stand for Hp(q, p) (when this is not likely to cause any confusion).
Let G be a simply connected solvable Lie group and r<=G a discrete
subgroup. Let A be an abelian normal subgroup of G which is closed
and connected. We assume that A/AnT is compact. Let p be a finite
dimensional representation of G in a vector space F. Let F (resp. F")
denote the space of F-valued C00 functions on G which are invariant under
the (right) action of T (resp. A T). Then F and F are modules over g (the
Lie algebra of G) in a natural manner. Moreover we evidently have an
inclusion
i: F'^F
of g-modules. In this situation we will now prove

116 VII. Cohomology Computations
7.23. Proposition. The map i induces an injection
i*: H"(a,F')^-H''{a,F).
If in addition p is unipotent on A, i* is an isomorphism.
Proof. We define a g-module homomorphism/: F-+F'- This is done
as follows: let/ef; then for xeG
J(f)(x)- J f(xa)da
A/Anr
where da is the Haar measure on A/A n T normalised such that
A/Anr
and the {A n r)-invariant function a i->/(x a) on A is treated as a function
on A/AnT. We claim that j(f)eF'- In fact if be A,
J(f)(xb)= J f(xba)da= J /(xa)<te=J(/)(xa)
^nr A/Anr
since da is translation invariant Also if yeT
J(f)(xy)= J f{xya)da= J f(xyay~l)da.
A/Anr A/Anr
Now the automorphism a i-> ya y~1 of A, we claim, is measure-preserving:
in fact, identifying A with its Lie algebra o (since G is simply connected,
so is A), we see that y leaves stable a lattice in the vector space o and must
therefore be measure-preserving; the transformation a\-+yay~l is also
orientation-preserving since yeG and G is connected. Thus we have
j(f)(xy)= J f(xyay-1)da= J f(xa)da=j(f)(x).
A/Anr A/Anr
Hence j(f)eF'- Also for any geG, we have
{Lgj{f))(x)=j(f){g-1x)= J f{g-1xa)da= J (Lgf)(xa)da=j(Lgf)(x).
A/Anr A/Anr
Thus j is a homomorphism of G-modules and hence of g-modules.
Clearly the composite map ;oi: F'^F is the identity. Thus i* is an
injection. This proves the first assertion.
Consider now the second assertion. We make a preliminary reduction.
Suppose 0 —> £j —> F —> E2 -* 0 is an exact sequence of G-modules and if the
mapS IV fl'(o,£',Hfl'(o,1fi1)
i$: H"(a,E'2)^H"{a,E2)

4. Solvable Lie Groups
117
are isomorphism, then so is the map i* since the sequences
(>-»£,-»*■•-»£2-»0
and
0-»£',-»<P-»J3-»0
are clearly exact. Moreover A acts unipotently on F if and only if it acts
unipotently on Et and E2. We are thus reduced to proving the proposition
in the case F is an irreducible G-module. Moreover one sees easily that it
suffices to consider representations over complex numbers. Since G is
solvable we are thus reduced to the case of 1-dimensional representations
of G trivial on A.
Since A is abelian and simply-connected we may identify A with R"
by an isomorphism carrying A n T into the standard lattice Z". For fixed
xeG and fsF (Note that/is a complex valued function), the function fx
on A defined by fx(a)=f(xa) is clearly periodic for the lattice Z". We have
thus a Fourier expansion
fx(a)=f(xa)= E Ca(x) exp27ii<ot,a>
«eZ»
where
Q(x)= J /C*a)exp — 2ni(u.,dyda.
A/Anr
Now since f(gy)=f(g) for gsG and yeT, we have
X Ca{x)exp2ni(x,ay=f{xa)=f{xay)=f{xyy-1ay)
aeZ"
= E Ca(xy)exp27ti<a,y_1ay>
aeZ"
= E Ca(xy)exp27ci<V*-1(a).a>
asZ"
where y* is the transpose of the linear transformation a\->yay~l of R".
Equating coefficient of exp27t i<y*(a), a>, we thus find that for aeZn
cc«(^y)=c),.-i(a)(x),
for all xeG. Clearly/€/ is in the kernel of/ if and only if
C0(x)= J /(xa)da
is zero. Now let Xt be the right invariant vector field on G whose restriction
to A corresponds to d/dx1 on R". Then we have for/eF, XifsF (since A
acts trivially on F) and if xeG,
d .
Xif{xa)=—f{<ptxa)\t_0

118
VII. Cohomology Computations
where q>t is the one parameter group corresponding to Xt. Thus
{Xif){xa)=—f((ptxa)\t^0=—f{xx-1(ptxa)\t=0
d
=—{ £ Ca(x)exp2rei<a.^ V.*«>}!.=
dt a6Z»
o
= -rr{ I Ca(x)exp(23Ti<a)a> + <x*-1(a))(p,»}(=o
= - E c«(x) exp 27t i <a, a> • <x* "'(a), e;>
ieZ"
where ef is the i-th element of the standard basis of R". Repeating the
calculation, we obtain,
(X?f)(xo)= I Ca(x)<x*-1(a),ei>2-exp27d<a.a>.
aeZ"
Thus if Ml =£ vf for r = (r„ .... r„)eR' and J = £ A"?, we have
i=I
(J/)(xa)= X Ca(x)||x*-1(a)ll2exp27CKa,«>.
aeZ"
Now, let K be the kernel of;. Then K is stable under A. Moreover if
feK, we have Co(x)=0 so that
f(x a)= X c«(x) exp 2 7t i <a, a>.
asZ»-(0)
For feK, let ^/ei"" be defined by setting for xeG
^/(xa)= E Ca(x)/||x*-1(a)||2-exp27ri<a)a>.
aeZ»-(0)
We must of course justify this: first, the series converges for each x
absolutely and uniformly in a since /is C00, (hence) the series £ QM
asZ»-(0)
converges absolutely and the ||x*_1(a)||, a=#0, remain bounded below for
aeZ". Secondly if xa=yb, x,yeG, a, beA, we have x—yba~l so that
9f(xa)= £ {Ca(x)/||x*-1(a)||2}exp27u<0(,«>
asZ»-(0)
= E {Ca(x)/||y*-1(a)||2}exp27C»<a,a>,
aeZ»-(0)
since ba~l acts trivially on R". Moreover we have
Ca(x)= J /(xr)exp-2.ri<<M>^
A/Anr
= J f (yba'11) exp-2ni(a,t}dt
A/Anr

4. Solvable Lie Groups
119
= I f{yt)exp—2ni(ci,ab 1tydt
AlAnT
= J f(yt)exp(tx, t) ( — 2ni)- {exp — 2.r.<a,ab~1)}
A/Anr
= Cx{y) exp — 2 n i <a, a b_1 >
so that
3/(xa)= I Cx{y)/\\y*-1(^W2^P27ti{oi,b} = ^f(yb).
U6Z"-(0)
Thus ^/is a well defined function on G and is easily seen to be of class C00.
We assert that @fis /"-invariant. For
S/(*y)= I Cct(xy)/||x*-V-1(a)ll2
U6Z"-(0)
= I Cy.-I(.)(x)/||x*-1y*-1(a)||2
oteZ»-(0)
= £ c^w/iix* -1 (^)|| 2=sr/(x)
06Z»-(O)
(since the series is absolutely convergent). Thus ISfeF. From the
definition it is clear that for/e/C,
A9f=f, 9Af=f.
Thus A: K—*K is an isomorphism of a-modules (note that A is abelian
so that A commutes with a). Consider now the exact sequence
0->K->F^Uf'->0.
To show that j* (and hence i*) induces an isomorphism in cohomology
it suffices to show that H"(a,K) = 0 for all p^O. Now let U be the
enveloping algebra of a. Then by definition (cf. Preliminaries § 3).
//"(ct.K)=Ext(-I(C,J«i:)
where C is considered as a module over U through the trivial action of a.
Now AeU acts as an automorphism of K and hence for the natural
[/-module structure on Ext£(C, K) (note that U is commutative), the
homothesy by A is an isomorphism. On the other hand A C = 0 so that A
acts trivially on Extf,(C,K). Hence H"(a,K) = 0 (cf. Preliminaries §3.1).
7.24. Definition. Let G be a connected simply-connected Lie group
and rcGa lattice. Let p be a finite dimensional representation of G on
a complex vector space F. Let Ad denote the adjoint representation of G
on its Lie algebra g as well as the complexification gc of g. We will say
that the representation p is I'-supported if p (r) and p (G) have the same
Zariski closure in Autc(i5). The representation p is r-admissible if p © Ad
(on F © gc) is T-supported.
Evidently a r-admissible representation is T-supported. The following
observations are immediate consequences of the definition.

120
VII. Cohomology Computations
1.25. Remarks. 1) If p is T-supported so is any quotient of a sub-
representation of p.
2) If p is T-supported so is every tensor representation obtained from
p and its dual p*.
3) If N is a connected normal closed subgroup of G such that under
the natural map it: G—*G/N, 7t(T) is discrete, then a representation p of
G/N is 7t(T)-supported (resp. 7t(T)-admissible) if pore is T-supported
(resp. T-admissible).
4) Let {pi}ieI be a finite family of quotients of G-submodules of the
(mixed) tensor representations of a representation p (resp. of p © Ad)
where p is a T-supported (resp. T-admissible) representation. Then LIP|
is a T-supported (resp. T-admissible) representation.
5) Combining (3) and (4) we see that we have the following result:
let G be a simply-connected (connected) solvable Lie group and r<=G
a lattice. Let N be a closed connected normal subgroup of G such that
rN is closed in G. Let n: G-> G/N be the natural map and Tt = 7t(T). Let n
be the Lie algebra of N identified with a subalgebra of g. (n is in fact an
ideal in g.) Let p be a T-admissible representation of G. Then the natural
representation of G/N on the cohomology group Hp{n,p) is
inadmissible. To see this we note that Hp(n, p) is a quotient of a G-sub-
module of LI Horn (.4* n, F) and apply (4) above.
k
We are now in a position to prove the following result due to
Mostow [7].
7.26. Theorem. Let G be a connected simply-connected solvable Lie
group and r^G a lattice. Let p be a finite dimensional r-admissible
representation in a complex vector space F. Then HP{T, p)^Hp{Q, p) where
p is used also to denote the representation o/q induced by p: G-* Autc(F).
Proof. Let F denote the space of F-valued functions on G which are
invariant under T. Then we have seen (Proposition 7.18, § 3) Hp(r, F)m
Hp(q,F), where F is considered as a g-module via p: for Xe&feF,
p-{X)f=Xf+p{X)-f
Clearly regarding F as constant functions on G, we have an inclusion
i: F^-*F
of g-modules. Thus we have to prove under our hypothesis on p that i
induces isomorphisms on the cohomology groups.
Suppose now that E^-*F is a g-submodule and E/F is denoted A,
we have an exact sequence of G-modules
0-> £,-!-► F-H-^-^o

4. Solvable Lie Groups
121
and a corresponding sequence of spaces of vector-valued functions:
Evidently the diagram
0-+E-+->F-!L^A-+0.
0-+E~^>F-*-+A-+0
0->E >F >A-+0
is commutative. Moreover the representations of G obtained in E and A
are clearly r-admissible. A simple induction argument using the five-
lemma thus reduces the proof to the case when p is irreducible. Since G
is solvable we may thus assume that p is 1-dimensional.
We will argue by induction on the length of the derived series for G.
First consider the case when G is abelian. If p is the trivial representation,
the result follows from Proposition 7.23 above (taking G = A). Suppose
then that p is nontrivial and irreducible. Then the representation p of g
is irreducible as well. Moreover since p(f) and p{G) have the same
Zariski closures, p\T is non-trivial as well. Now let yeT (resp. Xsq)
be any element such that p (y) 4= Identity (resp.ppQ=#0). Let A (resp. U)
be the group algebra of T (resp. the enveloping algebra of g). We have
considering C as a A (resp. U) module through the trivial representation
of r (resp.g) on C, H>(T, p)d==fExtjJ(C,F) (resp./F(g.p)d=lfExt&(C.F)).
Now ExtJ(C, F) (resp. Ext£(C, F)) is a module over A (resp. U) (note
that A and U are commutative) (cf. Preliminaries §3.2) the element
{y—l)eA (resp. XeU) acts on F as an automorphism of F. It follows
that the homothesy defined by y — 1 (resp. X) on H"(r, p) (resp. H"(q, p))
is an isomorphism. On the other hand y— 1 (resp. X) is the operator 0
on C considered as a A-[iesp. t/-)module. Thus this homothesy must be
trivial as well. This shows that Hp{r,p) (resp. tfp(g,p))=0 thereby
proving the theorem when G is abelian.
We now consider the general case. Since p is r-admissible, Ad is
T-supported. It follows that if {Dk(r)} is the derived series for T and
{Dk{G)} that for G, then Dk{G)/Dk{r) is compact (cf. Theorem 2.1,
Chapter II; note that [G, G] is a nilpotent Lie group).Thus Dk (G)/-Dlt (G) n r
is compact for all k. Let r be the integer such that Dr(G)=A+e while
Dr+1(G)=e consider now the inclusion
i: F^-*F
of g-modules. Let o be the ideal in g corresponding to A. We then have
the Hochschild-Serre spectral sequence for the two modules and a
homomorphism of these spectral sequences induced by i.

122 VII. Cohomology Computations
£2(0: fl'(a/o, H«(a, F)H Jf(a/a, /^(a, £)).
Consider now the module F' introduced in Proposition 7.23. We then
have a commutative diagram
F—i—>£
\/
F'
We have then a corresponding diagram of spectral sequences
Jf'(a/a, //"(a, J0)-^U/f'(a/o, Jf(a,£))
/f'(a/o,/f(o,r))
Now it evidently suffices to show that £2 (0 is an isomorphism. According
to Proposition 7.23, E2(i0) is an isomorphism: in fact H9(a,F)-> H9(a, F)
is an isomorphism. Thus it suffices to show that E2(J) is an isomorphism.
Now F' consists of all C00 functions / on G with values in F such that
f(gx)=f(g)
for all xeAr. F' may be regarded as the space of F-valued functions
G/A invariant under 7t(F) where n: G-> G/A is the natural map. This is
moreover compatible with the action of G on F' as a subspace F on the
one hand and as the space of F-valued CMunctions on G/A: for an
F-valued C00 function / on G/A, the action of Xeg is given as follows:
let n{X) be the projection of X on G/A; p(X)=0 for Xea since p is
necessarily unipotent on^c^ (G) (if r > 1); let pv denote the
representation of G/A on F obtained in view of this fact; then
P1(X)f=n(X)f+p1{n(X))-f-
Now from the definition pt it is clear that as a G/4-module,
Hq{a,F')^Hq(a,F),
the space of if7 (a, F)-valued functions on G/A invariant under n(r). Now
H"(a, F) considered as a G/A -module, it is easily seen, is 7t(r)-admissible.
Since G/A has smaller derived series length than G, it follows from
induction hypothesis applied to G/A and H"(a, F) that E2 (J) is an
isomorphism. Thus E2(i) is an isomorphism. We see therefore that i induces
isomorphisms in cohomology. This proves the theorem.
7.27. Corollary (van Est[l]). If G is a nilpotent simply connected Lie
group, fcG a lattice and p a unipotent representation of G, then Hp(r, p)

5. Semisimple Groups (Weil's Rigidity Theorem) 123
is isomorphic to Hp(q, p) where p is used to denote also the representation
of g induced by p.
7.28. Corollary (Nomizu [1]) (see also Matsushima [1]). // G is nil-
potent and simply connected and r^Gisa lattice then
H"{G/r,R)^H"{Q,R)
where Hp(G/r, R) is the p-th singular cohomology group of G/r with
coefficients in R;for the definition of Hp(q, R), R is considered as a module
over g via the trivial representation.
7.29. Corollary. If G is simply connected and solvable and r<=G is a
lattice, such that AdT and Ad G have the same Zariski closures Autc(g),
then
H"(G/r,R)^H"(Q,R)
(notation is self evident—suggested by Corollary 7.28).
7.30. Remark. It is fairly easy to give examples of (solvable) Lie
groups G when the map i*: H"(q, F)-+H"(q,F) induced by i: F-+F is
not an isomorphism. However, i* is always an injection. In fact the
averaging process defines a g-module projection n: F-+F as follows:
for feF, xeG,
nf(x)= \f{xg)dg.
G/r
(Here the function g i->/(x g) is r-invariant and hence may be regarded
as a function on G/r; dg is the Haar measure normalised in such a way
that G/r has volume 1.) Thus we have the inequality
dim H"(r, F)^dim Hp(q, F);
in particular if bp is the p-fh Betti number of G/r,
bp^dimH"{Q,Q.
Thus a knowledge of the cohomology of the Lie algebra gives us
some information about topological properties of G/r and also about
the cohomology of the group T in general in the case when G is solvable.
5. Semisimple Groups1 (Weil's Rigidity Theorem)
We now further specialise the considerations of §§ 1-3 to the case when G
is a semisimple linear Lie group and K is a maximal compact subgroup
of G. As in § 3 we assume that r has no torsion. We use the notation
of §§1-3.
' This account borrows heavily from Matsushima and Murakami [1],

124
VII. Cohomology Computations
We will first choose the supplement p to t in a canonical manner
and further introduce a natural metric along the fibres of E(p). On
K^G=X we have a G-invariant riemannian metric which defines a
riemannian metric on X/r as well. Thus considerations of § 2 can be
applied and we proceed to describe this in some detail.
G being semisimple, the Killing form A( , ) of its Lie algebra g is
well known to be nondegenerate and negative definite on f. We may
thus choose
p = {yeg;4(y.Z)=0forallZeI}.
Then we have
g = I©p, lnp = (0), [I,p]c:p, [p,p]c:I,
and A is positive definite on p (cf. Preliminaries §§1.5,1.6). Thus if 6: g->g
is the involution X + Y\-+X-Y(Xei, Yep) the form (X, Y}g= -
A(X,6(Y)) is positive definite on g. As in §3 let w0: G->J«C\G be the
natural map and let x0 = co0(e), e being the identity of G. Then co0 defines
an isomorphism of p on the tangent space 7^. at x0 to X. The restriction
of A to p is positive definite and hence defines a metric on the tangent
space TXo. Since the restriction of A to p is invariant under the adjoint
action of the compact group K on p, this metric on 7^ defines on X
a G-invariant metric.
Under our hypothesis, T operates freely on X as a group of isometries
so that X/r is a riemannian manifold in a natural manner with X as
the universal covering space. We orient X by choosing once and for all
an ordered orthonormal basis Yt YN of p thereby fixing an orientation
on TXo. G being connected, every element of G preserves orientations
on X so that, since r<=G, X/r is orientable and the projections of
Yj YN at x0 fix an orientation on X/r.
Now let < , >F be a positive definite scalar product on F which is
K-invariant and with respect to which p(Y) is symmetric for all Yep
(such a metric exists; cf. Preliminaries § 1.5). We then define a metric
along the fibres of E(p) as follows: let xeX/T be any point; choose any
geG such that now0(g) = x (n: X-+X/T is the natural map); then,
for v, w in the fibre at x the scalar product ax(v, w) is defined as
<P (g) </>«"' ("). P fe) </>«"' (w)>F
(q>g for geG was defined in § 3; q>g is an isomorphism of F on the fibre
over x).
If g' is another point in G such that re°c50(g')=x, we have
g' = k-g-y

5. Semisimple Groups (Weil's Rigidity Theorem) 125
where keK and ye T. Now from the definition of the bundle E(p), we have
<pg-1(f)=p(vr1<pg-1(f)
<Pg-1(w)=p(y)-Vg-1(w)
so that
<p(g')<Pg-1(f),p(g')<Pg-1(w)>F
= (P(k)<p;Hv),p(k)<p;1(w)>F
in view of our choice of < , >F. Thus our definition is independent of
the g in G chosen.
We have thus an oriented riemannian manifold M=XjT, a
representation p of the fundamental group T of M and a metric along thr fibres
ofE(p).
We have already computed the operator dp on A%(r, G, K, p)
(Proposition 7.21, § 3): for ueAp0(r, G, K, p),
(7-31) (d,*\ ,,+1='E(-irMn.+puuK l «,„■
The operator 8 on Ap(E{p)) defines an operator again denoted 8
on A"(r, X, p). Let 8p be the operator 8p: AP0{JT, G, K, pH^o"1^ G> K> P)
defined by the commutativity of the diagram
A'(r, x, p)—i—A'-1(r, x, p)
Al(r, G, K, p)-h—Al-1(r, G, K, p).
Our next aim is to compute this operator 8 . But for this we need to
compute the scalar product ($q1(£),®o 1(l)) °f tw0 f°rms £,»/ in
Ap0 (r, G, K, p) (one of which has compact support) where $0 denotes the
composite map
A'{E(p))-±+A'(r, X, p)-^Al{r, G, K, p).
We have for the scalar product the following
7.32. Lemma. For a suitable Haar measure d\i on G and l^neA&tf, G, K, p)
we have
(7.33) (*o1(0.*o1W)= E J<&. !,.*«, h\dn
li< — <ip G/r
where we let d/x denote also the measure on G/T induced by dp on G.

126
VII. Cohomology Computations
Proof. Using a partition of unity we can assume that £, and r\ have
support in an open relatively compact set U such that an open subset V
of X is mapped homeomorphically onto U under the map n: X-* XjT.
But when £, and n have support in U by replacing these by forms on X
with support in V we are reduced to the case when T= {e}. In this case
it follows easily from our definitions that
<«Pol(a,^o1('?)>= E <zh ,,.** .>
il < ••• < ip
where the functions <<!;;. f , r\tl, >f being K-invariant are considered
as functions on X. Thus the proof reduces to proving the following.
7.34. Lemma. There exists a Haar measure dy! on G such that for
every continuous function f with compact support on X, we have
(7.35) lfdm=lfo&0dn'.
X G
where dm is the riemannian volume form on X.
Proof. Let d\i be a Haar measure on G. Then the linear functional
f\->\foa>0dn
G
is a Borel measure on X which is G-invariant. Evidently dm is also a
G-invariant measure on X. It follows from this that for a suitable scalar
multiple d\£ = Xd\£ the Eq. (7.35) holds.
We need also
7.36. Lemma. Let f be a Cx function on G/T with compact support
and Y an element of g. Then
J Yfdn = 0
G/r
for any Haar measure dp on G.
Proof. Let {Yi}1^ign be a basis of g and let (o^ be the 1-form defined
by (W((yj)=^,j. Let Q = co1 a ••■ A(wn. Then, for a suitable scalar X
(7.37) \XYfdyL= \YfQ.
G/r G/r
On the other hand we have
(7.38)
Yf-Q = 0Y(fQ)

5. Semisimple Groups (Weil's Rigidity Theorem) 127
where 8Y denotes the Lie derivation with respect to Y (note that 6Yii=0).
But
(7.39) eY(f£i)=iYd(f£i)+diY(fQ) = d(iY{fQ)).
Now / has compact support, hence so has iY(fQ). Applying Stoke's
theorem we have
0= \d{iY(fG))= \0Y(fQ)= l(Yf)dn
G/r G/r git
which proves the lemma.
From the lemma and Leibniz's formula we deduce
7.40. Corollary. Iffltf2 are two C00 functions one of which has compact
support then for any 7eg,
ifi'Yf.d^- fA-Y&dfi.
G/r G/r
7.41. Definition. For & nEAp0(r, G, K, p) we set
We can now prove
7.42. Lemma. Let £eAp0(r, G, K,p). Ifp=0,8pZ = 0.Ifp£l, we have
(&PQtl i,.1=-E(n-P(n)K«1 w
*=1
Proof. Let n be the element of Al~l{T, G, K, p) defined by
% !,.,=-E(n-P(n))&«. i,-,-
(It is easily checked that n belongs to Ap0~l(r, G, K, p).) Then it suffices
to show that for any Cei4g-I(r, G, K, p) with compact support, we have
(r,,Q=(£,dQ.
Now in view of the Corollary 7.36, we have
Knu iP^ciu...,ip.1}Fdp.
G/r
= !<&., .,.„- nch h>Fdi*.
G/r
Also since p(Y,) are symmetric for lgfcgN with respect to the scalar
product on F, we have
<p(J*)<*« «„.,.£, ip-1>F=<4i1 ^...pW., ,,.,>,-

128 VII. Cohomology Computations
It follows that we have
fo.O= I- E <(Yk-p(Yk))^kil ip_.,C,, i^dy.
G/r t,.i<...<ip-i
= J E <4(1 ....(n+pWlC *,-.Vp
G/r fc,ii<"-<ip-i
= J E <&. ^.(-D"+1(^+p(^))^ j ;,Vp
G/r Ji< — <jr
= \(Z,dpOdvL.
G/r
Hence the lemma.
A straight forward calculation now gives us
7.43. Proposition. Let Ap=dp8p+dpdp. Then for £,eAp0{T, G, K, p) we
have N
v,fh «,= E(-n2+p(ia2)^ «,
+ E E(-i)"+1(-[^, o+ptfii.. 0«*h i i,-
Proo/ We have
(dp&p*X ir=£(-W+1(Yi.+P(Yj}(8pQh lu h
u=-l
= e E(-ir+l(n.+p(yj){-(n-P(ia)^ ?„ U
i»-l * = 1
= E Et-irfo+pajHn-pCWn, i. .,-
(^a., i,= E -(n-pdi))0p%t i,
= E-(rt-pra)(n+p(rt)K.. (,
t-i
+ E(-i)"+1(rt-p(>y)(^+p(>l))^1 Ju ,,-
Hence
(^o., «,= -j:(n2-p(W«. i,
+ E E(-D"+1([n. iy-p(rc, yJK<, ?u i,-
*=1 «=.l
Proposition 7.43 shows that Ap can be written as the sum Ap = AD+H
of two operators zdD and H,, defined as follows:

5. Semisimple Groups (Weil's Rigidity Theorem) 129
(A) (ADQk .,=-£!?£.. ,,
-I f(-ir+1[n..n]{*i1 %. h
(7.44)
(B) (H„£, ,,= Ep(n)2€i, i,
+ 1 E(-irV(L^,rj)&, i. ,-
It is easily seen that A D £ and Hp(£) again belong to 4g(r, G, K, p).
Evidently
Ap = AD+Hp,
We have moreover
7.45. Lemma. Let D, D*, lp and T* be the operators defined as follows
(D and T„ map Ap0(r, G, K, p) into A%+1(r, G, K, p) while D* and T* map
Ap0{r, G, K, p) into Ap0~\r, G, K, p)).
oil ^yiW^nA :. .,♦,
(A) r
o*ih i^-EC-n)^ i,-,
(7.46) k"1
Vi. i,t,='i;1(-irlp(n1)^ r. ,,tl
(B)
t;s., .,_.= !>(%., ,,.,.
Ifcen we have for £ »/e/lg(r, G, K, p)
0>&,)=(iD*,)
Moreover D* D + D D* = A D and T* Tp+Tp T* = Hp.
The proof, which is entirely analogous to those of Lemma 7.42 and
Proposition 7.43, is omitted. As a consequence we see that we have for
£eAl(r,G,K,p)
(ADZ,0=(DZ,DO+(DH,D* 0^0
(H„& 3=(T„& t„3+(t;6 T;a^o.
On the other hand since we have
(7.49) (V,O=(4>&0+(H,<;,O.

130 VII. Cohomology Computations
We conclude in view of 7.48 and 7.49 that we have
7.50. Corollary. For £eAp0(r,G,K,p), Ap£=OifandonlyifAD£=0
andH^=0. Or again Ap£ = 0 ifand only ifD£=D* f = T„f=T* f=0.
7.51. Remark. The formulae (7.44).(B) and (7.46).(B) obviously make
sense for elements £, of Hom(Ap p, F) as well. We denote these
corresponding operators by Hp, Tp and T* respectively. We will now justify our
notation. The operator Hp (and hence Hp) is not dependent on the
orthonormal basis Yt YN of p chosen. We will now establish this.
We complete the orthonormal basis Y1,...,YN into a basis Yt Yn of g
such that Yaet for N<agn and for N«x, j8^n, A{Ya, Yfi)= -5af (A is
negative definite on I).
Since I and p are orthogonal to each other with respect to the Killing
form A, we have in addition A(XL, Xj=0 for l^i^N<a^n. Finally
since [I, p] <=p, [f, fj erf, [p, p] erf, we have the following formulae
IY„YJ] = E c&n for 1-giJ-gN
■V<o.£n
(7.52) lYm,YJ = £ dtYj for lgi^N<K^n
1ZJZN
LYm,YJ= £ cl„Yy for N«x,p^n.
■V<y£n
Moreover the c^,l ^A, n, vgn are related to each other in the following
manner:
(7 53) cA/,= -ciA. lgA,/*,v=n
With this notation, we have for ^eHom(/l''p,F),
^W, \)
= ip(Yk)2£{Yil,...,Yip)
+ ii(-i)1+"i:p(cn..yj)«n.ii.-.^ v
(7.54) P w
11 = 1 *=1 « = 1V+1
= Ep(n)2«^.-.^
+ t t PWt(Yk,...,lYm,Yd \).
u-1«=-lV + l

5. Semisimple Groups (Weil's Rigidity Theorem) 131
Let a denote the canonical extension of the adjoint representation of f
in p to all the exterior powers. Let t denote the canonical representation
of f in Hom(A" p, F) obtained from a and the restriction of p to I. Then
we have
(Tra2s)ttv-,iy
= p(YJ(p(YJZ(Yil Yip»- £{2pWZ){Yh, ...,[7a, YJ,..., YJ
(7.55) ^ "-1
+2£ t(Yh,...,[ya) yj,.... yij5....[y„ yj,.... \)
u<v
+E€(^.-.[n[y.,yj],...,yj.
Now for a Lie algebra h let t/(h) be its enveloping algebra. We identify h
with a subspace of t/(h) and denote by the same letter a representation
of h as well as its extension to t/(h). With this convention elements of h
are regarded elements of [/(h). Let
c=£y2- £ Y? and c'=- £ Ya2.
1 = 1 « = jV + l a-jV + l
These are then elements of t/(g). The latter may be regarded as an element
of t/(I) as well. It is known that c (resp. c') is a central element of [/(g)
(resp. [/(I)) and that if {Y/}igf§iv and {Ya}N<llSn are respectively ortho-
normal bases for p and f with respect to A and — A, then
IV n IV n
i-l «=JV+1 / = 1 « = 1V+1
and
- £ y«2=c'=- £ y;2
a=N+l tt=N + l
(cf. Bourbaki [1], § 3, No. 7). From (7.54) and (7.55) we now get for
feHom(/lpp.F),
(7.56) 2H>pZ=2(p(c)-p(c'))oZ+r(c')(®-p(c')oZ-Zoa(c').
Clearly (7.56) shows that the operator Hpp depends only on p (and p,
of course) and not on the orthonormal basis Yt,..., YN of p chosen for p.
Combining now Propositions 7.12 (§ 4), 7.43 and Lemma 7.45 above
we obtain
7.57. Proposition. Let Qpp be the quadratic form on Hom^p, F)
defined by setting Qp{Q=(Hp.{£),Of. If Qpp is positive definite we can
find a constant C>0 such that for all £,eA^{T,G,K,p) with compact

132 VII. Cohomology Computations
support, we have >A £,£)>C{£,Q.
Moreover in this case every closed form £,eA^{T, G, K, p) such that
(<!;, £) < oo is a coboundary.
In particular if TcG is such that G/r is compact and p is such that
Qpp is positive definite, H"(r, p) = 0.
We will now investigate when QAd (where Ad is the adjoint
representation of G in its Lie algebra g) is positive definite. In the sequel we denote
7.58. Lemma. Let g= Llgj be the decomposition o/g into simple ideals
and f = LIl,, p = LIpi, (lj©Pj = 9f) be the corresponding decomposition of
I and p respectively. Then the quadratic form Qg is positive definite on
Hom(pr, gj for r+s and g, non-compact. Also on each of the spaces
Hom(pr, gr) the restriction ofQg equals QBr.
Proof. We assume as we may that the basis {Yx}1^Xin respects the
direct sum decomposition g= LIg,. Then c (resp. d) becomes a sum
5>, (resp.£C;) where c= £ V~ E tf (resp. c'r= - £ Ya2).
rel rel Y,epr Y.etr Y*€lr
It is then clear that Ad cr is zero on g, for r+s and a(c'r) is zero on p, for
r+s. We conclude from this that we have for £eHom(pr, qs) r+s
Qe(0= <2(p(cJ-p(<0)oZ-T(c'r + c'sm + P(c's)° £ + £ °o(c'r), 0B-
Now x(c'r)(0=£, o a(c'r) and x(c'J(0=p(c'j ° f. It follows that
e9(a=2<p(Cs-c;)(a,o6= E E <ady?zw, s(yt)>8
YiCp, Yjcpr
= E EIIK.W]!2-
Ytep. Yj€pr
Thus Qg(0=0 if and only if [_Yt, f (ty] = 0 for l^eps and r}epr. This means
that for all Yjepr, £(Y) commutes with all of ps. Now if gs is non-compact,
p, generates gs so that £, (Yj) must be in the centre of gs; and the centre of gs
is zero. That Qe restricted to Hom(ps, gs) equals Q^ is easily seen.
Lemma 7.58 reduces the problem of determining the null-space of Qg
(for a semisimple algebra g) to the special case when g is simple. For the
investigation of this case we will need the following two lemmas.
7.59. Lemma. Let g be a real non-compact simple Lie algebra and
9=1 ©P a Carton decomposition. Then I is abelian if and only i/dimg=3
and in this case g is. si (2, R).
The adjoint representation of I in p is irreducible and faithful.
Moreover AAX is skew-symmetric for Xet (with respect to < , >g). It follows

5. Semisimple Groups (Weil's Rigidity Theorem)
133
that if I is abelian, dim p = 2 and dim I = 1 so that dim g = 3. A comparison
with the Cartan decomposition of si (2, R) shows that g is isomorphic to
si (2, R) (see Preliminaries § 1.6).
7.60. Lemma. Let Qbea non-compact simple Lie algebra and g=I © p
be a Cartan decomposition. Let Yt,...,YN be a basis of p. Then for any
r, l^r^N, we can find rt rk, l^r^N such that rt = l, rk = r and
[yrj)yr(+1]+o.
Proof. Let E<=[1, AT] be the set of integers such that for ieE, Yt has
the property stated in the lemma. It suffices to show that £=[1,N].
Let E' be the complement of E in [1, N~\. Suppose £'=|=0. Let ht be the
Lie subalgebra of g generated by {Yi3 ieE'}. Then ht +0 is stable under
the adjoint action of {Yt,ieE}-in fact {Yt,ieE} commute with ht.
Since p generates g it follows that ht is an ideal in g so that ht =g. But
then {Yi3 ieE} commute with all of g. It follows that £=0. On the other
hand since Ylt..., YN generate p and p generates g as Lie algebra, there
exists Y, with [Yj, YJ + 0 and clearly ieE, a contradiction. Hence the
lemma.
We can now prove
7.61. Theorem. Let Qbea semisimple Lie algebra without compact or 3
dimensional factors. Then QAd is positive definite.
Proof. In view of Lemma 7.58 we may assume that g is simple.
Consider now the decomposition
Horn (p, g) ^ Horn (p, I) © Horn (p, p).
(In the sequel we use g and F interchangeably to denote the Lie algebra g.)
From the Eq. (7.56), it is easily seen that tfAd leaves Hom(p, I) and
Hom(p, p) stable; moreover the two sub-spaces are easily seen to be
orthogonal to each other for the scalar product we have defined on
Hom(p, g). Thus it suffices to show that Q^d is positive definite on
Hom(p, I) and Hom(p, p) separately. Now let £eHom (p, I) be an element
such that Qld{l;) = 0. According to Lemma 7.9, we then have in particular,
rAd(£)=0 i.e. we have [Y„ £(Y,)] = [Y,, £(Y,.)], lgi, j^N. Since £(Y;)ef
for all j, we have for 1 ^ k _: N,
<[% {(15)], Yk}F= -<Y., [Yt, £(¥$>?
= -<Yi,iYPaYkmF
= aYi,Z(Yk)lYj>F
= aYk,Z(YJ],Yj>F
= <Z(YMYk,YJ>F
= <tt(Yi),YJ],Yk>F
= -<lYl,Z(Yj)lYk>F.

134
VII. Cohomology Computations
Thus <[1^, t(Yj(], Yk}F=0 for all i,j,k. It follows that £{Yj) centralizes
all of p. Hence f (r})=0 for ally. Hence f=0.
Consider now Hom(p, p). The representation on this space is deduced
from Ad and a. a is the same as Ad restricted to f acting on p. Since g is
simple the adjoint representation of I in p is irreducible. Thus (Ad c' on p) =
ct(c') is a scalar. Clearly since o-(!Qis antisymmetric with respect to < , >f,
o(T) leaves the spaces of symmetric and antisymmetric endomorphisms
of p stable. Hence in view of (7.50) noting the fact that Ad(c) is a scalar
operator these two subspaces of Hom(p, p) are stable under Exk& as well.
Moreover the two subspaces are mutually orthogonal subspaces for the
canonical scalar product on Hom(p, p). It suffices thus to show that QAd
is positive definite on each of these subspaces.
Let £eHom(p, p) be an antisymmetric endomorphism of p (with
respect to the canonical scalar product on p). If £=|=0,
i It it
since a non-zero element of p cannot commute with all of p. Thus it
suffices to prove that for an antisymmetric £eHom(p, p),
(We use now the expression (7.46) for tfAd.) We have, using (7.52),
' *=EE<[iun.«iM].OT>.
i k
-IE<[n.[^.{(n)]].«yD>.
i k
=H<[n.«(m[^.{(n)]>.
i k
-EIE<[n.[^«ii)]]. r*>.<i;,OT>.
i It h
=E[n.«n)]|.2+ E <[n.w.«y»)]].n>B<«nxii>B
It l,k,h
(since £, is skew-symmetric)
=IE cn. «n)]|2+ e <k, «m en, rj>9 <«y^ ^>9
It i, It, k
HIEcn.fUMll2- E <^[cy».ia.«is)]>.<«^ii>.
It j.t.k
=E[n.«n)]||.2- E <[[ii.nKTOra>„.

5. Semisimple Groups (Weil's Rigidity Theorem)
135
(We have here used the Jacobi identity and the fact that for Yep, Zt, Z2eg,
<[7, ZJ, Z2>g=<Z1, [Y", Z2]>g.) It follows (writing i instead of h in the
second term on the right hand side) that
2E<[c^n].«n)].{(iD>.=Ecn.«n)]||^o.
i,k k
Hence QlA6 is positive definite on the space of skew-symmetric endo-
morphisms of p. (Note that so far we have not used the hypothesis that
dim g > 3.)
Finally, let £eHom(p, p) be a symmetric endomorphism. In view of
Remark 7.51, we may assume the orthonormal basis so chosen that Yj
are eigen-vectors for £'£(Yj)=A, Y{ with A;eR. Now T'Atd(^)=0 yields,
forlgi,y = N,
AJ[i^,yJ]=u,€(rJ)] = [yJ,{(i9]=Al[yJ,ia.
It follows that whenever [Yj, Yj]=t=0, A,= — X}. In view of Lemma7.60,
we conclude that X,= ±Xl for all i. Thus if £=t=0, p decomposes into an
orthogonal direct sum p = £©F where E={Yep\£{Y)=A1 Y} and
F={Yep\Z(Y)=-XlY) with £#0, F*0. Now clearly if Y,Y'eE,
[y, r] = 0. It follows that if Zef, and Y, YeE, <[Z, Y], Y\=<Z, [YiF])=0
i.e. ad(Z)(E)<=F. Similarly adZ(F)c=£ for all Zel. Hence adZadZ'
and ad Z' ad Z map £ (resp. F) into £ (resp. F) for Z, Z'el. Thus ad
([Z,Z'])(£) = ad([Z,Z'])(f)=0(=£nF). It follows that ad I is abelian.
Thus for f+O, T£d{0=0 if and only if g=sl(2,R) and {: p->p is a
symmetric endomorphism of trace 0 (and is hence unique upto a scalar
multiple) (Lemma 7.59). Hence the theorem.
When g = sl(2, R) one sees easily that for any symmetric
endomorphism £eHom(p, p) with trace 0, Qa(0=0. We thus obtain
7.62. Proposition. Let g be a semisimple Lie algebra, g" its maximal
compact ideal, g' the ideal containing all simple noncompact factors of
dimension >3 and {gjie/ the set of all simple ideals of dimension 3. Let
9=1© 9 be a Cartan decomposition of g. Then the null space of QB is
precisely Horn(p, g")© LI Sym(Pi) where Sym(pf) denotes the space of
iel
symmetric endomorphisms of p( which are of trace zero.
As a corollary we obtain
7.63. Theorem. Let G be a connected semisimple linear Lie group
and F a torsion free discrete uniform subgroup of G. Then if G has no
compact or 3 dimensional factors, Hl(T, Ad) = 0.
We will prove a result to include some cases where we allow 3
dimensional factors. But first we will relax the condition that G be linear
with the aid of

136
VII. Cohomology Computations
7.64. Proposition. Let G be a connected semisimple group and r a
discrete uniform subgroup. Let G' be the adjoint group of G and n: G—► G'
the natural map. Then n(]T)=r' is a uniform lattice in G'. Let G' = G[x G'2
where all the simple factors of G[ are non-compact and G'2 is compact.
Letp2: G'—► G'2 be the natural map. Then ifp2(r') is dense in G'2 the natural
map
tf1(r;Ad)-^tf1(r,Ad)
is surjective.
Proof. Let/: r—► g be a 1-cocycle on r with coefficients in the adjoint
representation (cf. Preliminaries §3.2). Let Z be the centre of G and
C = rnZ. Then for xeC we have for yeT,
f(x)=f{y x y~1)=f(y)+Ad yf(x y ')
=f{y) + Adyf{x) + AdyAdx-f(y-1)
=f(y)+Myf(x)-AdyAdx-Ady-1f(y)
=f(y)+Adyf(x)-Adxf(y)
= Adyf{x).
Thus Ady/(x)-/(x)=0. According to Corollary 5.18 (Chapter V) this
implies that f(x)=0. Thus any 1-cocycle / vanishes on C. Now for
yer,xeC
f{xy)=f{x)+Adxf{y)=f{y).
It follows that / defines a cocycle fy on V = TjC and / is the composite
of /"*—► V and fl. This proves the lemma.
We will now prove
7.65. Theorem. Let G be a connected semisimple Lie group without
compact factors and rczG a uniform lattice. Let g be the Lie algebra of G
and let E be the set of 3-dimensional ideals in g. For g'eE. Let g" be the
ideal supplementary to g'. Let G" be the Lie subgroup corresponding to g".
Let n': G-> G/G" be the natural map. If 7t'(r) is dense in G/G" for all g'eE
then HHr, Ad)=0.
Proof In view of Proposition 7.64 we may assume that G is the
adjoint group. It is thus linear. According to Theorem 6.21 (Chapter VI)
we can then find a normal subgroup /] of r such that T/7] is finite and /J
is torsion-free. Since
H^r.AdJ-^OI.Ad)
is injective we may replace r by /}. In other words we may assume
without loss of generality that r is torsion-free.

5. Semisimple Groups (Weil's Rigidity Theorem)
137
In view of Proposition 7.10, § 2, we are then reduced to showing that
if £eAl0{r, G,K, Ad) is such that AM£=0, then £=0. According to
Lemmas 7.45 and 7.49, JAd f=0 if and only if AB f=0 and tfAd f=0.
Let {g,},e/ be the set of all three dimensional ideals. For iel, let h,-
be the (unique) supplementary ideal of g, in g. Let p.=g, n p and pj=hf np.
Also let h = Q h, and p'=hnp. Now since HAd(^)=0, we have in view
of Proposition 7.62, £(p')=0 and £(7) is a p,-valued function for 7ep,
for all iEl. Moreover £j=£|P(: pf—>Pj is a symmetric endomorphism
of p,. For Yept and Y'ep'i we have (Corollary 7.50 and the Eq. (7.46))
0=D£(Y, Y')=YaY')-Y'-£(Y).
Since £(Y')=0, we conclude that Y'£{Y)=0. Since pj generates all of h„
it follows that the function £,{Y) on G/r is invariant under the action of
the normal subgroup Ht of G corresponding to h,-. Now since Htr is
dense in G (in view of our hypothesis), we conclude that £(7) being a
function on G/T invariant under Ht must be constant for all Yep,.
It follows that £(7) may be regarded as an element of pf. Thus £,■ itself
may be regarded as an element of Hom(p;, p,) which is symmetric and
of trace zero. Now for isl let (7,, Z,)€p,. be a basis of pf such that
&(10=A, Yt and ^,(Z,)=-A,Z,. On the other hand according to the
definition of A\(T, G, K, p) (Eqs. (7.15) (§3): we use the infinitesimal
version of (b')) we must have, for Zelj=lng;, Z=|=0,
zz(tq-z([z,tq)=iz,z(tq]
i. e. we must have
S([z,y.])=A.[z,y.].
Now ad Z is a skew symmetric endomorphism of p, so that we have
adZ(7i)=/iZj and adZ(Z,)= -nYt with n+0. It follows that we have
-A,/iZ,.=A,/iZ,.
leading to A,= -A,=0. Thus £=0. This proves the theorem.
7.66. Corollary. Let G be a connected semisimple Lie group without
compact factors and rczG be an irreducible uniform lattice. If G is not
locally isomorphic to SL(2,R), H1^, Ad)=0.
This follows from Corollary 5.21. Appealing now to Proposition 6.6
we have
7.67. Theorem. Let G be a linear semisimple algebraic group defined
over a numberfield kcR. Let GR be the R-rational points of G and G the
identity component of GR. Let fcG be an irreducible uniform lattice.

138
VII. Cohomology Computations
Assume that G has no compact factors and is not locally isomorphic to
SL(2, R). Then there exists a number field L with k eLeR and an element
geG such that g rg~1<=GL.
(Any connected linear semisimple Lie group G can in fact be realised
as the identity component of the R-rational points of a semisimple
algebraic group defined over a numberfield k.)
7.68. Remarks. Selberg [1] first proved that a discrete uniform
subgroup T in SL(n, Rl n ^ 3, is locally rigid (Definition 6.5). Theorem 7.63
for the special case when G/K carries a hermitian structure was first
established by Calabi-Vesentini [1]: their theorem is (in this special case)
stronger than Theorem 7.63. The analogues of Theorems 7.63 and 7.67
for non-uniform lattice are not yet settled. Garland-Raghunathan [1]
prove the analogues of Theorem 7.63 for non-uniform lattices in rank-1
semisimple groups. Margolis [1] has announced deep results but the
proofs have not appeared at the date of the writing.
Y. Matsushima [2] and [3] has considered the question of computing
the Betti numbers of XjT when this manifold is compact. Though
Matsushima's techniques bear a strong resemblance to the methods
used above we will not go into this question for reasons of space. Kaneuki
and Nagano [1] have (using the results of Matsushima) established that
if G is a semisimple group without compact or rank 1 factors then the
first Betti-number XjT vanishes. D. Kazdan [1] and S.P. Wang [2] have
proved this last result for any (not necessarily uniform) lattice T in G
(a semisimple group without compact or rank 1 factors). The techniques
employed in these papers however are radically different from those
used in this book and we cannot therefore unfortunately include these
results here.
A sufficient condition for the form Q"p to be positive definite—and
hence for the vanishing Hp(r, p)—where p is an irreducible non-trivial
representation of G is given in Raghunathan [2], In Raghunathan [1]
all pairs (G, p) for which Q* is positive definite are determined.
Mostow [8] has announced the following result. If G, G' are adjoint
groups without compact or rank-1 factors then any isomorphism q>: /"*—► F
of lattices T, F in G, G' respectively extends to an isomorphism of the
group G on G'. Mostow [9] combined with Margolis [2] yields a similar
theorem for G = S0(n, 1). These results when applicable are evidently
stronger than Theorem 7.63.

Chapter VIII
Discrete Nilpotent Subgroups of Lie Groups
An attempt to study the deeper properties of lattices in a semisimple
Lie group has led to a close study of discrete nilpotent subgroups of Lie
groups. Most of the results we establish are based on results due to
Zassenhaus [1] (at least in slightly weaker form). The first result we prove
has at first sight little bearing on the study of lattices in Lie groups; but
it turns out indeed that this result plays an important role in the
investigation of lattices.
For a solvable group H, let H=D°(H)=>D^H)^•••=>£>*(H)=(e)
Dk-\H)*{e), be its derived series: Z)'(ff) = [/)'-^H), D'^fff)]. Then we
call k the length of the derived series of H or more briefly the d.s. length
of H and denote it by d(H). With this notation we have
8.1. Theorem (Zassenhaus [1]). For each integer n, there exists an
integer q>(n) such that any solvable subgroup ofGL(n, Q has d.s. length
less than or equal to q>{n).
Proof. We argue by induction on n. Assume the existence of integers
(p(k\ l^fc^n-1 such that for any solvable H'cGL(k,Q, d{H')^(p(k).
Since for k<k, GL(fc, Q is a subgroup of GL^, Q we can assume that
<p(k)^<p(k') for k^k'. It follows then that if//' is contained in a product
GL^j Q x ••• x GL(fcr, Q, 1 ^kt<n, and is solvable then
d{H')^ Supcpikd^vin-l).
lgigr
Suppose now that H <=GL(n, C) is a solvable subgroup. Consider the
natural representation of H in C"= V. Assume first that this
representation admits an invariant subspace WcV. Let P be the subgroup of
GL(n, C) which fixes W. Let R be the subgroup of P consisting of
automorphisms which induce the identity on W and V/W. Then we have an
exact sequence
e -► R -U P -*-> GL(IT) x GL(K/W) -»■ e.
Then it is easily seen that R is a solvable group with d(R)=l i.e. R is
abelian. From the exact sequence above, we obtain an exact sequence
e-^Rr\H-^H-^n(H)-^e.

140
VIII. Discrete Nilpotent Subgroups of Lie Groups
Since n(H) is solvable, by induction hypothesis, dii(H)^<p(n — l). Thus,
since R is abelian, we have
d(H)^q>(n-l)+l.
We may therefore assume that V is an irreducible //-module. Let /ft
be a normal subgroup of H. Then V, when considered as an iJj-module
is completely reducible—in fact if EcV is a nonzero simple ifj-sub-
module, then for geH, gE is again a simple Hj-submodule of V and
clearly V= £ gE. Considered as an i/j-module, V can therefore be
geH
decomposed into a direct sum
of /fj-submodules Wt of K, each Wt being a sum of simple submodules,
all isomorphic to a nonzero simple i/j-module Mf. Then for each geH,
g Wt = Wj for some j. Thus if acts as a group of permutations on the set
A = {Wi\l^i^k}.
If St denotes the permutation group on k symbols, we thus obtain a
homomorphism
u: H-^Sk.
Let H2 be the kernel of u. Then H2 leaves WJ stable for each i; moreover
it is easily seen that the natural representation of H2,
H2<^>GL{V)
factors in the following manner:
H2-^l\GL(Wi)-^GL(V).
Suppose now that k+1, then dim Wt<n for all i so that d(H2)^q>(n -1).
Now H/H2 is isomorphic to a subgroup of Sk and clearly k^n; hence
H/H2^UcSn. U being solvable, it follows that d{U)^order t/gn!.
Thus we see that if fc+1, d(H)^d(H/H2) + d{H2)^q>(n-l)+n\.
The sequence
e -► SL (n, Q -► GL (n, Q -^ C* -► e
being exact so is
e-► tf n SL (n, Q-► tf-► det fl-► e.
It follows that d(H)Sd(H nSL(n,C)) + l. We may thus assume that
HcSL{n,Q.
8.2. Thus we are left to consider only the following case: H is an
irreducible subgroup of SL(n, Q such that V=C as a module over any
normal subgroup Ht, is a sum of simple submodules all isomorphic to

VIII. Discrete Nilpotent Subgroups of Lie Groups
141
one another. In particular if H^ is abelian, since every irreducible
representation of Ht is of dimension 1, Ht acts as scalars on V. Thus any
abelian normal subgroup of H is contained in the centre. Let C be the
centre of H. Let p: H —► H/C = G be the natural map. C consists of scalar
matrices of determinant 1. Hence C is a finite group and the order of C
is a divisor of n. Let U be a maximal abelian normal subgroup of G=H/ C
and let N=p~\U). Let Z be the centraliser of N in H. Clearly Z is
solvable. We claim that Z is abelian. If possible let d{Z) = k>\. Let
If Z=Zr. Then [Zt_ 2, Zk_ 2] = Zt_ t is an abelian normal subgroup of H
and is hence contained in C. Moreover Zk_2nN is contained in the
centre of N and is hence an abelian normal subgroup of H. Thus since
Zk_2 is not abelian, Zk_2 is not contained in N. It follows that p(N -Zk_2)
=1= [/. On the other hand [p(N ■ Zk_2), p{N ■ Zk_2)~] =e since N commutes
with Zk_2 and [N, N] <= C, [Zt_2, Zt_2] <= C. But U was assumed to be
a maximal abelian normal subgroup of G, a contradiction. Thus k = 1
i.e. Z is abelian. Hence Z= C.
Now H operates on N by inner conjugation and clearly the kernel
for this action is C. We now claim that N is a finite group of order at
most n3! n. To see this we first observe that V as an N-module decomposes
into a direct sum of simple submodules Mt all isomorphic to a fixed
simple N-module M. Since the representation of N in V is faithful so is
the representation in M. Let a denote this representation. Clearly since
V is a direct sum of copies of M, a(x) for xe N is a scalar if and only if x
operates as a scalar on all of V. Hence the set
{x|xeN, <r(x) is a scalar}
is precisely C. Now we can find n2 (n = dim KigdimM) elements xu ...,
x^eN such that o^x,)^,^ span (as a linear space) the entire algebra
End(M). Let Nt be the subgroup generated by {Xi}^,^ and C. Then
clearly Nt is a normal subgroup of N. Moreover for yeN, we have
yxjy-^XiajO-)
where at(y)e C; also yxf y-1 = xt for all i if and only if a (y) ff(x;) ff(y)_1=
ff(xj) i.e. if and only if o(y) commutes with all of End (M)—in other words
if and only if a(y) is a scalar i.e. ye C. We see thus N/C acts faithfully on
the finite set S= {(xf • c) 1 ^i^n2, ce C}. Since C has atmost n elements S
has cardinality at most n3. It follows that we have an exact sequence
e->C-►#->■ N/C
where N/C is a finite group of order at most n3! and C is a finite group
of order n. It follows that N has order atmost equal to n3! n. Now since H
operates on N with kernel C we have an exact sequence
e—>C—►//—>G—>e

142
VIII. Discrete Nilpotent Subgroups of Lie Groups
where G is a finite group of order at most (n3\n)l. This shows that
d(ff)=order (G) ■ Order(C)^(n3! n)! n
(the right hand side is a bound for the order of H).
This completes the proof of the theorem.
8.3. Corollary. Let G be a connected Lie group. Then there exists an
integer rG such that any solvable subgroup H of G has d.s. length d(H) less
than or equal to rG.
Proof. We can assume without loss of generality that G is simply-
connected. G admits a locally faithful representation.
p: G->GL(n, C).
The kernel of p is a central subgroup C of G. For any subgroup H of G
we have an exact sequence
e->flnC-.W->p(fl)->c.
If H is solvable so is p(H) and we have
d (H) g d (H n C) -I- d p (H)=1 + q> (n)
where cp(n) is defined as in the theorem.
8.4. Corollary. Let H be a subgroup of a connected Lie group G. Then
H is solvable if and only if every finitely generated subgroup of H is
solvable.
Proof. If H is not solvable in the derived series
H=H0=>H1=>H2=>-=>Hk=>Hk+l,...
none of the groups are trivial. In particular Hk+e for k=rG. Let xeHk
be an element different from e. Expressing x in terms of successive
commutators we find a finitely generated subgroup H' of H such that H'k+e.
But this is impossible since H' is solvable and k=d(G). Hence the
corollary.
8.5. Corollary. Any non-empty collection of solvable subgroups of a
connected Lie group has a maximal element.
Proof. We apply Zorn's lemma. It is sufficient to prove that if {#i}ie/
is a totally ordered family of solvable subgroups of a Lie group G, then
(J H( is solvable. This follows from Corollary 8.4.
ui
8.6. Corollary. Let G be a Lie group and H any subgroup. Then H
admits a unique maximal normal solvable subgroup.

VIII. Discrete Nilpotent Subgroups of Lie Groups
143
8.7. For a matrix AeM(n, R) as usual, let ||.4|| denote the positive
real number (Xay)i where ai} l = i,;'=n are the entries of A. We then
have the following well-known and elementary inequalities: for
A,BeM(n,R)
M + fl|| = MII + ||fl||
Mfl|| = Mil-11*11.
oc
Suppose now that A = 1 + 5 where ||5II <1. Then the series/-!- £ 5" ■(— 1)"
n-l
converges to a limit A' in M(n, R) and A'A — AA' — I; thus A' = A~l and
we have for XeM(n, R)
rxx\\ =
x+Z(-iy?x
= ||X||+£||51I|X||
We thus have
(2)
and similarly,
(3)
= ||A-||(H-||5||/(1-||5||))
= I|X||/(1-||5II).
M-1 x|| = ||A1|/(i-||£||
lix^-'ii^nxii/ci-i^li:
Suppose now that ,4 = 7 + 5 and B=I + n with ||5|| <1, ||/?|| <1; then wc
have
ABA'1 B~1-I = (AB-BA)A-1 B~x
= gri-riQA-lB-1
so that applying (3) twice we have
M^-1B-1-/|| = ||(5/?-/?5)^-1ll/(l-||'?ll)
= II5>?->?5II/(1-|I5II)(1-NI).
It follows that we have
(4)
(5) Mz-M-'zr'-JH =-211511 IMI/(i-H5ll)(i-IM).
The inequality (5) enables us to prove the following
8.8. Proposition (Zassenhaus). There exists a neighbourhood Q of e in
GL(n, R) such that the following holds: define Q{n) inductively as follows:
Q = Qi0) and fl(",= {[fl,fl("-1,]} = {afca-1fc-1|aefi, &efi0,"1)};
then i2(n)c=f2("-1) and given any e>0 we can find a positive integer n0 such
that for all AeQin), n>n0, \\A — I\\ <c.

144 VIII. Discrete Nilpotent Subgroups of Lie Groups
Proof. Let U = {X\XEGL(n,R); ||Jf-/||<i}. Let U° = U and we
define t/(r) inductively by setting
t/(r)={[t/, Ulr-1)]} = {aba-1b-1\aeU, beU1'-^}.
Then we will show that if A e Uln\ \\A -I\\ g(fy • £. (We can then evidently
take for Q any open subset contained in U and containing e.) We prove
our last assertion by induction n. Suppose that the assertion is proved
for all n<m. Let AsUm. Then we have
A = BCB~1C-1
with BeU and Cet/(m-1). Let B=I+£ and C=I+n; then we have by
choice of U, ||^||<i and by induction hypothesis ||»/|| "i(tr_1-i- In
particular \\t]\\ <i<l. Applying (5) now we have
\\A-I\\ = \\BCB-1 C~l-I\\
•g2i{||-||i,||/(l-||{||)(l-||i|||)
:g2||S||.||»,||/(l-i)(l-i)
which proves the proposition.
This proposition enables us to prove a stronger result which is
frequently needed in the sequel. We need two definitions.
8.9. Definition. For a subset S in a Lie group G, we define inductively
the sets S(n) for a positive integer n as follows: &0) = S and for n>0,
S(B)={[S,S(--1)]} = {afca-1fc-1|aeS,fc€S(',-1)}.
8.10. Definition. A sequence An of subsets of a Lie group G is said to
tend to e as n tends to oo if for every neighbourhood U of e in G there
exists an integer r such that for all m^r, Am<= U.
With these definitions we have the following
8.11. Proposition (Zassenhaus). Let Gbea Lie group and A a connected
nilpotent normal subgroup. Let n: G-^G/A be the natural map. Then there
exists an open neighbourhood Qc G/A of n(e) such that {[Q, ft]} cii and
if K is any relatively compact subset of G contained in n~\Q), the sequence
Kin) converges to e.
Let N be the maximum connected nilpotent normal subgroup of G.
Then A<=N so that we may replace A by N.
8.12. Suppose p: Gt—> G is a covering and assume the proposition
proved for (G, N). Let Nt be the identity component of p~1{N). Then the

VIII. Discrete Nilpotent Subgroups of Lie Groups
145
diagram
Gi/Nt—9-^G/N
is commutative. Let Qt be an open subset of Gj/A/j containing e such
that q(Q^cQ where QczG/N has the following property: for any
compact Kci7t'l(Q), Kin) tends to e as n tends to oo. Let Kx be any
compact subset of G such that 7r1(K1)c:.21. Then TtlpiK^cQ so that
Pi{K){n) tends to e as n tends to oo. Since p is a local homeomorphism,
we can find E„czGl for every integer n^O, such that E„ maps onto
p(K1)(") and for large n, E„aU where U is neighbourhood of e in Gt
such that p restricted to U is a homeomorphism onto the image. Then
E„ tends to e as n tends to oo. Now
K<">Cp-1(p(K1)<">)c=ZE„
where Z is the kernel of p. Since Z is central we have for n^O,
jc?" •=[*:„ £„-!];
for n large [K^, £„_,] e U so that we have
je}->c=[ji:1,ZE11_1]c=£11.
Thus K{"] tends to e as n tends to oo.
8.13. Conversely assume that Gl has the property, i.e. that we can
find an open neighbourhood Qt of e in GJ^ such that if K1c7ii1(Q1)
is any compact set, K^ tends to e as n tends to oo. We assume, as we
may, that Qt is connected and if x is in the kernel of q,xQ1r\Q1=f).
Suppose now that KcG is compact and
n{K)cq{Ql) = Q
i.e. Kcn~i(Q). Since fl is connected and so is N, n~i(Q) is connected
and by enlarging K if necessary we may assume that K is connected.
We can then find a compact connected set Kt in Gt such that eeKl
and p(K^ = K. Then JijCKj)era"1 (Q) = HQU where tf = kernel of q.
Since .r1(Ki) is connected, in view of our choice of Q1 it follows that
n1{K1)cQ1. Thus K^ tends to e as n tends to oo. Hence K^^niK^)
tends to e as n tends to oo.
8.14 The discussion in 8.13 enables us to assume that G is simply
connected. Then N is also simply-connected and there exists a locally
faithfully representation p: G—>GL(n,R) such that p(N) consists of
unipotent matrices. By 8.12 we may replace G by G/(kernel of p). In

146
VIII. Discrete Nilpotent Subgroups of Lie Groups
other words we may assume that G admits a faithful representation p
taking N into unipotent matrices. We may further choose p such that
p(G) is a closed subgroup of GL(n, R). We may thus assume that G
itself is a closed subgroup of GL(n,R) and A (=N, its maximum
nilpotent normal subgroup) consists of unipotent matrices. Now for a
non-negative integer k, let
Ek={v\vERn,{T-I)kv=0 for all TeN}.
Then we have Ek=>Ek_1 for all fc>0. Let r be the integer such that
£r=R" but E,.,!*R". Then for k<r, Ek+Ek+1. Let Fk be a supplement
to Ek_t in Ek. Then we have
R«=F1©F2©...©Fr.
For AeR+. Let Dx be the endomorphism of R" defined as follows:
Dkf=Xr~lf for feF(. Consider now the group
H = {g\gEGL{n,R); gE.cE,}.
Let
V={g\geH, g acts as identity on EJE^^
and S={g\geH,gFiczFi}. Then V is a normal subgroup of H and H
is the semidirect product of S and V. Moreover G czH and NczV. Clearly
in view of Proposition 8.8 we can find a neighbourhood W0 of e in
GL(n, R) such that \_W0, Wo]cW0 and if W= W0 • W0, W(B) tends to e
as n tends to oo. Let Wt= W0 nS. On the other hand we can identify S
with H/V under the canonical map jt': H-+H/V. We have clearly a
commutative diagram „. „, IJ/T/
G—*—»G/.V
where i is the inclusion and j is the induced map. .r'(Wi) is an open
subset H/V and let ft=j-1(jr'(W')). Suppose now that K is any
relatively compact subset of G such that n(K)cQ; then we can find a
relatively compact subset L of V such that KcWlL. Now for any
A, DxWlDj1= Wt and if A is chosen sufficiently large we have
DkLDlxcW0. Then DkKDlxcW0W0cW. Since Win) tends to e as n
tends to oo so does Kin)cDjlWin)Dk. Clearly, \_Q,Q~]aQ. This proves
the proposition.
As a consequence we have
8.15. Corollary. Let G be any Lie group. Then there exists a
neighbourhood QofG such that \_Q, ft] cQ and fi(n) tends to e as n tends to oo.

VIII. Discrete Nilpotent Subgroups of Lie Groups 147
8.16. Theorem (Zassenhaus [1], Kazdan-Margolies [1]). Let G be a
Lie group. Then there is a neighbourhood UofeinG such that ifT is any
discrete subgroup of G,rr\U is contained in a connected nilpotent Lie
subgroup of G.
For the proof, we need the following three lemmas.
8.17. Lemma. Suppose S is a system of generators for a group A.
Then A is nilpotent if and only if S(t) = e for all large k.
Proof. Let Ak be the subgroup of A generated by \J S(r). We then
claim that Ak is normal. For k large, Ak={e} and is hence normal. We
argue by downward induction. Assume then that An+l is normal,
and consider the exact sequence
e-^An+i-^A—*->A/An+l-->e.
Then n(A^ is evidently generated by n(Sin)). On the other hand A/An+l
is generated by n(S) and
{MS), 7i(S"y]} = n{lS, S^}=n(^+1))={e},
so that 7t(A,) is central in A/A„+x. Thus A„ is normal in A. Since
A„/A„+i is in addition central in A/A„+l, A is nilpotent.
8.18. Lemma. Let G be a Lie group, g its Lie algebra and exp: g -»G
the exponential map. Then there is a neighbourhood U ofO in g such that for
00
XeU if x=expX, the logarithmic series £((—l)*+1/k)(Adx —/)* con-
verges to SidX (here I denotes the identity endomorphism of q).
Proof. Introducing a basis for g, we identify Endg with M{n, R).
Since expad^f=Adx, the problem reduces to finding a
neighbourhood V of 0 in M(n, R) such that for YeV if exp7=y, the series
J((-l)*+7fc)(y-J)* converges to Y. If fl = {yeGL(n,R)|||y-J|| «5<1},
the series £ (( — l)k+l/k)(y—I)k converges uniformly and absolutely in Q
k=l
and hence defines an analytic function log: £i—>M(n, R). One sees
easily then that for yeQ, exp(logy)=y. From the implicit function
theorem, it follows that log is an open map in Q and we can find a
neighbourhood Q'czQ of e such that log|H. is also one to one. Clearly then
\ogQ'= V is a neighbourhood of 0 in g such that logexp Y= Y for all
Ye V. This proves the lemma.
8.19. Remarks. We will call a neighbourhood U of 0 in g such as
is described in Lemma 8.18, an L-neighbourhood of 0 in g. Suppose
now that H <=G is a Lie subgroup with Lie algebra h. Then t/nh is

148
VIII. Discrete Nilpotent Subgroups of Lie Groups
an L-neighbourhood of 0 in fj. This follows from the fact that the
adjoint action of H on h is the restriction of the adjoint action of H on g.
Suppose next that H is closed and normal in G. Then g/h can be identified
with the Lie algebra of G/H such that the diagram
G—*—>G/H
exp
'Ofo
9 —
is commutative (we have let jt stand for both the natural maps).
Moreover the endomorphisms Adjr(x) and adjr(X), xeG, Xeq are induced
by Adx and adX respectively in the following sense: the diagrams
9
adX
9
are commutative. From this one immediately concludes that n(U) is an
L-neighbourhood of 0 in g/h.
8.20. Lemma. Let G be a Lie group and g its Lie algebra. Let U be an
L-neighbourhood of 0 in g and let V'cU be a neighbourhood of 0 in g
chosen such that [Q,Q~]cQ, and Adx(V)cU for all xeii, where
fi=expK Then a subset 1 cVgenerates a nilpotent Lie subalgebra of q
if and only if S=expZ generates a nilpotent subgroup of G.
Proof. We argue by induction on dimG. Let k be the integer such
that &k)±(e) and &k+1)=(e). Then &k)cQ so that for each xe&k) there
is a unique XeKwith cxpX=x. Let Zk={XeV\expXeS(k)}. We claim
that for YeZ, XeZlk\ [X, 7] = 0. Let y=expK Since yeS, we have
yxy~' =x. Since yxy~x = exp Ady(X) and Ady{X)eU we conclude that
Ady(.Y)=* i.e. (Ady-l)p0=0. Now since y=exp7 and YeU, the
series J((-l)*+1/fc)(Ady-l)k(X) converges to &d{Y)(X)= -[X, Y].
Thus \_X, y]=0. Now S*k) commute among themselves. We conclude
again arguing as above that [_X, 7] =0 for X, YeZ1®. Now let h' be the
(abelian) Lie subalgebra of g spanned by Zik). Let H'cG be the Lie
subgroup corresponding to h\ Let H be the closure of H' in G and h
the Lie algebra of H. Consider now the centraliser 3 of h in g. Let Z be
the (connected) Lie subgroup of G corresponding to 3. Z is evidently
closed in G. Moreover since £<=3, Z=>S. Consider natural map
jt: Z-^Z/H. Since &k)±e, Z^+O so that dimZ//J<dimG. We also
denote by jt the natural map jt: 3-^3/h (the latter is identified with

VIII. Discrete Nilpotent Subgroups of Lie Groups
149
the Lie algebra of Z/H). Now U'=n(UnZ) is evidently an L-neigh-
bourhood of 0 in 3/h in view of Remark 8.19. Let V'=n(Vn$) and
&=expV. Then it is easily checked that we have [fl', Q']<=Q'
Adx(V')<=U' for xeii1. Now according to the induction hypothesis
n{Z) generates a nilpotent Lie subalgebra of 3/I). Since h is central in 3
and Z<=$, it follows that I generates a nilpotent Lie subalgebra of g.
The start of the induction when dimG= 1 is trivial. This completes the
proof of the lemma.
8.21. Proof of Theorem 8.16. Let Q be a neighbourhood of e in G
as in Lemma 8.20 and in addition such that LimQ^n) = e (cf. Corollary
8.15). Then (£ir\r)M<=rr\&n) tends to e as n tends to 00. Since T is
discrete this shows that (rr\ii)m=e for k large. It follows from
Lemma 8.17 that Tnfi generates a nilpotent subgroup of G. Thus
{X\XeU, expXeTnQ} {=A, say) generates a nilpotent Lie subalgebra
h of g. If H is the analytic subgroup corresponding to fj, fn fi=exp A <=.
exphc/J. Hence the theorem.
8.22. Definition. A neighbourhood U of e in a Lie group G is called
a Zassenhaus neighbourhood if it has the following property. Let «P
be any discrete subgroup of G. Then $ n U is contained in a connected
nilpotent subgroup of G.
8.23. Remark. Note that if Q is a Zassenhaus neighbourhood of e
and U cQ is any neighbourhood of e in G, then U is also a Zassenhaus
neighbourhood. Theorem 8.16 guarantees the existence of a Zassenhaus
neighbourhood.
8.24. Theorem (Auslander, see Wang [1]). Let Gbea Lie group and R
a closed connected solvable normal subgroup. Let n: G—► G/R be the
natural map. Let H be a closed subgroup of G such that H°, the identity
component of H is solvable. Let U = n(H), the closure of n(H). Then the
identity component U° of U is solvable.
Proof. A simple induction argument reduces the proof to the case when
R is abelian. Let Q e G/R be a neighbourhood of e such that for any
compact K a n~1(Q), Kln) tends to e as n tends to 00 (cf. Proposition 8.11).
Now the connected component U° of U is generated by any
neighbourhood of e in U°. Since n(H) is dense in U, it follows that the closed
subgroup generated by n{H) r\Q contains U°. Now if K is any compact
subset of n~l{Q)r\H, then Kln) tends to e as n tends to 00. Thus for n
large KMcH°. In view of Lemma 8.17 it follows that if HK is the group
generated by K and H°, H^H °is nilpotent so that HK is solvable. In view
of Corollary 8.4, the group generated by n~' {Q) n H is solvable. Hence
the group generated by n(H) n Q is solvable. It follows that U° is solvable.
Hence the theorem.

150
VIII. Discrete Nilpotent Subgroups of Lie Groups
8.25. Corollary (Auslander [3,4]). Let G be a Lie group of the form
K • R where K is a compact group and R is a connected normal solvable
closed subgroup on which K acts with a finite kernel. Let T be a lattice
in G. Then G/T and N/N n T are compact where N is the maximum normal
connected nilpotent subgroup of R.
Proof. Let U be the closure of n(r) where n: G-^G/R is the natural
map. Let U° be the identity component of U. Then U° being a connected
solvable subgroup of a compact group it is abelian. Moreover U/U°
is finite so that we may, by passing to a subgroup of finite index, assume
that rejr_1([/°). Now we can find a closed connected abelian
subgroup A of K such that n~1(U°)=A -R = H; H is solvable and T is a
lattice in H since TczHcG. Thus H/T is compact. Now from
Theorem 3.3 it follows that if Nt is the maximum normal connected
nilpotent subgroup of H, NJ^nT is compact. Since A is compact and
acts almost faithfully on R, NtczR. Hence N=Nt. Thus N/NnT is
compact.
8.26. Corollary (Bieberbach). Let r be a discrete group of rigid
motions of R" such that R"/r is compact. Then T contains a lattice of
translations of maximal rank.
8.27. Corollary (Wang [1]). Let G be a connected Lie group and R
its radical. Assume that G/R has no compact factors. Let T be a lattice
in G and n: G-^G/R be the natural map. Then n(T) is a lattice in G/R.
Proof. It suffices to check that n(T) is discrete. Let H be the closure
of n{T) and h the Lie subalgebra of the Lie algebra of G/R corresponding
to H. Then h is stable under the adjoint action of n(T) on the Lie algebra
g' of G/R. T has property (S). Hence n{T) has property (S) (Definition 5.1,
Chapter V). By Theorem 8.24 h is solvable. On the other hand according
to Corollary 5.16 (Chapter V), h is an ideal in g'. Hence h=0. It follows
that n(T) is discrete.
8.28. Corollary. Let G be a connected Lie group TcG a lattice. Let
R be the radical of G and N the maximum connected closed nilpotent
subgroup of G. Let S be a semisimple subgroup of G such that G = S-R.
Let a denote the action of S on R. Assume that the kernel of a has no
compact factors in its identity component. Let n:G—> G/R and ti':G—> G/N
be the natural maps. Then R/Rr\T and N/Nr\T are both compact.
Moreover n(r) and 7t'(r) are lattices in G/R and G/N respectively.
Proof. Let U be the closure of n(T) and U° its identity component.
Then clearly U° is solvable. Hence so is n~1{U°)= V. Since Kis normalised
by r, TcH, where H is the normaliser of V. Clearly VT is closed in H,
that is, if a: H-^H/V is the natural map, a(r) is discrete. Since T is a

VIII. Discrete Nilpotent Subgroups of Lie Groups
151
lattice in G, it is a lattice in H as well (Lemma 1.6). In view of Theorem 1.13,
Vr\T is a lattice in V. Now V contains R. U° = n{V) is normalised by
n(T) and since n{r) has property (S) in G/R, U° is normalised by niS^
where Si is the minimal connected normal subgroup of S such that
S/Si is compact (Theorem 5.20) since V=>R, V=(SnV)R. From the
preceding remarks we see that the identity component of (S n V) is
solvable and normalised by Sx. It follows easily that the identity
component M of S n V is compact. Thus V=M -R where M is compact
and normalises R. MnSi is finite and hence the action of M on R has
finite kernel. Corollary 8.25 now shows that N/N n T is compact. Hence
jr'(r) is discrete. This proves the corollary.
Our next result is a consequence of Theorem 8.16.
8.29. Theorem (C. Jordan [1]; see also Boothby and Wang [1]). Let
G be a connected Lie group. Then there exists an integer r—r(G) with
the following property. Let FcG be any finite subgroup. Then F admits
an abelian normal subgroup F0 such that Index F0 in F^r(G).
Proof. Let K be a maximal compact subgroup of G. Then any finite
subgroup of G has a conjugate in K. We can therefore without loss of
generality assume that G itself is compact. G being compact, we note
that any connected nilpotent Lie subgroup of G is necessarily abelian.
It follows that if Q is a Zassenhaus neighbourhood of e in G (cf.
Definition 8.22 and Remarks 4.23) then for any finite subgroup F of G, F n Q
generates an abelian subgroup of F. Since G is compact we can find a
neighbourhood jQ' of e such that
(i) g&g~l cQ' for all geG and
(ii) ii'cQ.
If F0 is the group generated by F nO1, clearly F0 is an abelian normal
subgroup of F. Let U be an open symmetric neighbourhood of e in G
such that t/t/-1c=fi'. Let r=\jx(G)ln(y)~] where n is a Haar measure
on G. We then claim that F/F0 has order at most r. In fact if Xj,..., xr+j
r+l
are any r+l elements in F then since ^^(x,t/)=(r+l)^(t/)>^(G),
;=i
we conclude that for some pair (i,j), l^i<j^r+l, XjC/nXj-tZ+fl
i.e. x;"'x;e[/[/""'efi'. Since Fo^.Q'r.F it follows that xy'x.eFo i.e.
xiF0 = xjF0. It follows that there cannot be more than r distinct coset
classes of F modulo F0. This proves the theorem.
8.30. Remark. Boothby and Wang [1] give quantitative estimates
for r{G) in terms of a certain integral on the compact group G. The
question whether these estimates are the best possible seems to remain
unsettled.

152
VIII. Discrete Nilpotent Subgroups of Lie Groups
We will next deduce with the aid of Theorem 8.16, the following
8.31. Theorem. Let G be a connected Lie group and H a subgroup
of G such that every finitely generated subgroup of H is finite. Then the
closure U of H in G is compact and the identity component U° of U is
abelian.
Proof. Let Ad denote the adjoint representation of G in its Lie
algebra g. Let B be the subalgebra of End(g) generated by Ad(iJ).
B being finite dimensional we can find a finitely generated hence finite
subgroup H' of H generating B. Let K be a maximal compact subgroup
of G containing H' and let f be the Lie subalgebra of g corresponding
to K. Then f is stable under Ad(.ff) hence under B. It follows that f
is stable under Ad(ff). This implies that H normalises K. If xeH is
any element and F the (finite) group generated by x, clearly FK is a
compact group and K being maximal FK=K. Thus FcK. Hence
HcK. It follows that [/is compact. Now let Q be a Zassenhaus
neighbourhood of e in G. Since U° is generated by any neighbourhood of e
in U we see that as a topological group, U° is generated by QnU.
Let Hx be the subgroup of H generated by Q n U. Then any finitely
generated subgroup of Hx is contained in a group generated by a finite
number of elements belonging to QnU and hence in view of
Theorem 8.16 is nilpotent. In view of Corollary 8.4, Hx is solvable hence
so is its closure U°. Since U° is compact U° is abelian. Since U is
compact U/U° is finite.
8.32. Remark. One sees easily moreover that U/U0 has order £ r{G)
where r{G) is chosen as in Theorem 8.29.
8.33. Corollary. Let G be a compact semisimple Lie group and H a
dense subgroup. Then H admits a finitely generated subgroup Hx also
dense in G.
Proof. We assume as we may that G is connected. Let 3F be the set
of all finitely generated subgroups of H. For Fe#" let F be the closure
of F and d(F) the dimension of F. Let F0€#" be an element such that
d(F0)=n = sup{d(F)\Fe&l Let Gx be the identity component of F0.
If xe H is any element and fi the group generated by x and F0, then
Fxe& and FX=>F0. Thus d{Fx)^d{F0)^d{Fx). It follows that Gx is the
identity component of Fx and is hence normalised by x. Hence H and
(therefore) G normalizes Gx. Let n: G-^G/GX be the natural map.
If Fx is any finitely generated subgroup of H containing F0,d{Fx)=
d (F0)+dim n{Fx) ■ (n{Fx) is the closure of n{Fx)). It follows that dim n{Fx)=0
i.e. n(Fx) is finite for every Fxe&r containing F0 hence for all Fe^
Applying Theorem 8.31 to n(H) we see that G/Gt is abelian and hence
equals e. Hence the corollary.

Chapter IX
Lattices in Semisimple Lie Groups — A Theorem of Wang
In this chapter we combine the results of Chapters V and VIII to obtain
some deeper properties of lattices in connected semisimple Lie groups.
We need a definition before we can formulate the results.
9.1. Definition. A subgroup // of a Lie group has property (P) if the
following holds: let Ad denote the adjoint representation of G in the
complexification Qc of its Lie algebra g; then every Ad(//)-stable sub-
space of gc is Ad (G)-stable. (Note that then any Ad //-stable subspace
of g is also Ad G-stable.)
With this definition we have
9.2. Proposition, a) Let G be a connected semisimple Lie group with
no compact factors. Then ifH has property (P) in G so does every subgroup
of finite index in H.
b) If G is a compact connected semisimple Lie group, a subgroup H
of G is dense in G if and only if every subgroup of finite index in H has
property (P) in G.
Proof. We may assume that G is the adjoint group in proving either
part. G then is isomorphic to a product of simple factors: G=f]Gj.
For proving (a) thus, we assume that G is the adjoint group Ad(G).
Let g be the Lie algebra of G and gc=g (g), C. We have then HcGc GL(gc)
(group of all complex linear automorphisms of the vector space gc).
Let //' be a subgroup of finite index in H. By passing to a smaller subgroup
if necessary we may assume that //' is normal in H. We assume as we
may that G is the identity component of the rational points G„ of an
algebraic subgroup G<=GL(n, Q defined over R. Let ft be the smallest
algebraic subgroup of G containing //'. Then ft is normalised by H.
If H n G = Gi then Gi is a Lie subgroup of G with finitely many connected
components. Let G? be the identity component of Gx. Then G° is stable
under inner conjugation by H. It follows that G? is normal in G. Since
HG°/G° is finite there is a maximal compact subgroup KcG such that
HcKG°(=G', say). The Lie algebra g' of G' is then //-stable, hence
G-stable. Since G has no compact factors, this implies that g'=g hence

154
IX. A Theorem of Wang
G' = G. Thus G° = Gi = G. But in that case every //'-stable subspace of gc
is also G,-stable so that our claim is proved.
We now prove (b). If H is dense in G so is every subgroup of finite
index and hence every subgroup of finite index has property (P). Suppose
now that every subgroup of H with finite index has property (P). Let Gx
be the closure of H and G0 the identity component of Gx. If Gx + G,
the Lie subalgebra g0 corresponding to G0 is not equal to g. Since g0
is evidently //-stable, G0 is a normal subgroup of G. Now //'=//r\G0
has finite index in H and leaves any subspace E of g containing g0,
stable. It follows that G0 = G. This proves (b).
We will now establish some lemmas needed later.
9.3. Lemma. Let Gbea semisimple Lie group. Let g be its Lie algebra.
Let gn be the sum of all the noncompact ideals in g and Gn the
corresponding analytic subgroup. Let HcG be a subgroup with property (P) in G.
Then we can find a subgroup Hx ofH which is finitely generated, has
property (P) in G, and is such that p(H) and p(//i) have the same closure in
G/Gn. Here p is the natural map p: G—>G/Gn.
Proof. Let GC=G/Gn and G'c be the closure of p{H). Let G'° be the
identity component of G'c. Since H has property (P) in G, G'c° is a normal
subgroup of Gc and is hence equal to {e} or is semisimple. It follows from
Corollary 8.33 (Chapter VIII) that we can find a finitely generated
subgroup H\ in H such that p(H[) and p(H) have the same closure in G/Gn.
As usual let Ad denote the adjoint representation of G in its Lie algebra
g. Let B be the associative subalgebra of End g generated by Ad(fl).
B being finite dimensional a finite subset SofH spans B as a linear space.
Let H" be the subgroup generated by S. Then the subgroup Hx of H
generated by H[ and //" has the requisite properties.
9.4. Lemma. Let G be a Lie group and H a normal subgroup. Let p:
G-^G/H be the natural map. Let r be a lattice in G such that p{T) is
discrete. Let /] be any discrete subgroup of G containing r. Then /?(/;) is
discrete in G/H.
This is essentially a restatement of Theorem 1.13 in a special case.
9.5. Lemma (Wang [2]). Let G be a semisimple Lie group and K a
compact subset of G. Then there exists a neighbourhood W of e such that
the following holds: let r be any discrete subgroup of G such that r r\K
generates a subgroup with property (P); then r r\ W={e}.
Proof. By enlarging K if necessary we assume that K=K'1 and esK.
Let Q be a neighbourhood of e such that there exists a diffeomorphism
«P: fl—► U eg, the Lie algebra of G, such that exp o $ is the identity on
Q and <P(e)=0 and further such that the following holds: if a subset S

IX. A Theorem of Wang
155
of Q generates a discrete subgroup of G, then $(S) generates a nilpotent
subalgebra ns of g (cf. Lemma 8.20). Let Ns be the nilpotent Lie
subgroup corresponding to ns. Clearly if ScS'cQ and S generates a nil-
potent group, ns<=nS' and NscNs,. Now let dim G=p and let flj be a
neighbourhood of e such that
x^x-'cfl forall xeKp{={ara2 ap|a;€K}).
Let r be a discrete subgroup such that S=Tr\K generates a subgroup
/"j with property (P) in G. Define inductively Sr as follows: S0 = Qi nf,
Sr={xyx-1|x€rnK, y€Sr_!}. By our assumption KqcKp for q^p.
From the definition of Sr we have then Sr <= i3 for 0 ^ r ^p. Setting nr=nSr
we have n0c=rt1c=---c=rtp. Since dimG=p, nr=nr+1 for some r with
0"gr^p. But this implies that Adx(n,)=nr for all xet nK. Since rnK
contains along with each x also x-1, this shows that n, is stable under
Ad^). Since /"; has property (P) in G and nr is nilpotent, nr=0. Thus
Sr—e. Hence rnQl = e. Hence the lemma.
Using Lemma 9.5 we can now prove
9.6. Theorem. Let G be a compact semisimple Lie group and H a finite
subgroup with property (P). Then H is contained in only finitely many
finite subgroups of G.
Proof. In view of Lemma 9.5, there exists a neighbourhood W of e
in G such that WnHl = {e} for any finite subgroup Hx of G containing
H. Let Wx be a neighbourhood of e such that Wj-1 WxcW. Then if n
is a Haar measure on G and n is the index of Hx in H, we have
l/n.MG/H)=MG/tf.)^W)
so that n^n(G/H)ln(W^. Thus it suffices to show that for each integer
p there exists only finitely many subgroups of G containing H in which
H has index p. To see this let Hn be a sequence of subgroups of G such
that Hn contains Hasa subgroup of index p. Assume that H is of order
q. Let hln,...,hkn be an enumeration of the elements of Hn where for
r^q, hrnsH and hrn=hrn+l for all n. We may assume by passing to a
subsequence that the hrn converge to limits gr as n tends to oo. Evidently
the {gr}, l^r^k form a group H0. Since W1hrnr\W1hsn=^ if r=t=s, it
follows that gr+gs if r+s. Now hrB-h5B converges to grgs; grgs=g,
and h,„ converges to gt. It follows that hrnhsnh,~1 converges to e so that
for all large n, hrn hsn = htn. Since the H„ are all groups of order k it follows
that the map pn\ H0-^>G defined by pn(gt)=h,n is a homomorphism of
H0 in G for n>n0, n0 suitably large. Evidently p„ converges to the
inclusion p0: H0^*G. Since H0 is a finite group we have Hl(H0, Ad o p0)=0.

156
IX. A Theorem of Wang
Hence by Theorem 6.7 we see that pn and p0 are conjugates for n large.
Now for heH, p„(h)=p0(h)=h for all n. Hence if uneG is such that for
all heH0 and large n,
KPoih)u-l = pn{h)
we have for n large,
unhu~l = h
for all he/J. Since H has property (P) in G one sees by projecting on
the simple factors, and using Schur's lemma that un is central in G. Thus
Hn=H0 for all large n. Hence there does not exist any infinite sequence
of distinct finite subgroups Hn of G containing H as a subgroup of index
p. It follows that the set of such subgroups is finite. This proves the
theorem.
We will next prove
9.7. Theorem (Wang [2]). Let G be a connected semisimple Lie group
and r a lattice in G. Assume that r has property (P) in G. Then r is
contained in only finitely many lattices in C.
Proof. Let G* be the adjoint group of G and let Ad: G-> G* be the
natural map. Now if r is a lattice in G, Ad r is a lattice in G* (cf.
Corollary 5.17). If I[ is a lattice containing r, AdiJ is a lattice containing
Ad/7 Moreover if Adr2=Adru then T^cAd-'Ad^). Clearly there
are only finitely many such r2 for a fixed /"J. Thus the map
^-►Ad/;
of the set of lattices in G containing r into the set of lattices in G*
containing Ad r has finite fibres. Thus we may assume that G=G* is the
adjoint group. Now let G*=Gnx. Gc where Gn is the product of all the
noncompact factors and Gc is the product of all compact factors. Let
p: G*—>Gcbe the projection onto Gc. Let G' be the identity component
of the closure of p(r) in Gc. G' is normalised by T and since T has
property (P) in G, by G. Let p-1(G')=G1. Then G* is a product Gt x G2 where
G2 is compact. Let pt be the Cartesian projection of G* on G(. Then
Pi(r) is discrete for i= 1, 2. Moreover Pi(r)p2(T) is a lattice containing
r. Now if /"i is a lattice containing r, /"i nGt is a lattice in Gt so that
p2(Ji) is a lattice (Theorem 1.13). Since G2 is compact ^(/i) is a lattice.
Let S be the set of lattices Tt in G containing T. Let S'={/""!/"" a lattice
in Gt containing p^r)} and S" = {r"\r" a lattice in G2 containing p2(T)}.
Let a: S^S' x S" be the map
a(r1)=(p,(r1))P2(r1)).

IX. A Theorem of Wang
157
Now if a(-Ti)=a(r2) then we have
Since T has finite index in pl{rl)p2{rl) it follows that a has finite fibres.
G2 is compact and semisimple. According to Theorem 9.6, S" is finite.
Thus it suffices to show that S' is finite. In other words we may assume that
we are in the following situation: G is semisimple Lie group isomorphic to
its adjoint group; T is a lattice in G such that if G=Gn ■ Gc, Gc is
compact and G„ has no compact factors and p is the Cartesian projection
p: G -* Gc, then p(r) is dense in Gc. According to Lemma9.3, we can then
find a finitely generated subgroup r0 of T such that p{r0) is dense in Gc and
r0 has property (P) in G. In view of Proposition 9.2, every subgroup of
r0 of finite index in r0 has property (P) in G. Let K be a compact subset
of G such that KnT0 generates r0. Then if Tt is any lattice containing
r, Tj^Tq so that /"inK generates a subgroup r{ of G such that every
subgroup of finite index in r{ has property (P) in G. Let W be a
neighbourhood of e in G such that for any lattice /"i in G containing T, /"i n W={e}.
Such a neighbourhood of e exists in view of Lemma 9.5. It follows that
if Wt is chosen such that Wf lWt<=W, the measure of G/rt is greater
than fi(Wt) where n is a Haar measure on G chosen and fixed once for
all. On the other hand the measure of G/T is an integral multiple of
that of G/ru the integer being the index of r in rt. Thus it follows that
the index of T in /"i has an upper bound. Now let m be this upper bound
and for l^p^m, let Dp={rl\rl a lattice containing T as a subgroup of
index p}. Evidently, it suffices to show that each Dp is finite. Let rn be
any sequence in Dp. We may (by passing to a subsequence if necessary)
assume that the sequence rn converges to a limit 0 (cf. Theorem 1.20).
We claim then that 0 = Tn for large n. To see this for each n consider
the action of rn on rjE Let /^' be the kernel of this action. Then r„ is a
normal subgroup of rn and since T has index p in rn,rj has index at
most equal to p\; it follows that T^'cT is a normal subgroup of T of
index less than or equal to p! in T.
Let rn0 = r^nr0. Then rn0 is a normal subgroup of 7^, of index at
most p\. Now let I be the (finite) collection of all subgroups of the
permutation group in r=p\ symbols. Then any normal subgroup Tq in r0
of index ^r can be obtained as the kernel of a homomorphism t: r0—>F
where F is a member of I. Now 7J, is finitely generated so that there are
only finitely many distinct homomorphisms of /"j, in F. Thus the set $
of subgroups of T0 of index less than or equal to r is finite. Since rn0 <=■ r0
for all n, it follows that Q 7^0 = /^ is equal to a finite intersection and is
n
hence a subgroup of finite index in 7^,. Now since /Jc^c^cr, and

158
IX. A Theorem of Wang
r„ is normal in rn, we have for 9nern and ye/^J,
Now let 6e0 be any element. Then we can find a sequence 6nern such
that 0„ converges to 6 as n tends to oo. Then for yeT,,,
Now since 6ny6~1er and T is discrete, this sequence must terminate
and we have for all large n,
Applying this to y running over a finite set of generators of 7^ (note that
To being of finite index in r0 is finitely generated) we find that for large
n, O'+i 6n centralises To. Now r0' has property (P) in G and has a dense
projection in Gc. Moreover the centre of G is trivial. It follows from
observing that Ad Gc:End(g) and using property (P) for the adjoint
representation of AdG in End(g), that 6~^l6n=e. Thus 6=9n for all
large n. Since rn^T for all n and the index of T in .r is p, 0=Tn for all
large n. Thus the set Dp is finite for all p, l^p^m. This proves the
theorem.
9.8. Remark. As a consequence of Theorems 9.7 and 5.7 we obtain the
following result.
Let G be a connected semisimple Lie group without compact factors.
Let r<=G be any lattice. Then r is contained in only finitely many
lattices in G.

Chapter X
Arithmetic Groups: Reduction Theory in SL(n)
and the Compactness Criterion
We begin the study of arithmetic subgroups of algebraic groups in this
chapter. We start with the construction of a fundamental domain for
SL(n, Z) in SL(n, R). Using this construction we establish a criterion
(Theorem 10.19) for the quotient of the real points of an algebraic group
defined over Q by an arithmetic subgroup, to be compact. This criterion,
conjectured by Godement, was proved independently by Borel-Harish
Chandra [1] and Mostow-Tamagawa [1]. The proof given here is due
essentially to the latter.
10.1. Notation. We will adopt the following notation for the first few
sections of this chapter. Let G denote the group SL(n,R) of (nxri)-
matrices of determinant 1. Let r denote the subgroup SL(n, Z) of all
integral matrices in SL(n, R). Also K will denote the special orthogonal
group, A the subgroup of diagonal matrices in SL(n, R) with positive
diagonal entries and N the group of all upper triangular unipotent
matrices.
10.2. Definition. A Siegel set in SL(n, R) is a set of the form S,,=
K-At-n where t is positive real number,
At={aeA\ait£tai+u+l(l£i<n)}
and n is a compact subset of N. When n is a compact set of the form
Nu={neN\\niS\£u{lg.i<j£n)}
u a positive real number, we denote StNu by simply Stu.
It is evident from the definitions that for a Siegel set S, KS=S and
for geAN, Sg is again a Siegel set.
An important property of A, is
10.3. Lemma. IfncN is a relatively compact subset of N, then the
set \J ana'1 is also relatively compact in N.
aeAt

160
X. Reduction Theory in SL(n) and the Compactness Criterion
Proof. Clearly ar\a 'ciV. For i<j and tieN moreover, we have
(ana~1)iJ=(aii/ajj)rijj. It follows that
Hana-W^tJ-'n-.j.
Since n is relatively compact, n<=Nu for a suitable ueR, m>0. Let u'=u ■
Sup(r"_1,1). Evidently then we have
\J ana~l<=.NU'.
Hence the lemma.
We now state our first main result.
10.4. Theorem. For t^2/|/3 and u^\, StuT=G.
Proof. We argue by induction on n. (For n = 1, there is nothing to
prove since G=(I).) Let geG. The function i"i->||gi"|| of R" into R+
(=[0, oo)) is a proper map (here for a vector v=(vu ..., v^eR", \\v\\ =
(rf H hi"^)*). It follows that this function takes a minimum m on the
set Z"- {0}. Let eeZ"-0 be a vector such that ||ge|| =m. Now if e=Xf
with AeZ and/eZ", we must clearly have X= ±1 in view of the
minimality of m. It follows that if n^/1 we can find yeT such that yet=e
where eu..., en is the standard basis of R". We find thus that for any
lattice point fe Z" - (0),
Bgycill^llg/B-
Let gy=g. It is clearly sufficient to show that there exist /eT such that
g'y'eS^. Now we assume the theorem proved for all G = SL(fc, R), k<n.
We can write ^=k'a'ri where WeK a'eA and n'eN. Set h=dri. It
suffices to prove that there exists y'eT such that hyeStu. Now h is in
the form
A-f X)
where T is an (n — l)x(n — 1) matrix. The determinant of T is a-1. Since
a' has all diagonal entries positive, a>0. Let /JeR+ be such that f}"~1 =
a-1. Then T=flT where T has determinant 1. Now by induction
hypothesis one can find a y"eSL(n — 1, Z) such that if we set
we have
h Y = k ■ a ■ n
where keK, neN and a is a diagonal matrix whose entries au are of the
following form: an=a, au=^-du with a£i/a|+t.+t'S2/"|/3 for 2^i<n.

X. Reduction Theory in SL(w) and the Compactness Criterion
161
Now the matrix / leaves et invariant. The element k'eK leaves the
norm || || invariant so that we have
ll*«i II = ll«' ^ii ^ ngyil = ll«'y-y II = II* y-VII
for all/,eZB-(0). Since y'1 maps Z"-(0) onto itself, we see that
||*e,|| .£||*/|| forall/eZ»-(0).
Choose for/any vector of the form e2+keu XeZ. We then see that
since neJV, ne1 = e1 and
n{e2+Ae1)=e2 + {/. + g)el
where £eR is a real number determined by n. Using the equation hy' =
k-a- n, we find then that we have
<%x=\\hel\\2£\\fid22e2+(k + ®allelV
= p2a'22+(l + <;)2a21.
Now the integer X can be chosen such that |(A + ^)|^j. It follows that
\a2xl^>$2 d22 = a222.
We find thus that we have
a11/a22^2/v/3.
Hence G = K ■ A2/yj ■ N ■ r. To complete the proof we need only observe
that N = Ni(rn N). This is again proved inductively. Writing any
element neN in the form
(1 x\
( )
\0 n'l
where ri is an (n — l)x(n — 1) matrix and x=(nu)2s,Sn we choose an
element yeTnN as follows: y is of the form
1 Z\
n
where / is the identity matrix and Z=(Zu)2gtgn and Z,, are all integers.
Now one sees easily that
<1 x\ /l Z\ /l x
\0 n'l \0 // \0 n'l
where (x'u)=(xit + Zu) for 2^/^n. An obvious inductive argument now
gives the required result since the {Zu}2SiSn are integers at our choice.

162
X. Reduction Theory in SL(n) and the Compactness Criterion
10.5. Corollary. r is a lattice in G.
Proof. It suffices to show that a Siegel set Stu has finite measure for
any Haar measure on G. For this consider the diffeomorphism
/: KxAxN^G
given by I(k, a,n)=k-an. Set AxN=B. The group KxB operates on
G: K operates as left translations and B as right translations and these
two operations commute. Now G is a homogeneous space for this action
of KxB and carries a unique (uptrascalar) Borel measure which is KxB
invariant (G being semisimple, it is unimodular; hence any Haar measure
dg on G is invariant under KxB). The map /', (k,b)\-+ k■ b, of KxB
in G is compatible with action of KxB on the two spaces G and KxB;
on the latter space KxB acts as follows (on the left): (k,b) {(k'b')} =
(k k', b' b~l). Now let dk, db be the left and right Haar measures on K
and B respectively. Then dkxdb is an invariant measure on KxB.
Hence its direct image in G under /' must be a Haar measure on G. Thus
since K is compact, it is enough to show that the set A, Nu has finite
volume with respect to the Haar measure db on B. The group B is a
semidirect product A-N,N being normal. A simple calculation shows
then that
db = p(a)dadn
where da (resp. dri) is a Haar measure on A (resp. N). Note that A being
abelian and N nilpotent, they are unimodular and p(a) is a character
on A defined as follows: each asA defines an automorphism of N viz.
nHanfl"1; this automorphism carries the Haar measure dn into a
positive scalar multiple p(a)dn of dn. It is easily seen by looking at the
action of A on the highest degree component of the exterior algebra of
the Lie algebra n of N that
p{a)=l\aii/ajJ.
Since Nu is compact to prove our assertion we need only show that
J p(a) da < oo.
At
If we set bi=aii/ai+li+1, p(a)=Y[brt' where rt are positive integers. Now
the bt form a system of coordinates on A. On the other hand the map
0"i)i£«-£n,">(exPyi)i£t<n 'S an isomorphism of the additive group R"-1
onto A. Here we have identified the product of (n — 1) copies of R+
(= positive reals) with A by means of the bt. This isomorphism transforms
the Lebesgue measure on R" into a Haar measure on A. Thus to prove

X. Reduction Theory in SL(n) and the Compactness Criterion
163
our contention it suffices to check that
logt
I! J esp r,ytdyi <co.
This is indeed true since rt>0 for \^i<n. This proves the corollary.
Appealing now to Theorem 1.12 (Chapter I) we have
10.6. Corollary. Let n: G^G/r be the natural map. Let xmeG be a
sequence of points in G. Then the sequence 7t(xmX meN, has no convergent
subsequence if and only if there exists 0mer=SL(n, Z), 0m=#J such that
xm 6m x~1 tends to the identity. Moreover for large m, 6m is unipotent.
For large values of m, the elements 9„ in the statement of the
corollary above are necessarily unipotent: in fact the characteristic
polynomial Pm(T)eC[T] of 6„ is integral and on the other hand must tend
to the characteristic polynomial of the identity matrix. Thus for large
m, 6m is unipotent.
Let Z„ = 0„—I where / is the identity matrix. Evidently Zm=#0,
ZmeM(n, Z) and xmZmx~l converges to zero as m tends to oo.
Conversely, let xmsG be a sequence such that for some sequence Zme
M(n, Z)—(0), x„Zmx~1 tends to zero. Then the characteristic
polynomial Pm of Z„ is a monk integral polynomial. Evidently Pm tends to
the characteristic polynomial of 0 and must therefore be equal to it for
all large m. Hence Zm is nilpotent for all large m. If 6m=I + Zm, clearly
9m+I and x„Omx~1 tends to e. We have therefore proved
10.7. Corollary. Let n: G—> G/r be the natural map. For a sequence
xmsG, n(xm) has no convergent subsequence if and only if there exists a
sequence ZmeM(n, Z)—(0) such that xmZmx~1 tends to zero.
10.8. Remarks. The last assertion of Corollary 10.6 holds for any L-
subgroup (Definition 1.21) in a semisimple group G as will be seen in
Chapter XI. However unlike the proof given above in the special case at
hand the proof in the general situation is far more complicated.
Now the space G/r can be regarded as a set of lattices in R" as
follows. Let L denote the set of all lattices in R". For geG, evidently gZ"eL.
The map gt-^gZ" of G in L passes down to a map/: G/r^> L. It is not
difficult to see that a sequence xneG/r converges to a limit in G/r if
and only if the sequence f(x^ of lattices in R" has a convergent
subsequence in the sense of Theorem 1.19. It follows then from that theorem
and Corollary 10.6 that we have
10.9. Corollary (Mahler's criterion). A subset Q<=G is relatively
compact modulo r if and only if there exists a neighbourhood UofOin R"

164
X. Reduction Theory in SL(n) and the Compactness Criterion
such that gZnnU=0 for gsQ. Equivalently for a sequence xneG, n(xn)
has no convergent subsequence if and only if there exists a sequence
eneZn — {0} such that x„e„ tends to zero.
10.10. Remark. One can prove Corollary 10.9 more directly from
Theorem 10.4. According to Theorem 10.4 if xmeG is any sequence we
have y„er such that
xmym=k„-am-e„
where amsA2l^, k„eK and 0me.Vj, K and N^ being compact, if n(xm)
has no convergent subsequence, am€A2/v^ must be an infinite closed
discrete subset of A2/v^. If we set f„=y„ et where el,...,enis the standard
basis of R" we have xmfm=kmamymel = kmamel. Since K preserves
norms, we have
II Xm/m 11 = 11 amei 11= (On-
It suffices therefore to prove that {am)n tends to zero as m tends to oo.
Now we have for l^j<n
{aj1a(2/yiy-i-(ajJJ
so that
n («j11= n Q/]ffl-i-(<Djj
=(2/|/3)"(B~1)/2
since the determinant of a is 1. Hence we see that we have a constant C
such that (aJulC. Thus if (am)n does not converge to zero, we can
(by passing to a subsequence if necessary), assume that (On is bounded
below as well i.e. we can find c>0 such that
c=(aJn=C.
From this inequality we obtain for l^j_n,
(am)J.^(2/l/3)-(J-1)-C
so that
1= J! (0J-^(2/l/3)-<"-1)("-2)'2-(am)BB.
This proves that (Onn is bounded above and since
(o^(2/i/3r'(aj..-
We find that all the (ajjj, melS, l^j^n are bounded above and
below by positive constants. It follows that the {aJmeN} lie in a
compact subset of A, a contradiction. The converse assertion is easy to prove.

X. Reduction Theory in SL(n) and the Compactness Criterion
165
10.11. Notation. We will use some changed notation for the rest of
the chapter. Let V be a vector space over C provided with a Q-structure
i.e. we are given a Q-vector subspace VQ of V such that dimQ VQ=
dimc Kand which generates Kas a C-vector space. Let G be an algebraic
subgroup of GL(K) defined over Q. Let L be a lattice in VQ i.e. a Z
submodule of VQ generated by a basis of VQ. Let GL={g\geG, gL=L}.
Finally we recall that two subgroups A, B of a group H are said to be
commensurable if A n B is of finite index in both A and B.
10.12. Definition. Let G<=GL(K) be an algebraic subgroup defined
over Q. A subgroup T of GQ{={geG\gVQ=VQ}), the Q-rational points
of G, is said to be arithmetic if there exists a lattice L in V such that T
is commensurable with GL.
If we identify V with C by means of a basis of L we obtain a natural
identification of G with an algebraic subgroup G' of GL(n, Q defined
over Q such that GL gets identified with G'Z=G' n GL(n, Z). A subgroup
P of G'z is a congruence subgroup if there is an integer m such that
F = {x\xeG'z and x=Imodm}.
Evidently /"" has finite index in G'z.
10.13. Proposition. Let GcGL(n, Q be cm algebraic subgroup defined
over Q and let Gz = GnGL(n,Z). Let p be a representation ofG defined
over Qona vector space V {defined over Q). Let Lbea lattice in VQ. Then
there is a subgroup r<=Gz of finite index such that p(r) L=L.
Proof, p being defined over Q, if we fix a basis of V and write pig)
as a matrix (p„v(g))„,, for gsG, we see that the p„„(g) are polynomials
in the entries gi} of g with coefficients in Q. Thus if
6^v(g.j-f5u)=P^v(g)-^v
then Q^ are polynomials without constant term. If m denotes the least
common multiple of the coefficients of all the polynomials g„v one sees
that
e„v(g.,-<yez
if gij—Sij is divisible by m. Thus the subgroup
T={ye Gz|y'=:/mod m}
is a congruence subgroup of Gz such that p(y) L = L for all yeT.
10.14. Corollary, (i) Let G<=GL(V)be an algebraic subgroup defined
over Q (V being a vector space defined over Q). Let r be an arithmetic
subgroup ofG and L a lattice in Vq. Then T contains a subgroup of finite
index leaving L invariant.

166
X. Reduction Theory in SL(n) and the Compactness Criterion
(ii) Let q>: G^*G' be an isomorphism of algebraic groups defined
over Q. If TcG is an arithmetic subgroup then <p{T) is an arithmetic
subgroup of G'.
(iii) Let q>: G—> G' be a homomorphism defined over Q of the algebraic
group G (defined over Q) in the algebraic group G' (also defined over Q).
Let r<=G be an arithmetic subgroup. Then (p(T) is contained in an
arithmetic subgroup of G'.
Proof r is commensurable with GL for a lattice L<=VQ (GL=
{ge G|gL=L}).Wemay therefore assume that T—GL. Identifying V with
C by means of a basis of L, we see that G is identified with an algebraic
subgroup of GL(n, Q defined over Q and T=GL with Gz. Now
according to the proposition there is a congruence subgroup of Gz which
leaves L invariant. This proves (i). The other two assertions are
immediate consequences.
Thus the notion of an arithmetic subgroup of G depends only on the
structure of G as an algebraic Q-group and not on any particular
realisation of G as a Q-subgroup of the linear group. Our aim now is to
give a criterion to decide when the quotient GJT for an arithmetic
subgroup r of the Q-group G, is compact. Towards this end we prove
first
10.15. Proposition. Let G<=GL(n,Q be an algebraic group defined
over Q and let Gz=GnGL(n,Z) and GR=GnGL(n, R). We assume
that G admits no nontrivial characters defined over Q. Then GcSL(n, Q
and the natural map
GJGZ-^ SL(n, R)/SL(n, Z)
is proper (here of course GR and SL(n, R) are considered as real Lie
groups).
Proof. Since G admits no nontrivial characters over Q, G<=SL(n,Q.
Let p: SL(n, Q—> SL(m, Q be a representation defined over Q such that
there exists in Q"a. vector v with the property
-G={geSL(n,Q\p(g)veCv}
(cf. Preliminaries §2.1). Then we have for gsG
p(g)v=x(g)-»
where %: G—>C* is a rational character defined over Q. It follows that
p(g)'v=v- Let q> be the orbit map.
gi->p(g) lv

X. Reduction Theory in SL(n) and the Compactness Criterion
167
of SL(n, R)in C™. Since G={g|geSL(n, Q, pig) v=v} setting T=SL(n,Z)
we see that GRr=(p~1(p{r)v). Now according to Proposition 10.13,
p(r)<=GL(m, Q) and contains a subgroup H of finite index leaving Zm
invariant. Clearly p(r)Zmc: Q™ and the sum L= £ p(y)Zm is evidently
equal to a finite sum among them so that L is a lattice stable under r.
It follows that we can find AeZ such that p(r)Xv<=L. In particular
p(T) v is closed in C". Thus GRr is closed in SL(n, R). GR being open and
closed in GKr we see easily that the map GjJGz—> GRr/r is a homeo-
morphism where GRr is given the induced topology from SL(n, R).
Since GRr is a saturated subset of SL(n, R) for the action of T the map
GRr/r^> SL(n, R)/r is a homeomorphism of GRr/r onto a closed subset
of SL(n, R)/r. Composing the two maps we see that the natural map
GR/GZ^> SL(n, R)/SL(n, Z)
is proper.
10.16. Proposition. Let G<=GL(n, Q be an algebraic subgroup defined
over Q. Let G° be the identity component of G and g the Lie subalgebra of
M(n, Q corresponding to G°. Let Ad: G—>GL(g) denote the adjoint
representation of G in g. Assume that G° (which is also defined over Q)
has no nontrivial characters defined over Q. The space g is spanned by
gz=g nM(n, Z) and hence has a natural Q-structure; with respect to this
Q-structure Ad is a homomorphism defined over Q. IfG* = Ad(G) and
GJ is the subgroup {AdglgeG, Adg(gz)=gz}, then G£ is an arithmetic
subgroup of G* and the natural map induced by Ad
A: GjGz-^Gi/Gi
is proper.
Proof. The group G° is a normal subgroup of finite index in G defined
over Q. One deduces easily from this that if the Proposition is valid for
connected G it is valid for all G. In the sequel therefore we assume that
G=G°. Now let p be a representation of GL(n, Q defined over Q on a
vector space F (over Q) such that there exists veFQ with the property
G={g|geGL(n,C), p(g)veCv}.
Since G does not admit non-trivial characters defined over Q, pig) v=v
for gsG.lt follows that (since G is connected)
Q={X\XEM(n,Q,p(X)v} = 0
where p denotes the representation g on F induced by p. Since p is
defined over Q and feFQ, g is defined over Q. Hence g nM(n, Z) spans g
as a complex vector space. It is now immediate that Ad is defined over Q
and hence G*=Ad G is an algebraic subgroup of GL(g) defined over Q.

168
X. Reduction Theory in SL(n) and the Compactness Criterion
In view of Proposition 10.15 and Mahler's criterion (Corollary 10.9) we
see that it suffices to prove the following. Let gmsG be a sequence of
elements in G such that we can find f„eZn—(0) with the property that
g„ fm tends to 0 as m tends to oo; then g* = Ad g„ cannot be relatively
compact modulo G£. Assume the contrary. Let y„eG2 be chosen such
that g*y*€£, a compact subset of G*. Now let Xlt..., Xt be a basis
for gz. Let B be the associative algebra generated by g in M(n, Q. Let
s=(sj,..., s^ be a finite sequence of integers and let /: s—>(1,..., f) be
any map. For a pair (s, /) as above let W(s, f) be the element
Clearly the W(s,f) as s and / vary span B as a vector space. Let
{W^= IV(S(,/i)|lgigr} be a finite subset of the W(s,f) spanning the
finite dimensional algebra B. Now let y„eG be an element such that
Adym=y*. Then g*yZ(Xd=gmy„Xiy~1g~1. Since g*y*eE we can find
a constant M>0 such that for 1 gig/
||g*y**,||:£M||.Yl||
where ||r|| for a matrix T is defined as Sup{||Tp||/||p|||0#p=
(»!,...,pjeC} and for x=(x1,...,x^)eC" ||x|| = { £ x?}*. Since for
rt, T2€M(n,Q we have || 7i r21| g || 7J || • ||r2||, it follows that we can
find a constant C>0 such that for 1 gigr
UmymWiy^g^\\^c.
Let W^(m)=ymW^y~1. Clearly the set {Wf'llgigr} spans B as a linear
space and we have
Moreover since y*€GJ and X(€M(n,Z) for lgigl, H?m)eM(n,Z). Now
consider the linear span in C of {W'j(m)/Jlgigr}. This space is stable
under B hence under g and therefore under G as well. By passing to a
subsequence and rearranging the Wtlm) if necessary we can assume that
{wr'/jigigp}
is a basis for the G-stable space spanned by
{HjW/Jl£i£r}.
Now gm^(m)/m=gm^(m)g-1gm/m and hence ||gm^(m)/m||-gC||gm/m||.
Thus gmWtlm)fm tends to zero for lgigp. Now let vm be the element
Mm)fm a Wim)/m a ■ ■ ■ a W5",/Me A'R". Since the linear span of
W*./Jlg.-gp}

X. Reduction Theory in SL(n) and the Compactness Criterion
169
is G stable, G-vmeCvm and since G admits no non-trivial characters
over Q, gvm = vm for all geG. Thus g„v„ = vm. On the other hand vm+0
since fm =#0. Moreover since Wfm)f„, 1 ^ i^p, are all lattice points, vm is a
lattice point in APW (for the standard lattice there). Thus || vm || is bounded
away from zero. On the other hand,
Vm = gmVm = gm W^fm A gm W2*fm A - A ft. W™fm
and since g„W^m)fm tends to zero for l^i^p, g„vm converges to zero,
a contradiction. This completes the proof of Proposition 10.16.
We can now deduce
10.17. Theorem. Let G<=GL(n,C) be an algebraic group defined
over Q. Let T be an arithmetic subgroup of G. Let G° be the identity
component of G. Assume that G° has no nontrivial characters defined
over Q. Let gmsGbea sequence and let n: GR—> GjJT be the natural map.
Then the sequence 7i(g„) has no convergent subsequence if and only if
there is a sequence 0mer of unipotent elements such that 6„ + e and
Sm^mSm1 tends to e (compare Theorem 1.12).
Proof. Let Gz=GnGL(n, Z). Since T is commensurable with Gz we
can find an integer r such that /eGz for all yeT and (feT for all
yeGz. Thus if n'\ GR—> GfJGz is the natural map 7t(gJ has a convergent
subsequence if and only if n' (g J has. It follows from these remarks that
we can assume that T=GZ. Now in view of Proposition 10.16 combined
with Mahler's criterion (Corollary 10.9) we see that n(gj) has no
convergent subsequence if and only if there exists 0+X„€qz=q r\M(n, Z)
such that gmXmg~l converges to zero. Here g denotes the Lie subalgebra
of M(n, Q corresponding to G. Now the characteristic polynomial of
X„ is the same as that of X'm=gmXmg~l. Since X'm tends to zero, the
characteristic polynomial P„(t) of X„ must converge to r". On the other
hand XmEM(n,Z) so that P„(t) has integral coefficients. It follows that
Pm (t)=f for all large m. Hence for a suitable m0, X„ is nilpotent if m ^ m0.
Now let Ym=Xm0 for m<m0 and Ym=Xmfor m^m0. Then YmeM{n, Z)ng
and is nilpotent for all m. Clearly we can find an integer q such that for
all m, exp(qI^)=0meSL(n, Z). It follows that 0m is unipotent, belongs to
Gz and we see that gmdmg^1 converges to e. And 0m=#e since X„+0.
Hence the theorem.
We can now deduce
10.18. Theorem. Let GcGL(n,C) be a reductive algebraic group
defined over Q. Let G° be the identity component of G. Let T<=G be an
arithmetic subgroup. Then the following two conditions are equivalent
1) GjJT is compact.
2) Every element of Gq is semisimple and G° admits no nontrivial
characters defined over Q.

170
X. Reduction Theory in SL(n) and the Compactness Criterion
Proof. Since (G°)„ is of finite index in G„ we reduce the proof easily
to the case when G= G°.
(2)=*-(l). This is an immediate consequence of Theorem 10.17. In fact
if Gjjr is noncompact one can find a sequence g„eG such that n(gm)
has no convergent subsequence. But then we can find unipotents 0mer
such that 9„+e and gmdmg^,1 converges to e. Since r<=GQ, 9„eGq.
Hence the implication.
(l)=*-(2). Suppose xeGQ is a non-semisimple element. Then we have
x=s-u where s is semisimple and u is unipotent s, ueGQ. Since u is
unipotent we can find an integer r such that j/e Gz. Thus Gz contains a
unipotent element. Now since T is commensurable with Gz, GJGZ is
compact. It suffices to show that Gz contains no nontrivial unipotent
element. Now an element ge G is semisimple if and only if its orbit under
inner conjugation in G„ is closed in GR. If gsGz, the orbit of g under Gz
belongs to Gz and is hence closed. Since GR/GZ is compact we see that
the GR-orbit of g is closed as well: in fact if £ c: G is compact set such
that EGZ=GR and g„=enyn, ensE, yneT is any sequence in G such that
gnSgn1 converges then yngy~l converges; but then yngy~l =ygy'1 for a
fixed ysGz for all large n. Since en has a convergent subsequence the
assertion follows. Next if p is a rational representation of G defined
over Q, then for yeT<=Gz, p{y) leaves a lattice invariant in the
representation space where r is a suitable subgroup of finite index in Gz.
It follows that the eigenvalues of p(y) are algebraic integers. Also p{T)
maps into a matrix with entries in Q. Thus if p is a character, p(y) is a
rational algebraic integer and hence p(y)e {± 1}. Now Gjr is compact
and since p is defined over Q, p(G,)c.R*. It follows that p(G^)<={± 1}.
Since G is connected G„ is Zariski dense in G. This implies that
p(G)<={±l} and since G is connected, p(G)=l. This proves the
implication (1)=>(2).
A slightly more general version of Theorem 10.18 is the following
10.19. Theorem. Let G <= GL(n, Qbean algebraic subgroup of GL(n, Q
defined over Q and r an arithmetic subgroup. Then Gjjr is compact if
and only if every unipotent element of r belongs to the unipotent radical U
of G and G° admits no non-trivial character defined over Q.
Proof. We can assume that f=Gz=GnGL(n,Z). In fact if y is a
unipotent not in U, y*£ U for any k and for any yeT there is a k such
that y*€ Gz. Now U is defined over Q. Hence by the results of Chapter II,
UJU n Gz is compact (Remark 2.13). Also we can find a reductive
subgroup M defined over Q such that G=M -U, a semidirect product.
M being reductive Theorem 10.18 applies to M. We conclude then
MJMZ is compact if and only if M has no nontrivial characters over Q

X. Reduction Theory in SL(n) and the Compactness Criterion
171
and MQ contains no nonsemisimple element. Suppose now GJGZ is
compact. Let x be any character on G; then exactly as in the case when G
is reductive one concludes that x is trivial on G. Now let n be the
projection of G on M = G/U. According to the Corollary 10.14, there is a
subgroup r of finite index in Gz such that n(r)<=Mz. Clearly then
r<=Mz-Uz(Uz=UnGz). Thus MZUZ has finite index in Gz. Since
UjJUz is compact GjJGz is compact if and only if MJMZ is compact.
If MJMZ is compact MQ contains no unipotent element. If yeGQ is
unipotent, n(y) is unipotent in MQ. Hence if GJGZ is compact, %{y)=e
for all unipotent ysGz. Conversely, if every unipotent element of GQ is
in l/Q, Mq cannot contain unipotents. Also if GQ has no nontrivial
characters defined over Q, nor has MQ. Thus in this case MjJMz is
compact. Hence GjJMz Uz is compact. It follows that GjJGz is compact.
From Proposition 10.14 we deduce also the following result.
10.20. Theorem. Let G, G' be two Q-groups and r: G—> G' an isogeny
defined over Q and r an arithmetic subgroup of G. Then r(r) is an
arithmetic subgroup of G'.
Proof. The proof is easily reduced to the case when G and G' do not
admit any nontrivial Q-characters. Let G* and G'* denote the adjoint
groups of G and G'. We then have a commutative diagram
G —'—> G'
AdJ JAd'
G*—^G'*
where r* is again defined over Q. Since the map Ad induces a proper
map of GJGZ on G%IG% it follows that Ad~1{Gz) contains Gz as a
subgroup of finite index. Here we have fixed a realisation of G as a subgroup
of GL(n, Q. We then conclude from the definition of arithmetic groups
that if r*<=G* is an arithmetic group, Ad_1(r*) contains an arithmetic
subgroup of G as a uniform subgroup. Now let T, F be arithmetic
subgroups of G and G' respectively. Let T* be an arithmetic subgroup
of G* and let r* = r*(r*). Now r* is an isomorphism and fn Ad_1(r*)
(resp. r'nAd-1 (/""*)) is uniform in Ad-Hr*) (resp. Ad_1(r*)); since
r*oAd=Ad'or, the desired result follows.

Chapter XI
The Results of Kazdan-Margolis
This chapter is devoted, for the most part, to some work of Kazdan-
Margolis [1] of a fundamental nature. We formulate the main results in
the framework of what we called L-subgroups (Definition 1.21) whereas
Kazdan-Margolis considered only lattices; the proofs however are
essentially those given by them. The present treatment enables us to
secure a certain amount of unification in the theory of arithmetic groups
on the one hand and lattices on the other. Theorem 11.6 which, among
other things enables one to sharpen the results of Kazdan-Margolis, is
due to the author. Readers may notice that the present account has
borrowed from Borel [5].
11.1. Notation. Throughout this chapter G will denote a connected
linear semisimple Lie group without compact factors. We fix a maximal
compact subgroup K of G. Let g denote the Lie algebra of G and ! the
subalgebra corresponding to K. Let p be the orthogonal complement of
I in g with respect to the Killing form A( , ) on g. Let 8 be the Cartan
involution defined by the decomposition g=fffip and for Xeg, let
II^H2 = -A(X, 0(X)) and for X, Ye& let B{X, Y)= -A{X, 0(Y)). Then B
is a positive definite form on g, invariant under the adjoint action of K
on g.
With this notation we will first prove
11.2. Lemma. Let Q be a neighbourhood of e in g. Then there exists a
constant a>l with the following property: given any totipotent subgroup
HofGwe can find geQ such that for Xel) (=Lie algebra of H) and any
integer r>0, we have
wAdfixn^ofwxw
and
||Adg-'(x)||=a-'||X||.
Proof. Let o be a maximal abelian subspace of p and o* be the dual
of o. Then Ad (a) consists entirely of semisimple endomorphisms with
real eigen-values. For Aea*, let
gA={Xeg I Ad a(X)=k(a) ■ X for all aea).

XI. The Results of Kazdan-Margolis
173
Let <P = {A\Aea*, A=t=0, gA=|=(0)}. Introduce a lexicographic ordering on
a* and let <P+ denote the set of positive elements in <P with respect to
this ordering. Let
Then rt is a nilpotent subalgebra of g and the subspaces {gA|Ae$ + } are
mutually orthogonal with respect to B. Let N be the Lie subgroup of G
corresponding to rt. Then N is a maximal unipotent subgroup of G.
Moreover any unipotent subgroup H can be conjugated into N by an
element of K (see Preliminaries). Since || || is invariant under AdK, to
prove the lemma, we may assume that H=N. Let A be the Lie subgroup
of G corresponding to a and let exp: g ->•G be the exponential map. Let
!31={A'ea|expA'e£2}.
Then flt is a neighbourhood of 0 in a. Let C be the cone {X\X(X)>0 for
xe$ + } in a; C is nonempty since #+ contains a basis A of a* such that
any element of $+ is a nonnegative linear combination of elements of A
(see Preliminaries § 1.7). Let leCnfi, andg=exp X. For YegA, xe$+,
and reZ,
Adgr(7)=er;iw-y.
Thus if 7= £ ^ yAeg\ is any element of rt,
-.60+
Adgr(7)= £ ^wy1.
It follows that if we set tx=lnf{eMX)\/.e<P+}, we have for every integer
r>0,
a-2r||Adgry||2 = a-2r £ e^^H 7*||2
-U0 +
^ I lir-YHIH2
">a2r Xe_2'"'-('¥,ll>'All2^l|Adg-ry||2a2r.
Since i(Ar)>0 for Xe<P+, a>l. This proves the lemma.
11.3. Corollary. Let H<=-G be a connected unipotent subgroup of G
and E<=H a compact set. Then given a neighbourhood Ut of e in G, we
can find g€G such that gEg~la Uv
This follows from Lemma 11.2, once one observes that the
exponential map of the Lie algebra h of H on H is a homeomorphism
and H is closed in G.
We now continue with the notation introduced in the proof of
Lemma 11.2. Thus a is a maximal abelian subspace of p. Let z(a) denote

174
XI. The Results of Kazdan-Margolis
the centraliser of o. We also set
n-= LI g\ b- = z(o)©rt-
we then have a direct sum decomposition
g=rt+ © b" = rt+ © z(o) © n~.
n- is also the Lie subalgebra corresponding to a maximal unipotent
subgroup N~ of G. With this notation we next prove
11.4. Lemma. Assume that G has no compact factors. Let fjc.g
be a nilpotent subalgebra of G. Then there exists g in G such that
Adg(t,)nb- = (0).
Proof. We assume as we may that G <= GL(n, R) and g <= M(n, R). Let
Qc be the complex linear subspace of M(n, Q spanned by g and let Gc
be the corresponding complex Lie subgroup of GL(n, Q. Then Gc is
an algebraic subgroup of GL(n, Q. Moreover let b£ denote the C-span
of b~ in M(n, Q and B£ the complex Lie subgroup of Gc corresponding
to bf. Then Bq is an algebraic subgroup of Gc. Let H <=G be the Lie
subgroup corresponding to fj. Let i/cc:Gc be the Zariski closure of H
in Gc and let bccgj, be the corresponding Lie subalgebra. Then Hc
being connected nilpotent and algebraic decomposes into a direct
product Hc = Tc ■ Uc of a torus 7^. and a unipotent subgroup Uc. Since
Hc is defined over R, so are Tc and Uc. It follows that the Lie algebra
fjc of Hc decomposes into a direct sum ^ + 1^ where t^ and Uf. are
generated as vector spaces over C by ucng=u and tcng=t
respectively. Evidently it suffices to show that there exists geG such that
Adg(tjc)nbc =(0). Now since HC<=GC is algebraic and nilpotent, we
see that
Adg(hc)nbc =Adg(tc)nbc ffi Adg(uc)nbc-
Thus it is sufficient to show that there exists gsG such that
Ad g(tc)n b^ =(0) and Adg(uc)nbc =(0). Now the set
. S={g€G|Adg(uc)nbc=(0)}
= {g€G|Adg(u)nb-=(0)}
is evidently the complement of an analytic set. It is non-empty since u
can be conjugated into rt+ and rt+nb_ =(0). Thus S is open and dense
in G.
Next let & denote the family of all nontrivial subtori of 7^. defined
over R. Then !F is a countable set. Let 7£e.F and t'c<=tc be the
corresponding Lie subalgebra. Let t' = t'c n g and
il(72)-{geG|Adg(f)«=b-}.

XI. The Results of Kazdan-Margolis 175
Evidently A (7^) is an analytic subset of G. Further if we set b=z (a) ffi rt+,
then b and b_ are conjugates and bnb_=z(o). It follows that
g'= () Adg(b~) is an ideal in g contained in z(a). Since z(a) is reductive
gsG
and has a compact semisimple part, g'=(0) so that A(JQ^G. Thus
S(73={geG|Adg(tW-}
is open and dense in G. G being locally compact and & being countable
theset Sn{f)S(T$ (=£,say)
is non-empty (Baire's theorem). Let geE. Then Adg{tc)nb^ =(0); in
fact if Adg(tc)nbc =l=(0) it must contain Adg(t'c) for some t'ce^, a
contradiction. This proves the lemma.
11.5. Corollary. Assume that G has no compact factors and let beg
be a nilpotent Lie subalgebra of g. Then given C>0, there exists geG
SUcHthat ||AdgaO||^C||X||
for all Xe\).
Proof. In view of Lemma 11.4 we can find g^sG such that
Adg1(b)nb_=(0). Since we can find constants p, q>0 such that
p||*||g || Ad giWUgg ||*||
for all Xeg, replacing C by a different constant if necessary, we need
only prove the lemma for b1=Adg1(b). Now consider the projection
% of g onto rt+ given by the direct sum decomposition g=rt_ ffi z(o) ffi rt+.
Since ker7r=b~, n restricted to b^ is injective. Thus we can find a
constant m>0 such that for Xebj
|| TrOOU^m || X||.
Thus if XEt)u
X = X_+X0 + X+
where X_en~, X0Ez(a) and X+exi+. Now arguing as in Lemma 1,
we can find aea such that all the eigen-values of a on rt+ are greater
than zero. If we set g2=expa, we see that
Adg"2X=Adg"2X_ +Adg"2X0+Adg"2X+
and Adg2 leaves rt", z(a) and rt+ stable. It follows that for Xe^
||Adg5X|| ^||Adg-2X+||^or'||X+||=or'||7t(JO||^inor'||A-||
where a is the smallest of the eigen-values of Adg2 in the space rt+; by
our choice a > 1 so that if n is chosen so large that m a" > C and we

176
XI. The Results of Kazdan-Margolis
set g=g"2, then for all Xs\)t, we have
IIAdgA-HHC ||.y||.
This proves the lemma.
11.6. Corollary. We assume that G has no compact factors. Given a
constant C>0, there exists a compact set E of G with the following
property: i/tjcg is any nilpotent subalgebra of g, there exists geE such
that for all Xe^, ||Adg.Y||fcC|X||.
Proof. For each positive integer p, let Lp denote the subspace of
the Grassmann manifold of p-dimensional subspaces of g consisting of
all nilpotent subalgebras of dimension p in g. It suffices evidently to prove
the corollary for all E,eLp. Lp is a closed subset of the Grassmann
manifold and is hence compact. Now for each E,eLp, we can, according to
Corollary 11.5 above find an element x(h)eG such that
\\Adx(MX)\\*2C\\X\\
for all Xefy. Now, from the definition of the topology on the
Grassmannian it follows immediately that there exists an open
neighbourhood U^ of h in Lp such that
||Adx(h)*||=C||X||
for all X eh', h'€ l^. The collection {U^| [j eLp} is an open covering of Lp.
Since Lp is compact we can find a finite set rjj,..., rjr of elements of Lp
such that t/iiu---ut/^=Lp. Evidently the finite (hence compact) set
£={x(rjf)|lgi^r} has the required property.
Now, let exp: g—>G be the exponential map. For r>0, let
Br={.X'eg|||.X'||<r} and let Ur=expBr. We choose and fix once for
all a constant r0 such that the exponential exp: g—>G when restricted
to Bro is a diffeomorphism of Bro onto the open set Uro. Let $: Uro—>Bro
be the inverse of the exponential map and for xsUro, let |x|= ||$(x)||.
With this notation we will now prove
11.7. Theorem (Kazdan-Margolis [1]). Assume that G has no compact
factors. Let c> 1 be any constant. Then there exists a constant a,0<a<ro,
and a compact set E<=.G with the following properties:
(i) AdgBa<=Bro and gUag-l<=UrQfor all geEvE~\
(ii) If T<=.G is any discrete subgroup we can find an element geE
such that , ,. . .
\gxg "I'Sclxl
for all xeTn Ua.
Proof. Choose £ to be a compact set as in Corollary 11.6 above
taking for C, the constant c. Let a>0 be a constant such that (i) stated

XI. The Results of Kazdan-Margolis
177
above holds and further Ua is a Zassenhaus neighbourhood i.e. Ua
possesses the following property: if xt xkeUa generate a discrete
subgroup of G, then {«P(X()| l^i^fc} generates a nilpotent subalgebra
of g. (Definition 8.22: the existence of such a neighbourhood Ua of e in G
is guaranteed by Theorem 8.16.) We assert now that a and E chosen as
above have the required property. Let fcGbe any discrete subgroup
of G and let fj be the subalgebra of g generated by $(rn t/J. Then in
view of our choice of E, we can find geE such that
||Adg(J0||^c ||X||
for all Xef). Taking X=${x) with xefnU, we have
||Adg<P(x)||£C||<P(x)||.
Now Adg$(x)eBro since $(x)eBa. Moreover,
gxg-1=expAdg(«P(x))€C/ro
so that
^(g^g_1)=Adg«P(x).
Thus we have
Igxg-^clxl
for all xeUanr. This proves the theorem.
We can now deduce
IIS. Theorem (Kazdan-Margolis [1]). Let G be a connected linear
semisimple Lie group without compact factors. Then there exists a
neighbourhood W of e in G with the following property: given any discrete
subgroup r ofG, there exists geG such that grg~1nW={e}.
Proof. Fix a constant c> 1 and choose a constant a>0 and a compact
subset £cG as in Theorem 11.7. Let 6>0 be a constant such that
Adg-^B^cB,, for all geE. Set W=Ub=&ipBb. We claim W has the
property required in the statement of the theorem.
To see this define for each geG a constant a(g)>0 as follows:
a(g)=b if Wngrg~i = e; if this is not the case, a(g) is the greatest
positive real number such that Ua{g)ngrg~1={e}. (Such a constant
a(g) exists since the intersection of W with gTg'1 is a finite set.) Clearly
it suffices to show that the upper bound b0 of the set {a(g)\geG} is
equal to b. Suppose then that b0 is less than b. We can find geG such
that a(g)>c~1b0 (note that c>l by our hypothesis). In other words
if we set r=grg~l then Pr\W^{e} and for xefnH, x#e,
Ix^C"1/^. Now according to Theorem 11.7, we can find g'eE such
that |g'yg'_1|^c|y| for all yer'nUa. Consider now the group
r^gT'g'-1. Evidently r"=g'gTg-1^'1. Suppose now xer'nW,

178
XI. The Results of Kazdan-Margolis
then x=g'yg'~l for some ye/"" since y=g'~lxg' and g'eE, it follows
that yeUa. Thus we have from our choice of g',
|x| = |g'yg'-1|^C|y|.
It follows that \y\<x so that yeW. But then if x+e
\x\^c\y\>c-c-ib0=b0.
Since by our hypothesis a(g'g)<b, P'n W + (e). On the other hand the
inequality above shows that a(g'g)>b0, a contradiction. This proves
the theorem.
11.9. Corollary. Let G be a connected semisimple Lie group without
compact factors and \i a Haar measure on G. Then there exists a constant
M>0 with the following property. For any discrete subgroup r<=.G, the
total volume of the homogeneous space G/r with respect to the measure
on it induced by \i is greater than or equal to M.
Proof. In view of Corollary 5.17 of Chapter V we may assume that
G is linear. Now choose W as in Theorem 11.8 and let V<=G be a
neighbourhood of e in G such that K=K_1 and VV~lcW. Let T be
a discrete subgroup of G. Since G is semisimple, it is unimodular and
hence the Haar measure n is invariant under inner conjugation. We
may therefore replace f by a conjugate r'=gTg~1. In view of
Theorem 11.8, then, we can assume that rr\W—{e}. Consider now the
natural projection %: G^G/r. Setting n(V) = M, it clearly suffices to
show that the restriction of % to V is 1-1. In fact if n(v)=n{v'\ v, v'eV,
we can find yer such that vy—v', i.e., y=v" v~lsVV~l<=- W. Hence
ysrr\W={e} so that v = v'.
A second interesting consequence of Theorem 11.7 is the following
result due essentially to Kazdan-Margolis [1]. Kazdan-Margolis
formulate and prove the result for lattices; their proof however practically
carries over to cover the present formulation. We recall the definition
(Definition 1.21) of an L-subgroup. Let G be a locally compact group.
A discrete subgroup r<=G is an L-subgroup if it has the following
Property(L): let n: G—>G/r be the natural map; for a sequence
{an\neZ+} in G, {n(an)\neZ+} has no convergent subsequence (in G/r)
if and only if there exists {Oner-(e)\neZ+} such that {andna-1\neZ+}
converges to e (equivalently for a neighbourhood U of e in G if K(U)=-
{geG|grg-1n U = {e}, then n(K(U)) is relatively compact in G/r).
11.10. Theorem. Let G be a connected linear semisimple Lie group
without compact factors and r<=.G an L-subgroup. Then there exists a
finite set Sl<=.r—{e}of unipotent elements in T with the following property.
Given any neighbourhood W of e in G, we can find a neighbourhood W<=.W

XI. The Results of Kazdan-Margolis
179
of e such that if for geG, gTg~l n W^{e}, then fng"' W'g contains a
non-trivial unipotent element conjugate (in T) to an element of Sv (Note:
Sl could be empty; in that case the theorem asserts that there is a
neighbourhood W of e such that no element of T other than e has a conjugate
in W.)
Proof Let c> 1 be any constant. Choose a compact set £cG and
a constant a>0 as in Theorem 11.7. For a discrete subgroup A of G
we will say an element hsG is "adapted to A" if (i) hsE and (ii) for
x€AnU„,\hxh-l\^c\x\.
Let b>0 be a constant such that Adg(Bb)cBa for all ge£u£-1.
(Evidently this implies that gUbg~l<=Ua for all ge£u£-1.) Let ZczG
be a compact subset such that Zr=>K(Ua) = {g€G\grg~ln Ua = {e}}:
note that n(K(Uaj) is compact so that such a Z exists. Let S be the
(finite) set S={yer\ZynEUbE-lZ*Q).
Let S'cS be the set of non-unipotent elements and S, the set of non-
trivial unipotent elements in S. The orbits of elements of S' under inner
conjugation do not contain identity in their closure (cf. Preliminaries
§ 1.7). We can therefore find a neighbourhood W0 of e in G such that
the elements of S' have no conjugates in W0. We assume as we may,
that W is of the form Ub. for a suitable b' and that Ub. cUbr\W0. Let m
be the smallest integer such that c™b'>b. Let PV be a neighbourhood
of the identity such that Wc Ub, (= PV' c t/fc n W0) and
hP-Vh-'cl/,.
for all he£p = £ ■ £ £ for all pgm. We assert now that the W chosen
p times
above has the requisite properties.
We choose inductively a sequence h„,n^0, of elements of £ as
follows: h0 is any element adapted to AQ = grg~1; assume h0h1,...,hn
chosen and define inductively Ak,l^k^n + l by setting Al = hQAQho '
and Ak = hk_lAk_lhkll; then h„+l is an element of £ adapted to A„+1.
With this notation, we claim that AHnUb = (e) for all large n. To see
this we first observe that if xeAH is such that hnxh~leUb, then
xeA-'C/^t/.sothat ^xh-i^e]xU
it follows thus that xeUc-ib<=Ub. Once again if we set g0 = h0,gn = hng„_l
inductively, it is easily seen that for xeA0, either gnxg~1^Ub or
xeUb and |x|^c"("+1) fo. Thus if n0 is chosen so large thatc"°+1-
Inf{|x||xezl0n Ub,x + e}>b we have clearly for n^n0,Ann Ub = {e). Let
/c = inf{«|zlnnt/1(={e}}. Since hWh~lcUb for all heE", p<,m, we note
that k^m+l. Now setting 'g,- = g,-g, we have fn'g^1, L/f)'g)c_1 = (e) so

180
XI. The Results of Kazdan-Margolis
that from our choice of Z it follows that there exists xeT such that
Next let y=|=e be any element of T such that
'sk~2y'gk-2^ub-
Then we have
'gk-iyx = hk_1-'gk_2y'g;_12-K}1-'gk_l-xeEUbE-l-Z.
It follows thus that x~1yxeS. Now 'gk-2y'Sk-2€^b- ^ follows that
gyg~1eUb and \gyg~1\<c~ik~1). b^c~mb<b' and W'= Ub, by
assumption. Hence gyg~1e\V. Now if x~1yxeS', y cannot have a conjugate
in W0 hence a fortiori in W^W0. It follows that x~1yxeS1. (In
particular y is unipotent.) This proves the theorem.
11.11. As hitherto, let G be a connected linear semisimple Lie group
without compact factors and FcGa L-subgroup. Suppose now WJ,'
is a fundamental system of neighbourhood of e in G. For each integer n,
let Wn be a neighbourhood of e in G with the following properties
(i) w^w;.
(ii) If gTg-i n W^ 4= (e), gTg~1r\Wn contains a nontrivial unipotent.
It follows therefore that if we can find a sequence gHsG and a sequence
yner such that yH=¥e and g„ ^g"1 converges to e then we can find
unipotent 0n=t=e in r such that g„ d„ g"1 converges to e.
11.12. Corollary. Let Gbe a connected linear semisimple group without
compact factors and fez G an L-subgroup. Let n: G —> G/r be the natural
map. Then for a sequence ansG, n(an) has no convergent subsequence if
and only if there is a sequence ofunipotents Qnsr~{e) such that an8na~i
converges to e (compare Theorem 10.17).
11.13. Corollary. Let G,T be as in Corollary 11.12. Assume further
that G/r is not compact. Then there exists a nonempty finite set S^f-je)
ofunipotents and a neighbourhood Wofe in G with the following properties.
Ifx(+e)er has a conjugate in W, then there exists yeSt and asT such
that ax a-1 and y generate a nilpotent subgroup of T. Moreover there
exists a unipotent element z( + e) in T commuting with x. In particular r
contains a unipotent element 8+e.
Proof. We take for W in Theorem 11.10, a Zassenhaus
neighbourhood of e in G (cf. Definition 8.22) and let St and W be chosen as in
that theorem. In view of Corollary 11.12, one sees that (since G/r is
not compact) Sj+0. Let geG be such that gxg~lsW. Then we can
find jSer and yeSt such that gPyP~lg~leW (Theorem 11.10). Since
W is a Zassenhaus neighbourhood and W^W,gxg~1 euidgPyP~1g~i

XI. The Results of Kazdan-Margolis
181
generate a nilpotent subgroup of G. If we set a = /?_1, y and ax a""1
evidently generate a nilpotent subgroup of F. Consider now the
sequence yneF defined inductively by setting y0 = PyP~1 and for n>0,
yn=xynx~1y~1. If n is the highest integer such that yn=¥e, evidently
yn is unipotent and commutes with x.
11.14. Corollary. Let fcG be a discrete subgroup such that G/r is
compact. There is a neighbourhood W of e such that no element of F
different from e has a conjugate in W. In particular every element is
semisimple.
Proof, F is evidently an L-subgroup. It suffices to show that every
element of F is semisimple (cf. Corollary 11.12). For this again we need
only show that the orbit under inner conjugation of an element yeF
is closed (cf. Preliminaries §1.7). In fact let yeF be any element and
g„sG a sequence such that gnyg^1 converges. Since G/r is compact
we can find a convergent sequence x„eG such that gn=xnyn for suitable
yneF. It follows that Vnyy"1 converges and since F is discrete in G,
the sequence yn y y,T * must in fact terminate. Thus the limit of the
sequence gn y g~i = x ym ■ y ■ y~i x~' where x is the limit of the sequence
x„ and m is sufficiently large. Hence the corollary.
11.15. Remarks. In view of Theorem 1.12, a lattice is an L-subgroup.
Thus Theorem 11.10 as well as all the corollaries proved above hold
in particular for lattices. Also, according to Theorem 10.17, an arithmetic
subgroup in a semisimple group is an L-subgroup. Thus Theorem 11.10
applies also in this case though Corollaries 11.12 and 11.15 are indeed
trivial direct consequences of Theorem 10.17 itself. However Corollary
11.13 is of considerable interest in this case as well. As will be seen in
a later chapter, this is a crucial result for our method of developing the
theory of arithmetic groups further.
Our next result may appear somewhat technical but has, as we will
see, some interesting consequences.
11.16. Theorem. Let GcGLfn, C) be a semisimple algebraic group
defined over R. Let G be the identity component o/G,=GnGL(n,R).
Assume that G has no compact factors. Let FcG be an irreducible lattice.
Let /Icf be any subgroup and <PcF a maximal unipotent subgroup
ofT normalised by A. Let A (resp. F) be the Zariski closure of A (resp. $)
in G. Let M be a maximal reductive R-subgroup of A. If$^{e} there
exists a connected algebraic solvable subgroup So/G with the following
properties.
(i) A normalises S.
(ii) S n F is Zariski dense in S.
(iii) [S,S]cFczS.

182
XI. The Results of Kazdan-Margolis
(iv) The kernel of the action ofM on Sis central in G. Also, the kernel
of the action ofMon F is finite.
We will first deduce some consequences of Theorem 11.16 before
we proceed to its proof.
11.17. Corollary. Let G be a connected linear semisimple Lie group
without compact factors an r<=G an irreducible lattice. For deT, let Ge
(resp. Q denote the centraliser of 6 in G (resp. f). Let peT be an element
normalising a non-trivial unipotent subgroup of T. Then the eigen-values
of Ad p are all algebraic integers. Moreover if Ad p is not unipotent,
the characteristic polynomial ^eRLXn of Adp has a monie rational
integral factor PpeZ[X~] which is coprime to (X— 1). Further if peT is
such that Gp/rp is non-compact, p normalises a non-trivial unipotent
subgroup of t.
Proof. The natural map Gp/rr\Gp^>G/r is proper (Lemma 1.14).
Now assume that Gp/rp is non-compact. Then we can find an€Gp and
unipotents 0„( # e) in T such that an 8n a~1 converges to e (Corollary 11.12).
Let Q be a Zassenhaus neighbourhood of e in G (Definition 8.22) and
let N=dimG. Choose an integer n so large that
anpkBnp-ka^{=pkandna-lp-k)sQ
for O^k^N. Let g be the Lie algebra of G and Xeq the unique element
such that adX is nilpotent and expX=d„. Then clearly,
pk 6np ~k=exp Ad pk{X). Let u be the Lie subalgebra of g spanned by
{AdpkQOIOgfc^N}. Since dim q=N,u is normalised by p: Adp(u)cu.
If U is the Lie subgroup of G corresponding to u, then p normalises U.
We now claim that U is a unipotent subgroup of G. To see this we
assume as we may that G=GnGL(n,R) where G<=GL(n, Q is an
algebraic R-subgroup of GL(n, Q. Let $t be the group generated by
{pkdnp~k\0^k^N}. Then $t is a nilpotent and hence unipotent
subgroup of G (our choice of an, 6n and Q guarantees this). Let 0l be the
Zariski closure. Then tp^U as is easily seen. Since 0 is unipotent
so is U. Let $2 = [/nf. Then $2 is a non-trivial unipotent subgroup
of r normalised by p. This proves the last assertion.
Now let p er normalise a non-trivial unipotent subgroup of T. Let
A be the cyclic subgroup of T generated by p. Let M, A, S etc. be
defined as in Theorem 11.16. Let S (resp. F) be the identity component
of SnGL(n, R) (resp. FnGL(n,R)). Let s (resp. f) denote the Lie
subalgebra of g corresponding to S (resp. F). Now F/F n r is compact
(Theorem 2.1) so that the Z-span L of exp~l{Fnr) in f is a lattice
in (the vector space) f. Clearly Adp leaves L stable. If ApeR[X] denotes
the characteristic polynomial of Adp acting on f, evidently Ap is a
monic integral polynomial. Next consider the quotient group S/F and

XI. The Results of Kazdan-Margolis
183
let n: S-+S/F be the natural map. We identify s/u with the Lie algebra
of S/F. Now since F/FnT is compact, n(r) is a discrete subgroup
of S/F. Since S/F is abelian, we see once again that the Z-span of
exp_17r(r) is a discrete subgroup L of the vector space s/f. Let f'cs/f
be the subspace spanned by L. The endomorphism Adp of g leaves
s and u stable and hence induces an automorphism a(p) of s/f which
leaves f and L stable. Clearly the characteristic polynomial Bp of a(p)
acting on f is also integral. Now if A: s—>s/u is the natural map and
f'=n~l(?) then one sees immediately that the characteristic polynomial
of Adp acting on f" is precisely ApBp=Ppt a monic polynomial in
Z[X~\. Evidently Pp is a factor of xp. Let g=g ® C and f" be the C-
subspace of g spanned by f". Then f" is stable under the adjoint action
of A. Let Z be the centre of G. Then M0=M/MnZ acts faithfully
on f" (Theorem 11.16). It follows that every algebraic representation of
M0 is the quotient of a representation contained in a tensor product
of representations equivalent to t or its dual where t is the adjoint
action of A0 on f" (cf. Preliminaries §2.1). It follows in particular that
if the eigen-values of x(x\ xeA, are algebraic integers, so are the
eigenvalues of t'(x) for any representation t' of A. (Note that for xeA, the
semisimple part of x belongs to M.) Also if t' is trivial on An Z, x'(x)
is unipotent if and only if t(x) is. Now taking x = p, and r'=Ad, we
see that the eigen-values of Adp are algebraic integers and Adp is
unipotent if and only if t(p) is; and x(p) is unipotent if and only if
iJeZ[\Y] is a power of (X— 1). This proves the corollary.
11.18. Corollary. Let G be a connected semisimple Lie group without
compact factors and TcG a lattice. Then there is a neighbourhood V of e
in G with the following property. An element p er has a conjugate in
V if and only if p is unipotent.
In the space of all monic polynomials in R[X] of degree N (a fixed
integer), the subset of all (monic) polynomials with a monic integral
factor different from X— 1 is closed. This fact combined with
Corollary 11.17 yields the desired result, when T is irreducible. The general
case follows from this by considering the projections of T in the quotients
G/H for those connected normal subgroups H such that HT is closed.
We will now take up the proof of Theorem 11.16. Our proof of the
theorem depends on a result of a general nature (for facts from
representation theory used, see Jacobson [1, Ch. VII]).
11.19. Proposition (Garland-Raghunathan [1]). Let GcGL(«,C) be
a connected linear semisimple algebraic group defined over a subfield k
of C. Let V be a unipotent k-subgroup of G. Let Z(U) be the centraliser
of U in G and let V denote the unipotent radical of Z(U). Let M be a
maximal reductive k-subgroup of Z(U). For a subgroup WcG let Hk =

184
XI. The Results of Kazdan-Margolis
H n GL(n, k). Then the kernel of the action of Mk on \k is a normal
subgroup ofGk. In particular i/Z(U) does not contain any proper connected
normal subgroups of G, this kernel is finite. If fc=R or C, Mk is closed
in Gk and the identity component of Mk has finite index in Mk (for the
euclidean topology).
Proof. \ is Zariski dense in V (cf. Preliminaries §§2.2, 2.3). The
general case therefore follows from the case fe=C except for the last
assertion. Let B be the centraliser of M in G. Since M is reductive so
is B (cf. Preliminaris §2.6). Clearly UcB. Let U'sU be a maximal
unipotent subgroup of B. Let Z(U') denote the centraliser of U' and V
be the unipotent radical of Z(U'). Evidently McZ(U')cZ(U). Now
Z(U)=M-V and Z(U')=>M. Hence Z(U')=M-(Z(U')nV). It follows
that V'=Z(U')nVcV. Thus it suffices to show that the kernel of the
action on M on V is normal in G. Let T be a maximal torus of B
normalising U'. Let X(T) denote the group of characters on T and we order
X(T) so that the Lie algebra u' of U' is spanned by the positive root
spaces. Now the Lie algebra g of G can be decomposed into a direct sum
where W is a finite set of elements in X(T) and g*1 is the sum of all Ad B-
stable subspaces of g having X as the highest weight. Let
^={v\veq\ Adt(v)=X(t)v for reT}.
Then it is easily seen that the Lie algebra z(u') of Z(U') is precisely
LI flA. Let 0 denote the trivial character on T. Then g0 = g° is the
centraliser of B in g and is hence a reductive subalgebra. This sub-
algebra evidently contains m, the Lie subalgebra corresponding to M.
Since Z(U')=MV, we have z(u')=m©»' where »' is the Lie
subalgebra corresponding to V. Since we have
and g0 is reductive, m= g0. Moreover, one checks easily that »'=
LI Qx- Suppose now that for xeM, Adx(p)=p for all ve gA for all
Asy-(O)
X s !P — (0). Since Ad B commutes with Ad x for x e M, it follows that Ad x
acts trivially on the Ad B-submodule of g generated by { gJAe W—(0)}.
Now for X € W, gA generates g*1 as a B-module. Thus we see that if M'
is the kernel of the action of M on »', M' commutes with exp Qx for
XeW—(0). Clearly M' is normal in M. Since m= g0 and evidently
{exp Qx\Xe V> generates G, M' is normal in G. If Z(U) does not contain

XI. The Results of Kazdan-Margolis
185
any proper connected normal subgroup of G, M' being algebraic is
finite (hence discrete and central) in G. Since V is unipotent, xeM acts
trivially on V if and only if Ad x is trivial on »'. The last assertion follows
from the fact that M is algebraic. This proves the proposition.
11.20. Proof of Theorem 11.16. Let Z(F) denote the centraliser of $
in G and N the unipotent radical of Z(F). Let Z(F)=Z(F)nGL(n,R)
and JV=NnGL(n,R). Let H = Z(F) F and H = Z{F)F. (H is then a
subgroup of finite index in HnGL(n, R).) Let U be the unipotent
radical of H and t/=UnGL(n,R). We have evidently JVcI/cfl.We
will now show that the map H/Hnr-*G/r is proper or equivalently
that HT is closed in G. Let fc$ be a finite set of generators of the
group $ (note that F/$ is compact and F is nilpotent: (cf. Theorem 2.18)).
Suppose now that hneH, hn=znfn, znsZ{F\ fnsF,ynsr are sequences
such that hn yn converges to a limit xeG. Since F/<P is compact we can
assume (by passing to a subsequence if necessary) that there is a sequence
</>„e<P such that /„</>„=/„' converges to a limit in F. Replacing yn by
q>~1 yn and hn by zn/„' we can thus assume that we have yneT, hn=znfn,
zneZ(F), fneF such that /„ and hnyn converge to limits feF and xeG
respectively. Now for \j/eW, let i//n=fn"Af~l. Then we find that the
sequence
(K Vn)"1 «A„Ci„ y„)=y„"Vn_1 z„_1 «A/„_1 *„/„ y.
=Vn"Vy„
converges to the limit xfi//f~1x~1. Since r is discrete, for all large n,
V^VVn=Vir+1i>/''>'n+i s*5 tnat Vn+iVlT1 commutes with all \j/eW. Since
W gerates «P, yn+1y~1eZ(F) (note that <P is Zariski dense in F). We
thus find that for all large m we have ym=6m-y where dmeZ{F)nrcz
H nTcH and yeT. It follows that the limit x of the sequence hmym
(=hm6m y, for m large) equals y ■ y where y is the limit of the sequence
h'm=hm6meH. This proves the assertion.
Next consider the natural map n: H-*H/U=H'. Let A be the
identity component of the closure of n(H nr)=F in H'. According to
Theorem 8.24, A is solvable. Let S0=re-1(/1). S0 is a connected solvable
subgroup of G (the group U is connected). Since S0 (H n T) is closed in H,
the map S0/S0nr—>H/Hnf is proper. It follows that the map
s0/s0nr^c/r
is also proper. Let S be the Zariski closure of S0 r\T in G. We will then
show that S has the properties required in Theorem 11.16.
The group A normalises 0. It follows that A normalises F hence also
Z(F). Thus A normalises H and hence HnT.AsA normalises U as well,

186
XI. The Results of Kazdan-Margolis
we conclude that A normalises S0 and S0nr. It follows that A (and
hence A) normalises S.
We will next prove that S is connected. For this consider the exact
sequence
e-+U-+H-=-»H/U-+e.
We deduce from this sequence an exact sequence:
e — UnS- S — n(S) — e.
Since UnS is unipotent, it is connected. Thus it suffices to show that
n(S) is connected. Now n(S0 nl~) is Zariski dense in n(S). On the other
hand n{S0nr) is dense in A in the euclidean topology. It follows that
A <=n(S). Thus n(S) is the Zariski closure of the connected Lie group A.
It follows that n(S) is connected. This proves that S is connected.
Next, the group S0 being connected and solvable, [S0, S0] is
unipotent. It follows that [S0n/^ S0nr] = «P' is unipotent. It follows that
$'$ is a unipotent subgroup of T normalised by A; in view of the
maximality of *, *' <= «P=F n F We thus see that [S0 n F, S0 n T] <= F n F
Taking Zariski closures, we conclude that [S,S]c:Fc.S.
We have now only to establish that the kernel B of the action of M
on S is central in G. B being a normal subgroup of M, it is reductive.
Evidently Bc:Z(F). Let g (resp. g=g®RC) be the Lie algebra of G
(resp. G). Let s0 be the Lie subalgebra of g corresponding to S0 and
*o=so ® C <= g. Since S0 is Ad J-stable so are *0 and s0. It follows that B
stabilises *0 and moreover the representation a of B on *0 is defined
over R. Now let*0=*J ©*5 be the decomposition of *0 as the direct sum
of the space a£ of B-invariants and its unique supplement a$: note that
B being reductive, *0 is a completely reducible B-module; ag is the sum
of all non-trivial simple B-submodules of a0. Since a is defined over R,
one sees easily that we have
s0 = So + So
where So=s0n»o and s5 = son0j. Let u (resp. u) be the Lie subalgebra
of s0 (resp. *0) corresponding to U (resp. U). Let n: s0 -*s0/u (resp. n:
a0—►a0/u) denote the natural map. Now u is a J-stable subspace of s0
defined over R. Hence it is also B-stable. Let t denote the representation
of B on a0/u. This representation again is defined over R (with the natural
R-structure on a0/u defined by the R-subspace s0/u). We now claim that
7t(si)=s0/u. To see this consider the homomorphism n: S0—►S0/U:
the group S0 is the Lie subgroup of G corresponding to a0. S0 and U
are stable under inner conjugation by B. We obtain thus an action of B
on S0/U compatible with n and the action of B on S0. Now B acts trivially
on rr\S0 since fnS0cS. It follows that B acts trivially on n(fnS0).

XI. The Results of Kazdan-Margolis
187
Now in the commutative diagram
S0—'-—>S0
the maps i and; are injective. Since n(rnS0) is dense in A = S0/U, B
centralises j{S0/U). It follows that for the representation t of B on s0/u,
we have x(x)(v)=v for xeB and i)€s0/uc«0/u:»0/u (resp. s0/u) is the Lie
algebra of S0/U (resp. SJU) and the action of B on S0/U induces the
representation t on *0/u. Since s0/u spans *0/u as a complex vector space,
t is the trivial representation. It follows that A(»o)=»0/u, and n(»g)=0
i.e. *5 <= u. Hence A (*q)=s0/u and sg <= u. Now Sq is a Lie subalgebra of s0.
Let S{j be the corresponding Lie subgroup of S0. Then we see immediately
that 7t(Sj)=S0/t/. Sj is a connected subgroup of S0 centralising B. Now
let T denote the centr aliser of B in S0. From the definition of Sq it is clear
that 6q is the Lie algebra of 7. Thus T-oSq- On the other hand consider
the exact sequence
e-* TnU-> T-5-»b(T)-» e.
Now Tn U is connected being unipotent and real algebraic. On the
other hand we have n(Sq)<=n{T)<=S0/U=n(SJ) so that n(T)is connected.
Thus T is connected and hence T= S'0.
Now from the definition of Sq and s$ we see that [*i, sj]cso- Thus
sJ is stable under Ad S„ ■ Since Sj => S0 n f, sg is Ad(S0 n Testable. Since
So is connected and solvable if sj =1=0 we can find 7=1=0 in sjjcu such
that Adgr=y for all unipotents geSo- Now Sq is a closed subgroup
of S0 containing S0 nf. Moreover the map (t, g)i->exp t X • g of R x S£
into S0 maps R x Sq homeomorphically onto a closed subset of S0. It
follows that the sequence {exp n Y | n e Z+} has no convergent subsequence
modulo S0nT in S0. Since the map S0/S0nr—>G/r is proper we
conclude from Corollary 11.12 that we can find unipotents 6n=¥e in r
such that (exp n Y) 0n(exp—n 7) tends to e as n tends to oo. Let Q be a
Zassenhaus neighbourhood of e in G. We assume now that !ln$
generates the group $. This assumption can be secured by replacing the
group r by a conjugate: cf. Lemma 11.2 ($ is finitely generated). Since
<P<=S0 consists of unipotents expnYcp exp—hY=(/> for all q>e<Pr\Q.
Now for large n, expn Y0nexp-n YeQ so that dn and $ generate a
unipotent subgroup of f. Forming successive commutators of 6n with
elements of $nO, one then sees easily that we can find unipotents
(/>„ =M in r centralising $ n O and hence $ and such that exp n 7 (/>nexp
— nY converges to e. Since (pneZ{F), q>n are elements of H. Since Xeu,

188
XI. The Results of Kazdan-Margolis
expn Yet/ so that rc ((/>„)=7t (exp n Ycpnexp —nY) converges to e. It
follows that n{q>n)eA for all large n i.e. q>„eS0 for all large n. But then
exp n Yq>„ exp—n Y=q>n, a contradiction. This shows that s5=sj=0 i.e.
B commutes with all of Sj = S0. Now Bc:Z(F) is reductive and S0
contains N the unipotent radical of Z(F) (note that NcU and UcS0).
In view of Proposition 11.19, to prove that B is central it suffices to show
that Z(F) contains no connected normal subgroup of G. Equivalently,
it is enough to prove that Z(F) contains no connected normal subgroup
of G. The lattice T being irreducible, every unipotent element deT has a
nontrivial (unipotent) image in every quotient G/G1 of G by a proper
connected normal subgroup Gt of Gt (Corollary 5.23). It follows from
this that Z(F) cannot contain any connected normal subgroup of G. Thus
B is finite and central.
Now let A0 (resp. Mo) be the identity component of A (resp. M).
S being a connected solvable normal subgroup of the connected group
^o"Mo> [^o> S] consists of unipotents. It follows that we have
[Jo.Snr]^
where A0=A0 nf. Evidently A0 is Zariski dense in A0 so that [A0, S] <=F.
Thus A0 operates trivially on s/f. Now let D (resp. D0) be the kernel of
the action of M (resp. Mq) on f. Then D/D0 is finite. To prove the last
assertion in the theorem therefore we have only to show that D0 is
finite. The group D0 being normal in "Mq, it is reductive. On the other
hand it operates trivially on both f and a/f. Hence its action on s is
trivial i.e., D0<=B. Thus D0 is finite. This completes the proof of
Theorem 11.16.

Chapter XII
Semisimple Algebraic Groups (Summary of Results)
This chapter supplements §2 of Preliminaries. No proofs are given
except for Propositions 12.8-12.10; for these results no convenient
reference seems to be available. Detailed proofs of most results here can
be found in Borel-Tits [1]. Where this is not the case, other references
are given.
12.1. The Cartan criterion. Let G be an algebraic group and p:
G->GL(n) be a faithful representation. Let g be the Lie algebra of G
and we let p denote the representation of g in M(n, C) as well. Let
Ap{ , ) be the bilinear form trace p(X)-p(Y) on g. Ap( , ) is non-
degenerate if and only if G is reductive. More generally the nullity of
Ap{,) (i.e. the subspace {Xeq\Ap(X, Y)=0 for all Feg)) is precisely
the Lie subalgebra of g corresponding to the unipotent radical U of G.
In particular if Ap( , ) is identically zero, G is unipotent.
12.2. Maximal split tori and roots. Let /tcCbe any subfield. Let G
be a connected linear semisimple algebraic group defined over fc. Then
the maximal fc-split tori of G are all conjugate under elements of Gk.
The dimension d of one such torus is the fc-rank of T. If <2=0, the group
is said to be anisotropic over fc. G is anisotropic over k if and only if
every element of Gk is semisimple. We fix a maximal fc-split torus T in
G. Let X(T) denote the free abelian group (the group law is denoted
additively) of characters on T. For ?.eX(J), let
8i={pefl|Adrp=A(r)p for all reT}.
The Lie algebra g has a natural fc-structure for which the gA are all fc-
subspaces. Also [&\ g,i''] ^&i+*'■ Let k<P = {<xeX(r)\Qa+0, a nontrivial}.
The elements of k<P are called fc-roots of g. If a is a root so is —a. For
oi.Ek<P, g" is called the root space corresponding to a.
Let Z(T) (resp. N(T)) denote the centraliser (resp. normaliser) of T.
The group N(T)/Z(T) is finite and is called fc-Weyl group G (with
respect to T) and is denoted kW°. This group acts on T and hence also
on X(T). Let AT(T)R = A'(T)(8)ZR and Y(T)R be the dual of AT(T)R. Each
element AeX(T) evidently defines a hyperplane Jf?x in Y(T)R. A Weyl

190
XII. Semisimple Algebraic Groups (Summary of Results)
chamber of G (with respect to T) is a connected component of the (open)
set Y(T)R-(Jjfa. Also, let X(T)C=X(T)®ZC. Evidently X(T)cz
X{T)R<=X(T)C.
12.3. Simple roots. We fix a Weyl chamber C of G (with respect to
T). A fc-root aet$ is positive (written a>0) if it takes positive values
on C. We set t$+ = {aet$|a>0}. aet<P is said to be simple if it is
positive and cannot be expressed as the sum of two positive roots. Let kA
be the set of simple fc-roots. Then tJ is a linearly independent subset
of X(T) and contains a basis of X(T)R (over R). In particular cardinality
of )A is r=fc-rank of G. Moreover if (pek$ is any root, q>= £ ma{(p) a
ae<4
where all the ma{(p) are integers of the same sign. We will say that a simple
fc-root a is a component of q> if ma(qt>)±0.
12.4. Parabolic subgroups. An algebraic subgroup B of G is parabolic
if G/B is complete. A second characterisation is that B is a complex
analytic subgroup and for the euclidean topology on G, G/B is compact.
We will now define parabolic fc-subgroups called standard parabolic
subgroups. We fix T, «P, C etc. as above. Let u= LI g" (resp. u" =
U. g"); then u (resp. u~) is a nilpotent Lie subalgebra of g. Moreover,
-«>o
as a Lie algebra u is generated by {g"|aetJ}. For a subset Y<=kA we
define a subgroup By of G called the standard parabolic subgroup
associated to !P as follows. Let Ty= f] (kernel a) and let Tv be the
identity component of Ty. Lettiycu be the linear span of {sf\<pek<P+,
q> has a simple component in kA — W}; then tiy is a Lie subalgebra. Let
Uy be the Lie subgroup of G corresponding to tiy. Uy is a unipotent
fc-subgroup normalised by Z(J^). Let BV=Z(TP)\JV. Moreover Z(Ej,)
is 1^ ■ My where My is a connected reductive fe-subgroup of By which
admits no nontrivial characters over fe and My n Ty is finite. Uy is the
unipotent radical of By. Let tv denote the Lie algebra of Ty and jy
that of Z(Ty). Then fa is the sum of t and {g*|(/)e«P'(!P)} where
A maximal connected solvable subgroup of G is called a Borel
subgroup of G. (This may not be defined over fe.) A subgroup B of G is
parabolic if and only if it contains a Borel subgroup. In particular a
Borel subgroup is parabolic.
12.5. Conjugacy. Let B be any parabolic fe-subgroup of G. Then it
is conjugate by an element of Gk to a unique By, V <= kA. Clearly if W <= W,
By <= By.. In particular B^ is minimal among the By. It follows that any

XII. Semisimple Algebraic Groups (Summary of Results)
191
minimal parabolic fe-subgroup of G is conjugate to B^. The unipotent
radical LL, of B^ is a maximal unipotent fe-subgroup of G. Any maximal
unipotent fe-subgroup N of G is conjugate by an element of Gk to U^.
In particular a unipotent element in Gk has a conjugate in U^t (under
an element of Gk).
12.6. The Lie algebras br and ur. For a, /Jet«P, g" and g^ are mutually
orthogonal with respect to the Killing form A( , ) unless a+ /?=(). g"
and g~" are dual to each other under A(,). Also the Killing form is non-
degenerate on the Lie algebra $(T) and $(T) is orthogonal to all the
g", aet«P. One concludes from these facts that by and tiy are orthogonal
complements of each other with respect to the Killing form. The Killing
form A( , ) of g when restricted to t is non-degenerate. We identify
XC(T) with the dual of t in the natural fashion; for AeX(T), let Hxet
be the unique element oft such that for Het,
A(H,Hj)=X(H)
where X denotes also the linear form ).: t—>C induced by ).: T—►C*.
For ycjj let
«P(!P)={(/)et«P+|(/)= X"1*' a> m**Q for some <x€kA — W]
as A
and let
<p*= £ (p-dimg*.
Then "P*(fQ=0 for ae"P while "P*(/y>0 for <xekA-W. It follows that
a(#y«)>0 for all ae$(!P). Thus we see that by is the sum of the eigen
spaces of adffy« corresponding to non-negative eigen-values. uv is
normalised by by and let a denote the adjoint representation of br on
Uy. Now by=3¥© X 9"- Als0> for *e9*> (pe«P("P)u«P'("P),adXis
ctedW)
nilpotent so that tr<7(X)=0=A{H$, X). For Xet, on the other hand,
tra(X)= £ ^ma*-<p(X)=A(H$,X). We see thus that
{Xeby|trace«7(X)=0} = {Xeblf,|y4(Hlf,.)/f)=0}.
Let by be this Lie subalgebra of by. It is evidently of codimension 1 in
by and ffy, belongs to the orthogonal complement of by with reference
to the Killing form. On the other hand the orthogonal complement of
by in by is easily seen to be »y=Uy © CHV,; this last subspace is
evidently a subalgebra of g. iiy is orthogonal to all of by so that any element
X in by such that A(X, X)=\ and orthogonal to bv is necessarily of the
form vHv,+Z where Zeuy and v= ± 1. Consider now the orbit of Hv,
under AdUy. Since Uv is unipotent, this orbit is closed (cf. Prelimina-

192
XII. Semisimple Algebraic Groups (Summary of Results)
ries§2.2). Moreover for ueUy, Adu(Hv»)=Hv, if and only if u=e (the
eigen-values of adHy« in u^ are all non-zero). Thus the orbit has the
same dimension as Uy. On the other hand the set uy={Hv,+X\Xeuv}
is an irreducible subvariety of vv containing this orbit. Consequently uy
is precisely this orbit. We collect these facts together in the following:
12.7. Proposition. bv={X€bv\tracea(X)=0}. Let Hsb^H^u^ be
any element such that A(H,X)=0 for all Xebv. Then the eigen-values of
ad if in b are of the form {n- z, /ieR+} for a fixed zeC*. In particular
if adH has a real eigen-value, then all the eigen-values of adH in bv
(resp. uv) are of the same sign (resp. non-zero and of the same sign).
Moreover if we assume that all the eigen-values of ad H in bv are non-
negative then by is precisely the sum of all eigen-spaces corresponding to
non-negative eigen-values of ad H.
12.8. Proposition. Let G be a semisimple algebraic group and BcGa
complex Lie subgroup of G. Let g be the Lie algebra of G and b the sub-
algebra corresponding to B. The following conditions on B are equivalent.
a) B is parabolic.
b) b is the normaliser of a subalgebra u such that u is the Lie algebra
of the unipotent radical U of the algebraic group B.
c) The orthogonal complement u' ofb in g with respect to the Killing
form is a subalgebra such that adX is nilpotent for all Xeu.
That a) implies b) and c) is easily deduced from the structure of
parabolic groups described in 12.4 and 12.5.
We will next show that b) implies c). Since b is the normaliser of u
in g, B is the connected normaliser of u in G and is hence algebraic.
It follows from 12.1 that u is precisely the orthogonal complement of
b in b with respect to the Killing form. Let u'={X\A(X, Y)=0 for all
Yeb}. Evidently ucu'. u' is adu-stable. It follows from Engel's theorem
(cf. Preliminaries § 1.2) applied to the action of u on u'/u that if u'=|=u,
we can find Xeu', X$u, such that [Y, X]eu for all Yeu. Evidently then
X normalises u i.e. Xsb. But bnu'=u so that Xeu, a contradiction.
Thus u=u'. Hence the orthogonal complement of b in g with respect
to A ( , ) is u. This shows that b) implies c).
Assume now that c) holds. Let U' be the Lie subgroup of G
corresponding to u'. Since adX is nilpotent for all Xeu' it follows that U'
is unipotent algebraic. Let U* be a maximal unipotent subgroup of G
containing U' and B* the Borel subgroup containing U*. Let b* (resp.
u*) be the Lie algebra of B* (resp. U*). Then b* is orthogonal to u* and
hence to u' (with respect to A( , )). It follows that b=>b* so that B=>B*,
so that B is parabolic (cf. 12.4). This completes the proof of
Proposition 12.8.

XII. Semisimple Algebraic Groups (Summary of Results)
193
12.9. Real semisimple Lie algebras. Let g be a real semisimple Lie
algebra and g=g®RC. Let beg be a subalgebra of g and b1 the
orthogonal complement of b in g with respect to the Killing form. Let b
(rcsp. b1) be the C-span of b (resp. b1) in g. Evidently b1 is precisely the
orthogonal complement of b in g with respect to the Killing form of g.
In view of these facts and Propositions 12.7 and 12.8 we obtain
12.10. Proposition. Let g be a real semisimple Lie algebra and b a
subalgebra such that the orthogonal complement nofb in g with respect
to the Killing form has the following property: n is a Lie subalgebra and
ad X is nilpotentfor Xe n. Then b is the normaliser ofn. Let a denote the
adjoint representation ofb on n and b={Xeb|trace<7(X)=0}. Let Y(#n)
be any element in the orthogonal complement of b in b- Then b is the
sum of eigen-spaces corresponding to all the non-negative or all the non-
positive eigen-values of ad Y. The subspace b is of codimension 1 in b. Also
n is the sum of the eigen-spaces corresponding to all the positive or negative
eigen-values of ad Y Moreover let b'cg be a subalgebra with the following
property: if n' is the maximal ideal in b' such that adX is nilpotent for
all Xe n' then b' is the normaliser of ri in g. Then n' is the orthogonal
complement of b' in g.
Suppose now that g=I©p is a Cartan decomposition of g, I being
the algebra. Let b<=g be a subalgebra of g as in Proposition 12.10. Let
n={X\A{X, Z)=0 for Zeb}. Then adX is nilpotent for Xen. Now for
Xe p, ad X is semisimple. It follows that p n n=(0) or taking orthogonal
complements with respect to A, g = I+b. We claim then that b =
Inb©pr.b©n. To see this, taking orthogonal complements again, we
will first show that n=(p+n)n(I+n)nb. Since b=>n, (p+n)nb =
(pnb)+n and (I+n)nb=(Inb)-(-n. It suffices thus to show that if
X+Y+Z=0 with Xelnb-Yepnb and Zen, then X=Y=Z=0. Now
for Xetnb, \_X, Y]epnb and [X, Z]en. Since pnn=0 it follows that
[X, 7] = [X,Z] = 0([X, Y+Z]=-[X,X] =0). Since X and Y commute
and both adX and ad Y are semisimple so is ad(X+ Y)= — ad Z. But
for Zen, ad Z is nilpotent. Thus X = Y=Z=0. The same argument
shows that b=I n b©p n b©n is indeed a direct sum. Now ad X is skew
symmetric for Xel so that trace<7(X)=0 for Xetnb, i.e. Inbcb.
Evidently neb. Thus b=n©Inb©bnpnb. Since b is of codimension 1 in
b, bnpnb=Fis of codimension 1 in pnb. Moreover bnp is orthogonal
to all of n©I n b. Thus the orthogonal complement F' of F in p n b with
respect to A is orthogonal to all of b. The form A on p n b is positive
definite so that F' is a 1-dimensional subspace R- Y, Yepnb of pnb.
We assume Ychosen such that the eigen-values of a{Y) are positive and
A (Y, Y)= 1. We have thus proved

194
XII. Semisimple Algebraic Groups (Summary of Results)
12.11. Proposition. There is a unique element Yep rib such that
(i) A(Y, Y)=l, (ii) b is the sum of all the eigen-spaces corresponding to
the non-negative eigen-values of ad Y and (iii) A(Y, Z)=0for all Zeb.
12.12. The reflections. We now go back to the notation of 12.1-12.7.
When !P consists of single element aekA, we see T((I)=T(1. Let N(TJ be
the normaliser of Ta and let N. denote the group N(TJ n N(T).
Evidently, ZCDcN.ciVCD. The group N„/Z(T) is a group of order 2. It
can be identified naturally with a subgroup of kW°. We denote the non-
trivial element NJZ(T) by s„. sa is called the reflection with respect to
a (the linear automorphism of t induced by sa is the map H\-+H —
{2a.(H)/a.(HJ}■//„). The elements {sa,<xekA} generate the Weyl group
kW°.
12.13. Bruhat decomposition. The natural map N(T)-*kW° maps the
fc-rational points N(T)k onto kW°. We fix a set kWcN(I\ of
representatives of the group kW°. Then
Gk=Vk tW Z(T\ Uk.
Suppose for <xekA we denote by S, the element of kW representing
saekW°. Then if for some xekA,
g=us!azv
where u, veUk and zeZ(Tt), we can also write
g=uls'azvl
with t^eU^ and u^ZiT^nV,,.
12.14. fc-rank 1 groups. If G has fc-rank 1, t,d consists of a single
element a. In this case it is known that either k$+ = kA=(tx) or k$+ =
{a, 2 a}. tW° is a group of order 2 and its generator sa acts on t as the
automorphism H\-+—H; also s'(1UsJ1"1=U". If geGt, either geP0t=
Z(T)t-Ut or g=us'azv with H,t,eUt and zeZ(T)k; this expression is
unique.
Using the results stated above we will now prove a lemma which
turns out to be important in the sequel.
12.15. Lemma. Let G be a connected algebraic semisimple k-group of
k-rank 1. Let N and N' be two maximal unipotent k-subgroups of G. Then
either N = N' or NnN'=(e).
Proof. Let P, T, Z(T), t«P, kW, N(T) etc. be as above. According to
12.7 we may assume that N = ll. and that we can find geGt such that
N'=gU0g_1. The group G has fc-rank 1. Hence kA = {<x}. According to

XII. Semisimple Algebraic Groups (Summary of Results) 195
Proposition 12.14 we have for gsGk
g = us'az-v or geB0
where u,ve\J^k and zeZ(T)t. It follows that either N' = U0 (this is the
case if geBj or -T, ,TT ,_• _. TT_ _•
Since u-1U^u=U^ and U^nU^fe) the desired result follows. As an
immediate consequence we have
12.16. Corollary. Let Gbea connected semisimple algebraic k-group of
k-rank 1. Let N be a maximal unipotent k-subgroup. Let gsGk. Then if
gNg"'nN=|=(4 g belongs to the normaliser P of N.
12.17. Corollary. Any unipotent element 6eGk is contained in a unique
maximal unipotent k-subgroup of G.

Chapter XIII
Fundamental Domains
Our aim in this chapter is to describe a "nice" fundamental domain for
a special class of discrete subgroups of semisimple groups. Once again
we will establish our theorems in the set up of L-subgroups so as to
enable us to treat lattices and arithmetic groups simultaneously. (We
must however add that one can perhaps give independent shorter proofs
in each of these separate cases.)
We begin with a general theorem on L-subgroups of semisimple
groups. (L-subgroups were introduced in Chapter I: Definition 1.21.)
13.1. Theorem. Let G be a connected linear semisimple Lie group
without compact factors and r<=.G an L-subgroup. Let U(r)be a maximal
unipotent subgroup of r and U the Zariski closure of U (r) in G (cf.
Preliminaries §2.3). Let Z(U) be the centraliser of U in G. Let V be the
unipotent radical of Z(U) (i.e. the maximal unipotent normal subgroup
ofZ{U)). Then Z{U)/Z{U)n Tand V/Vn Tare compact. MoreoverZ{U)=
M ■ V where M is a closed normal subgroup ofG and VcU.In particular M
and V commute. Further ifT is irreducible and [/(f) 4= {e}, M is central in G.
Proof. Since U(r) is unipotent so is U. In view of Theorem 2.1,
U/U(r) is compact. Also in view of the maximality of U(r), U(r)= t/nT.
The group [/(F) is finitely generated (Theorem 2.10). Let A <=. t/(F) be a
finite set of generators. Clearly then Z(U) = GA (=Centraliser of A).
Now U being unipotent, we can find geG such that gAg~l eft where Q
is a Zassenhaus neighbourhood of e in G (Lemma 11.2). We replace r
by its conjugate gTg~l and assume that A<=Q. Using the argument in
Lemma 1.14, one sees that the map
GJG4nr^G/r
is proper. Thus if GJG4nr is not compact, we can find xneGA and
unipotents, 0ner, 0n+e, such that xn0nx~1 converges to e (in view of
Corollary 11.12). Now xnA x'^ii for all n and xn0nx~1e£i for large n.
It follows that for large n, 0n and A generate a unipotent subgroup of T.
Since A generates UnT and UnT is maximal unipotent in r, 9„eUnr

XIII. Fundamental Domains
197
for large n. But then since xneGA = Z(U) we have x„0nx~l = 9„, a
contradiction. We conclude therefore that GJGA n r is compact. Now
Z(U) being (real) algebraic, Z(U)=M -V where V is the unipotent
radical of Z(U) and M is a reductive algebraic subgroup of Z(U). Now
the kernel Mt of the action of M on F is a normal subgroup in G
(Proposition 11.19). If M° is the identity component of Mu M{* is a
connected semisimple Lie group without compact factors. The identity
component M° of M is thus necessarily of the form M°=M^ • T where
T is a connected reductive algebraic group acting on V with a finite
kernel. Finally M/M° is finite. From these considerations we conclude
applying Corollary 8.28 that V/VnT is compact. Clearly V and U
commute so that VnT and UnT generate a unipotent subgroup of T.
It follows that VnTczUnT. Since VnT is Zariski dense in V
(Theorem 2.1), VcU. Thus every element of Z(U) commutes with V. In
particular M acts trivially on V so that M = Mt is normal in G. The
last assertion follows from Corollary 5.21 together with Proposition 11.19.
This concludes the proof of Theorem 13.1.
We next consider a rather restricted class of L-subgroups of a semi-
simple group. The restrictions we impose are suggested by certain salient
properties of arithmetic subgroups of Q-rank 1 semisimple algebraic
Q-groups.
13.2. Definition. Let G be a connected linear semisimple group.
Let rcG be an L-subgroup of G. r is said to have Property (Rl) if the
following holds: every unipotent element 0+e in r is contained in a
unique maximal unipotent subgroup of r.
With this definition we have
13.3. Theorem. Let G be a connected linear semisimple group without
compact factors and r^-G an L-subgroup with Property (Rl). Then the
set of conjugacy classes of maximal unipotent subgroups of r is finite.
Proof. When G/r is compact, there is nothing to prove. We assume
therefore that G/r is non-compact. Now according to Corollary 11.13,
we can find a non-empty finite set S<=T — {e} consisting of unipotents
such that for any unipotent x( + e) in r there exists aer and yeS such
that ax a-1 and y generate a unipotent group. For yeS, let A{y) be the
unique maximal unipotent subgroup of r containing y. Let A be any
maximal unipotent subgroup of r. Let xeA be any element not equal to e.
Let aer and yeS be chosen such that a xa_1 and y generate a unipotent
subgroup 0 of r. Let ¥ be a maximal unipotent subgroup of r
containing <P. Since axoi-'e^cf and a.xa~ley.Aat.~l, we conclude that
azda-1 = f. On the other hand, ye*PnA(y) so that Y = A(y). Thus
ixA a-1 =A(y). We have therefore proved that every maximal unipotent

198
XIII. Fundamental Domains
subgroup A of r is conjugate to A (y) for a suitable yeS. Since S is finite,
the theorem follows.
13.4. Notation. Let G be a semisimple Lie group and g its Lie algebra.
Let Yeg be an element such that ad Y is semisimple and has all
eigenvalues real. For such an element Yeg, we denote by u(Y) (resp. b(Y))
the subspace of g spanned by the eigen-spaces corresponding to the
positive (resp. non-negative) eigen-values of ad Y. u(Y) and b(Y) are
Lie subalgebras and b(Y) is the normaliser of u(Y). We denote by a
the adjoint representation of b(Y) on u(Y) and set b(Y)={xeb(Y) trace
<7(x)=0}. Then b( Y) is also a Lie subalgebra of g. We denote by U{Y), B{Y)
and D(Y) respectively the Lie subgroups corresponding to u(Y), b(Y)
and b(Y). U(Y) is connected and unipotent and is hence closed. B(Y)
is the identity component of the normaliser Bj(Y) of U(Y) and has
finite index in the latter. Clearly B(Y) is also closed in G. Finally D(Y)
is the identity component of the group
D1(Y)={geB1(Y)|detcr(g)=±l}
where we have let a stand for the representation of Bj( Y) on u(Y) as well.
Let g(Y) be the centraliser of Y in g and G{Y) the Lie subgroup of G
corresponding to g(Y). G{Y) is the identity component of the group
{xeG|Adx(Y)=Y}
and is hence closed in G. Let M(Y)=G(Y)r\D(Y). It is then easily seen
that B(Y)=G(Y)- t/(Y)andD(Y)=M(Y)- t/(Y);alsoG(Y)nN(Y)={e}.
The group M(Y) is also connected. If A{Y) denotes the 1-parameter
subgroup {exptY|teR}, A(Y) is closed in G and G(Y)=A(Y)-M(Y).
Also for ceR let
4c(Y)={expi.Y|i.<c}.
In view of Proposition 12.10, we have B{Y)=B{Y1), U{Y)=U{Y1)
and D(Y)=D(±Yj) where Yt is any element in b(Y)—n(Y) orthogonal
to b(Y) with respect to the Killing form A( , ) on g. Moreover if g=l©p
is any Cartan decomposition of g with! as the algebra, we can choose Yt
to belong to the space p and such that D(Y)=D(Y1). Let K be the
(maximal compact) subgroup of G corresponding to I Now the centraliser
g(Yj) of Yj is stable under the Cartan involution and contains a Cartan
subspace a of p (cf. Preliminaries § 1.7). If we choose a basis Xt,..., Xt of
a with Yj = Xt and take the lexicographic ordering given by this basis,
we have u(Y)=u(Yj)cn (in the notation of Preliminaries § 1.7). Also
b (Y) => a -(- n. It follows now from the Iwasawa decomposition that we have
13.5. Lemma. K ■ B{Y){=K- B(Yt))= G.
The next result is again somewhat technical.

XIII. Fundamental Domains
199
13.6. Lemma. Let n be any relatively compact subset of U{Y). Then
given any neighbourhood Q ofe in G we can find a constant ceR such that
exp t Y • g • exp — t YeQ
for all gen and t<c.
Ift0eR and co<=-D(Y) is any relatively compact subset, the set
{exprY-gexp — tY\geo), t<t0}
is relatively compact in D(Y).
Proof The exponential map is an analytic isomorphism of u( Y) on
U{Y). Moreover exp tY exp X exp — ry=exp(Adexpi.y(X)). It suffices
therefore to show that given any relatively compact subset n'^u(Y) and
e>0, we can find ceR such that for t<c and Xen', we have
||Adexpt7(X)||<e
where || || is a norm induced by a scalar product < , > on g with respect
to which ad Yis symmetric. Now Ad exp t Yis a symmetric endomorphism
and the eigen-values of ad Y on u(Y) are all positive. It follows that the
highest eigen-value of Ad exp t Y acting on u(Y) tends to 0 as t tends
to — oo. Thus for a suitable c, ||Ad exp tY(X)\\ <s for all t<c and Xerj'.
To prove the second assertion of the lemma, we first observe that
any relatively compact set co of D(Y) is contained in a set of the form
jjj • n2 where nt (resp. n2) is a relatively compact subset of M(Y) (resp. U(Y)).
Since exprY-(ab)-exp — rY=(expi.Y- a exp — tY) (exp tY • b -exp — tY)
it suffices to show that each of the two sets
/7i = {expi.Y-n-exp —tY\t<t0,nen1}
and
^'2 = {expi.y-»i-exp —ry|r<t0,j7iei72}
are relatively compact. That n\ is relatively compact is immediate from
the first assertion. Since exprY centralises n2,n'2=n2. This completes
the proof of the lemma.
We now introduce a second rather special property for L-subgroups.
13.7. Definition. Let G be a connected linear semisimple Lie group.
Let r c G be an L-subgroup. T has Property (R 2) if the following holds:
let U(r) be a maximal unipotent subgroup of T and U the Zariski closure
of [/(/") in G; let N(U) be the normaliser of U in G and N°(U)=
{geN (U) | Int g preserves the Haar measure on U}. Then N°(U)/N°(U)nr
is compact.
13.8. Proposition. Let Gbea connected linear semisimple Lie group
without compact factors. Let r be an L-subgroup of G with Property (R 2).

200
XIII. Fundamental Domains
Let U(r) be a maximal unipotent subgroup of T and U the Zariski closure
of U(r) in G. Then there exists 7eg ( = Lie algebra of G) such that
U= U{Y). Also D{Y) has finite index in N°{U) (where N°(U) is defined
as in the definition above). Moreover i/"g=f©pisa Cartan decomposition
of g with f as the algebra Y can be chosen to be an element of p.
Proof. Let a denote the adjoint representation of N(U) on the Lie
algebra u of U. Let Ut be the unipotent radical of the (real) algebraic
group N(U). Since a(x) is unipotent for xeUt, t/jcN0^). Evidently,
on the other hand Ut => U. Consider now the natural projection
q: tf0(l/)-»N°(l/ytf,.
Let Sj be the closure of q(N°{U)nr) in N^C/yt/j and S0 the identity
component of St. Let S=q~1(S0). Then S=> C/t => U, S is normalised by
N°{U)n r and S{N°{U)n r) is closed in N°{U). Since N°{U)/N°{U)nr
is compact, it follows that S/SnT is compact. Now according to
Theorem 8.24, S0 is solvable and hence so is S. Consider now the group
Z(U). According to Theorem 13.1, Z(U)/Z(U)n r is compact. Moreover
Z(U)=M ■ V where Fis the unipotent radical of Z{U), M is semisimple
and without compact factors and V/VnT is compact Combining
Theorems 2.1 and 5.5, one sees that since S is normalised by
Z(C/)nr(cN°(t/)nr),
it is normalised by Z(U) itself. Now let S* be the Zariski closure of S
in G. Then H=Z(U)r\S* is a solvable normal subgroup of Z(U)
containing V{V^U: Theorem 13.1). Since Z(U)/V is semisimple,
Z(t/)nS*=Kc[/. Thus every semisimple element xeS* acts non-
trivially on U. It follows that an element xeS is unipotent if and only
if Ad x acts unipotently on the Lie algebra u of U. It is then easily seen
that if S' is the maximum connected nilpotent normal subgroup of S,
every element of S' is unipotent. Now S'=> Ut and S'/S'nT is compact
(Theorem3.3). Since UnT is maximal unipotent in r, S'nr = Ur\r
and hence S'=U. It follows that S'=U1 = U i.e. U is the unipotent
radical of its normaliser. The proposition now follows from
Proposition 12.10.
13.9. Proposition. Let G be a connected linear semisimple Lie group
without compact factors. Let T^-G be an L-subgroup with Property (R2).
Let U(r) and U'(r) be maximal unipotent subgroups of T and U and U'
their respective Zariski closures in G. Let Y and Y' be elements of p such
that U=U{Y) and U'=U'{Y). Let rj(Y) {resp. ri(Y')) be a compact subset
of D{Y) (resp. D{Y')) and t0eR any constant. Then if U{T) and U'{r) are
not conjugates in T we can find tt = tx(t0, rj(Y), rj(Y'j) such that
KAti(Y)r,{Y)rnKAtn(Y')r1{Y')=Q.

XIII. Fundamental Domains
201
If [/(/") and U'(r) are conjugates, there exists tl = tl(t0,t](Y\n;(Y'))
such that for yeT if
KAh(Y)r,(Y)nKAt0(Y')r,(Y')y*V,
y U{Y)y~1 = U{Y'). Moreover ifn{Y)-(rnD{Y))=D{Y), there exists teR
such that
KAJY')n{Y')<zKAt(Y)n{Y).r.
Proof According to Lemma 13.6, K ■Ato(Y')n{Y')=Q0{Y')^EA,u{Y')
for a suitable compact subset E of G. Let S(Y) (resp. 5(7')) be a finite
set of generators of [/(O (resp. U'(r)). Appealing to Lemma 13.6 again,
we see that we can find a compact set C c G such that
gS^g-^C
for all geE •A,o(Y'). Now let W be a Zassenhaus neighbourhood of e
in G (cf. Definition 8.22) and V a neighbourhood of e such that
xVx~l^W
for all xeC. Now let tteR be chosen such that
gS(Y)g-^V
for all geKA,i(Y)r](Y)=Ql(Y): such a choice of tt is possible in view
of Lemma 13.6. Suppose now that geft0(y') and yeT are such that
h=gyeQl{Y). From our choice of tu we have hS{Y)h~l <=.V. On the
other hand h y-1 ■ S{Y') y h~l =gS(7') g~l c C. It follows that for (peS{Y')
if we set (p'=y~1q>y, we have hq>'S(Y)(p'~lh~lzzW. Since FcPVand
W is a Zassenhaus neighbourhood, q>'S(Y)q>'~1 and S(Y) generate a
nilpotent and hence unipotent subgroup of r. Since S(Y) generates U(r)
and [/(/") is maximal unipotent, we conclude that q>' normalises U{r).
We find thus that q>' and U{T) generate a unipotent group as well.
Thus (p'eU{r) for all (peS{Y'). Since S{Y') generates U'(r) we conclude
that y-1 U'(Dy = U(D. Taking Zariski closures, y U{Y) y~1 = U{Y'). In
particular therefore we conclude that the intersection
KAtl(Y)r,(Y)nKAJY')r,(Y')r
is non-empty only if U{r) and U'(r) are conjugates in T.
Assume now that r]{Y)-(D{Y)nr)=D(Y). Since y U(Y)y~1 = U(Y')
we conclude that yB{Y)y-1=B(Y') and yD{Y)y-1=D(Y'). Clearly we
have Ady-^Y^XY+Z where Zeb(Y') and A>0 (that X is greater
than zero is seen from the fact that trace a{Y')>0 where a denotes the
adjoint representation of B(F') on u(F) and a similar assertion is true
for Y). One sees easily moreover that exp t{X Y+Z)=exp tX Y• x, where

202
XIII. Fundamental Domains
x,eD(Y). We conclude thus that we have
KAto(Y')r1(Y')y = Kyy-1Ato(Y')r1(Y')y<zKyAto(AY+Z)D{Y)
=KyAXt0(Y)-D(Y).
Now we can write y = k-b where beB(Y) and keK (Lemma 13.5). We
therfore find setting Xt0=s, that we have
KAt0(Y')r1(Y')^Kb-As(Y)-D(Y)r
=Kb.As(Y)rj(Y)-r.
Now b=exp rY-0 where 9 e D (Y) and D (Y) is a normal subgroup of B (7).
If we set t=s+r we obtain the inclusion
KAt0(Y')n(Y')<zK-A,(Y)-n{Y)-r.
This completes the proof of the proposition.
We are now in a position to prove the main result of this chapter.
13.10. Theorem. Now let Gbea connected linear semisimple Lie group
without compact factors and T an L-subgroup with Properties (R1) and
(R2). Let K be a maximal compact subgroup of G. Then we can find a
finite set &<=■% ( = Lie algebra of G) and constants c, c'eR with the
following properties.
(i) Every element Ye3t is nonzero, ad Y is semisimple and has all
eigen-values real.
(ii) For each Ye9t, U(Y)nT is a maximal unipotent subgroup of T
and D{Y)/D{Y)nr is compact.
(iii) IfY, Y'e0iand Y*Y',KAc(Y)D(Y)nKAc.{Y')D{Y')r=0. Also
KAc(Y)D(Y)nKAc.(Y)D(Y)y*Q for yeT if and only //yeD1(Y)r.r.
(iv) For each Ye 31, we can choose a relatively compact subset
n(Y)<zD{Y) such that n(Y){D(Y)nr)=D(Y) and (J KAc{Y)n{Y)r=G.
YeSt ,
(v) // Q= (J KAe(Y)n(Y), the set {yer|fiynfi+0} is finite.
YeSt
Proof. The set of conjugacy classes of maximal unipotent subgroups
of r is finite since r has Property (Rl) (Theorem 13.3). Since each
maximal unipotent subgroup of r is of the form U(Y)nT for some Yeq
(with ad Y semisimple and having all eigen-values real), (Proposition 13.8:
note that r has Property (R2)) we can choose a finite set 31 satisfying (i)
and (ii) and in addition having the following property: every maximal
unipotent subgroup of T is conjugate to U(Y)nT for a unique YeSt.
That for given ceR we can choose c'eR satisfying (iii) follows from
Proposition 13.10. (v) is an immediate consequence of (iii) since for

XIII. Fundamental Domains
203
yeDt(Y)nr, KAc(Y)r,(Y)nKAc,(Y)r,(Y)y±Q if and only if K-rj{Y)n
Kr]{Y)y*Q and moreover the set KAc{Y)r]{Y)-KAe.(Y)ri(Y) is
relatively compact in G. We have therefore only to prove that we can find c
and r\(Y) to satisfy (iv) of the theorem as well. We first choose the r\(Y) as
any relatively compact subsets of D(Y) such that t](Y) ■ (D(Y)nr)=D(Y)
for all Ye&. Consider the set
ii*=[JKA0(Y)r1(Y).
We will show that the set {geG\g$Q* T}=E is relatively compact
modulo r in G. (It suffices to prove this since K • B{Y)= G for any Ye9t)
To prove this assertion we argue as follows. We observe first that
there is a neighbourhood W of e with the following property. If g=
k • a • x, (keK,aeA(Y), xeD{Y)) is any element such that g9g~1eU for
some 9eU{Y)nr, 9+e then aeA0{Y). To see this let || || denote the
norm \\v\\ = — A(v, 9v) (9 is the Cartan involution associated to K).
Any element of U(Y)n r can be written in the form exp Z with Zeu(y)
and ||Z|| ^ CifZ+0 where C is a fixed constant independent of the element
in U(Y) n r. We may assume that the same constant C serves for all Ye3t.
Let M>0 be a constant such that for any xerj(Y), Ye9t,
||Adx(Z)||^M||Z||
for all Zeg: such a constant M exists since 9t is finite and the r\{Y) are
relatively compact in G. Now we have for Zeg, with exp Z=9e U{Y)nr
g exp Zg~l = exp Ad g(Z)=exp(Ad k Ada Ad x(Z)).
Now x=yq> with yer](Y) and q>eD{Y)nr. Thus we find setting
ff=(p9(p-\g' = k-a-y
g> ffg1-1 =exp(Ad k Ad a Ad y(Z)).
Now Ad k preserves the norm on g and if a4A0(Y), we have
||Ada(Z')||^||Z'||
for all Z'eu(7). We see thus that we have
g'0'g'-1 = expZ"'
where || Z^H^M ■ C and Z'" is nilpotent. Now consider a
neighbourhood V of 0 in g such that exponential is a diffeomorphism of V onto
its image and Fis of the form {Zeg|||Z|| <e<M • C}. One then sees that
since Z'" is of the form Adfc(Z"), Z"eu(7), if e is chosen sufficiently
small, g' 0' g' ~1 = g 9 g~1 = exp Z'"$ W= exp V.

204
XIII. Fundamental Domains
Let W c W be a neighbourhood of e in G such that the following
holds: if grg_1n W' + {e}, grg-1n W contains a non-trivial unipotent
(Theorem 11.10). Let F={geG\grg~1r\ W'=e}. F is relatively compact
modulo r since r is an L-subgroup. We will now show that £cF.
Let geG—F. We can then find a unipotent QeT, 0+e such that gQg~le W.
Now 0 being unipotent, we can find yeT and Ye9t such that
0' = y-10y€t/(7)nr.
Clearly if h=gy, hffh~1eW. In view of our choice of W, this implies
that gy=k-a-x, keK, aeA0(Y) and xeD{Y) i.e. gyeQ*-T. Thus
G-Fcfi*• r so that FcG-fl*r=£. This proves our contention. The
proof of Theorem 13.10 is thus completed.
For the convenient formulation of results in the sequel we make the
following definition.
13.11. Definition. Let G be a connected linear semisimple Lie group
(which may admit compact factors). A discrete subgroup T of G such
that G/r is non-compact is a rank-1 discrete subgroup if the following
holds: let Gt be the maximal compact normal subgroup of G and
7t: G^G/Gj be the natural map; then n(JT) is an L-subgroup with
Properties (Rl) and (R2).
The result below is practically a reformulation of Theorem 13.10.
13.12. Main theorem. Let G be a connected linear semisimple Lie
group and r<=-G a rank-1 discrete subgroup. Then we can find a finite
set 9t<=-Q (=Lie algebra of G) and constants c, c'eR with the following
properties:
(i) Every element Ye3t is non-zero and ad Y is a semisimple endo-
morphism with all eigen-values real.
(ii) For each Ye<%, D(Y)/(D(Y)nr) is compact.
(iii) For each Ye9l,thereisarelativelycompactopensubsetr\(Y)zzD(Y)
such that foryer and Y, Y'sdl
KAc(Y)r,(Y)ynKAc,(Y')r1(Y')^
if and only if Y=Y', yED1{Y)nT and K-n{Y)ynn{Y')*Q.
(iv) (J KAc{Y)rj{Y)r=G {where r]{Y)czD{Y), Ye& are chosen as
re.*
in (iii)).
(v)IfQ={jKAc(Y)r,(Y),theset
{yer\QynQ+0\
is finite.

XIII. Fundamental Domains
205
13.13. In the sequel we fix once for all a connected linear semisimple
Lie group G and a rank-1 discrete subgroup TcG. We choose once for
all a set #c g, the Lie algebra of G, constants, c, c'eR and for each Ys9t
relatively compact open subset r](Y)^D(Y) as in Theorem 13.12 above.
For Ye0l, let Q0(Y)=KAc(Y)rj(Y). From our choice of the rj{Y) we
see then that the set ii0(Y) is an open subset of G. Now consider the set
Q0 = {jQ0(Y). This is a disjoint union of open subsets of G. In fact
from (iii) of Theorem 13.12 we see that S20(Y)n£i0(Y')=Q if Y+Y'.
Now for Ye9t, we define on Q0(Y)T a C00 function q>Y as follows. Let
xEQ0(Y)r; then x = fc-expry<j;y where ksK, t<d, £er]{Y) and ye/".
Here t=t (x) it is easily seen is a C00 function of x. In fact if x = k exp t Y •
£-y=k'expt'Y£'y, then y'y~1eD(Y) according to the main theorem
above and one deduces easily from this that t=t'. Thus if we set q>Y{x)=t
where x=kexptY£y as above then q>Y is a C00 function on Q0(Y)r.
Moreover it is easy to see that for x eKAc(Y) r\ (Y\ f(pY{x)+0 where
f is the left invariant vector field defined by Y Hence dq>Y is non-zero
on Q0{Y) and since q>Y is T-invariant as is evident (—d(pY) is non-zero
on ii0(Y)r. Here for a real valued C00 function f df denotes the
differential off (note that dq>Y(f)(x)=(f cpY)(x)). Putting together the
q>Y we obtain a T-invariant function cp on Q0r. Let p: G^G/r be
the natural projection. q> defines by passage to the quotient a C00
function i// on p{Q0). Now the function (—i//) has no critical points in
p(fi0). Also the set A = G/r—p{Q0) is relatively compact, hence compact.
We can thus find a C00 function u which coincides with — i// outside a
relatively open compact neighbourhood B of A. Now it is easy to check the
function u thus obtained maps G/r properly into [C, oo) for some CeR
and has no critical points outside the (compact) closure of B. Elementary
facts from Morse theory (see for instance Milnor [1]) now give us
13.14. Theorem. For r large, u~l\C, r] is a compact submanifold with
boundary whose interior h-1[C, r) is diffeomorphic to the manifold G/E
The function u can be assumed to be invariant under the action ofK on the
left.
Only the last assertion requires further clarification Note that the
functions q>Y are K-invariant on the left. One finds therefore that u is
K-in variant outside KB. To secure global K in variance one can for
instance average over K.
13.15. Corollary. T is finitely presentable.
Proof. G/r is diffeomorphic to the interior of a compact manifold
with boundary W. In particular G/r has the same homotopy type as W.
Thus the fundamental group T0 of G/r is isomorphic to the fundamental
group of W and is hence finitely presentable (Theorem 6.15). Since the

206
XIII. Fundamental Domains
mapG^G/r is a covering map with r as the "deck-transformation"
group, r is a quotient of r0. The kernel H of the natural map /J,—>T is
isomorphic to the fundamental group of G. Now G has the same homo-
topy type as the compact connected manifold K so that H is finitely
generated (it is also abelian since G is a topological group). It follows that
r is finitely presentable.
13.16. Corollary, r has a torsion free subgroup of finite index.
This follows from the corollary above and Theorem 6.11.
13.17. Corollary. Let F be a torsion free subgroup ofT. Then M/F is
diffeomorphic to the interior of a compact Cx manifold with boundary
(hereM=K\G).
This is an immediate consequence of Theorem 13.14. Since F is torsion
free it acts fixed point free on M=K\G (in fact for geG Kr\g~1F g is
discrete, compact, and torsion free and is hence (e)). The C00 function u
on G/r defines by composition with the projection a: G/F^G/r a C°°
function u' on G/F with similar properties as u. u' will be K-invariant if
u is, and according to Theorem 13.14 we may assume this. Thus u'
defines a C00 function/on M/F. Since K is compact and a is proper one
sees easily that /is proper and has no critical points outside a compact
set. Now we may apply the same arguments as above.
A final consequence of the main theorem is the following
13.18. Theorem. A rank-1 discrete subgroup r of a semisimple Lie
group G is a lattice.
Proof (compare proof of 10.4). We consider for each Ye St the inclusion
i: B(7)->G. This map is evidently compatible with the action on the
right of B(Y) on the two spaces. We define a Borel measure v on B(Y)
as follows: let n denote a Haar measure on G; then for a measurable subset
E<zB(Y), v(E)=n(K-i(E)). Since Ki(E)b=Ki{Eb) for beB{Y), it
follows that v is invariant under the right action of B(Y). Thus v is a
scalar multiple of the right invariant Haar measure on B(Y). To prove
the theorem therefore it suffices to show that the (right-invariant)
Haar measure of the set Ac{Y)n(Y) is finite. Now B(Y) is a semidirect
product of the one parameter group A(Y)={expt Y} _00<t<aa and D{Y).
We identify A{Y) with R through the isomorphism t\->exptY. Let q>:
Rx.D(Y)^>B{Y) be the analytic isomorphism (of manifolds)
<p(t, X)=exptY-X.
The pull-back of v by q> is a measure v0 on Rx D(Y). If dt denotes the
Haar measure on R and X the right-invariant Haar measure on D(Y) a

XIII. Fundamental Domains
207
simple calculation shows that the measure v0 takes the form
dv0 = e"'dt-dX
where p denotes the sum of the eigen-values of ad Y on d(Y). Clearly
p > 0. Also one sees easily that
<p-1(4(y)^(y))=(-oo,c)x^(y).
c
Since n(Y) is compact and J ept dt < oo, our theorem follows.
— 00
We will now show that the main theorem can be applied to describe
fundamental domains in two interesting cases:
(A) Let G be a connected linear semisimple Lie group with an
R-rank 1 factor and Gt its maximal compact normal subgroup; let
n: G->G/Gt be the natural map; let reG be a lattice such that n{r)
is irreducible (in the sense of Definition 5.20).
(B) G is the identity component of GR, the group of R-rational
points of a semisimple linear algebraic group G defined and of rank 1
over Q and r <= GQ is an arithmetic subgroup of G.
We will first deal with (A). We have
13.19. Theorem. Let Gbea connected linear semisimple Lie group and
Gj its maximal compact normal subgroup. Let n: G—>G/G1 be the natural
map. Assume that G has a (R-) rank 1 factor. Let T^-Gbea lattice such that
n(r) is irreducible. Then r is a rank-1 discrete subgroup ofG.
From the definition of a rank-1 discrete subgroup one sees that to
prove the theorem we may assume that G1={e) i.e. that G has no compact
factors.
Let G* denote the adjoint group of G and/: G-> G* the natural map.
Then/(/") = T* is a lattice in G*. Let H* be a factor of G* of rank-1 and
q>: G*—>#* the natural projection. From the definition of irreducibility
one sees easily that r* is also irreducible. Let H be the kernel of q>. Then
H nT* = e (cf. Corollary 5.21: note that the centre of G* is trivial). Now
if A <= r is a unipotent subgroup,/maps A isomorphically onto a unipotent
subgroup A*=f(A) of A*. Since (/>|p, is injective, (/>o/maps A
isomorphically onto a unipotent subgroup of H*. Now H* is the identity
component of the R-rational points of an algebraic semisimple group H*
defined and of rank 1 over R. It follows from Corollary 12.17 that
q> °f(A) is contained in a unique maximal unipotent algebraic subgroup
U of H* defined over R. If A t is a unipotent subgroup of r containing
A, q> °/(/dj) is a unipotent subgroup of H* containing q> of {A). It follows
that(pof(A1)^lJ. Let.
U(r)={x\xer, x unipotent, (pof(x)eU}.

208
XIII. Fundamental Domains
Then any unipotent subgroup At of r containing A is contained in the
set U(r). We will now show that [/(/") is a unipotent subgroup of /".
Let U(r) denote the group generated by U(r). Evidently for xeU(r),
q> o/(x)eU. It suffices thus to show that U{r) consists entirely of unipotent
elements. To prove this we observe first that the group U(r) is solvable.
This follows from the exact sequence
0-> C/(T)n kernel/— U{r)^f(U{r))-^e.
In fact f(U(r)) is isomorphic to q> °f(U(r)) and the latter is contained
in U, a nilpotent group. Let G be a semisimple algebraic group defined
over R having G as the identity component of its R-rational points. Let S
be the Zariski closure of [/(/") in G and S° the identity component of S.
Evidently S is solvable. Every unipotent element xeS belongs to S°
(cf. Preliminaries §2.3). It follows that the set .V = {x|xeS, x unipotent}
is a subgroup of S (Preliminaries § 1.2). Since [/(/") cN, U{r)<=N. Thus
U(r) consists entirely of unipotents. We have therefore proved that r has
Property (R1).
Let U(r) be a maximal subgroup of r and U the Zariski closure of
[/(/") in G. Let a denote the adjoint representation of N(U) (the normaliser
of U) on the Lie algebra u of U and N°([/)={xeN([/)|det<7(x)= ±1}.
We will now show that N0(U)/N°(U)nr has finite measure. The group
[/(/") being Zariski dense in U, we can find a compact set E <= U such that
E ■ U(r)=U (Theorem 2.1). Now, let Q be a Zassenhaus neighbourhood
of e in G (cf. Definition 8.22). Let fi'cfi be a symmetric open
neighbourhood of e such that Q'3<=Q. According to Lemma 13.6, we can find
gsG such that g• Eg~1 <= Q'. We replace the group r by its conjugate
g-T-g*1 and thus assume that there is a compact subset E<=UnQ'
such that E{Ur\ r)= U. Let q: [/—► U/Un r be the natural map. Suppose
now that Tt, T2eN°(U) and xeii' are such that T^T^yer. It suffices
to show that yeN°(U) (cf. Lemma 1.15). Now let n be a Haar measure
on U. Then if BcU is any subset such that n{B)>n(E), q\B cannot
be injective i.e. we can find yet/nT such that Byr\B+</> and y+e;
or again, equivalently, B~1Br\Unr+e. Taking for B the set
Tf1(UnQ')T2 (resp. TtUnii'Tf1) we conclude that we can find
y2 + e (resp. yt + e) in UnT such that T2y2T2~l€Q' (resp. Tf^^^Q').
Consider now the elements y^y-1 aQd Vi- These elements generate a
discrete subgroup of G. On the other hand
Tf^yy.y-^T^Tr'iT.xT^iTf'x-'T^T.exii'x'^Q
and
Tf'y.T^ii.
It follows that yy2y~l and yl generate a nilpotent and hence unipotent
subgroup $ of r. One concludes from this that U(r) is normalised by y:

XIII. Fundamental Domains
209
in fact since yC/f^Oy""1 is the unique maximal unipotent subgroup
containing yy2y"~1, yj.e#<=y t/(r)y_1- On the other hand [/(F) is the
unique maximal unipotent subgroup containing yt. Thus yU{r)y~1 =
U{r). Thus yeN(U). Moreover if F is a measurable set in U such that q
maps F bijectively onto U/U(r), q maps yFy-1 also bijectively onto
U/U(r\ It follows that n(F)=n(yFy-l\ Since fi(F)+0, this proves that
yeN°(U). This shows that N°(U)/N°{U)nr has finite measure.
We will now prove that N°(U)/N°(U)nr is compact. According to
Theorem2.12, the Z-span of exp~1(Unr) is a lattice L in the vector-
space u. Let u=u®C; we identify GL(u) with GL(n, C), n = dim. u, by
means of a basis of L and introduce the following notation.
A =a(N°(U)nrlH = a(N°(U)),
D* = Zariski closure of A in GL(n, C),
//* = Zariski closure of H in GL(n, C),
U*=unipotent radical of H* (=Zariski closure of a(Unr%
N*=unipotent radical of £>*,
D' = identity component of D*,
D = identity component of D* n GL(u),
A' =D'nA, <P=Dr\A.
In view of Theorem 13.1, it suffices to show that H/A is compact. Since A
leaves Lstable and A (resp. A') is Zariski dense in D* (resp. D') D* (resp. £>')
is a Q-subgroup of GL(n, C). Also, if p: D'-*GL(m, Q is any Q-represen-
tation, p(J')cGL(m,Q) and for xeA' all the eigenvalues of p(x) are
algebraic integers; in particular if p is a character p(A')<=-{ + \} and in
view of the Zariski density of A' in D', p(D')={±l}; and since D' is
connected in fact p(D')=\. Thus D' does not admit any non-trivial
characters defined over Q. Since A<=H D*cz H*, U* is normal in D*
(note that t/*cD*). It follows that l/*cJV*. We conclude now from the
maximal unipotencc of UnT in F that any unipotent element xeA
belongs to a(Ur\D (note that the kernel of a is unipotent) and hence
to N*. Moreover H (resp. 0) has finite index in H* n GL(u) (resp. A).
Since A is a lattice in H (see Theorem 13.1), and $cDc//, $isa lattice
in D. Finally $cDnGL(n,Z), a discrete subgroup of D. We see
therefore that ^ is an arithmetic subgroup of D (it has finite index in
D n GL(n, Z)). It follows that D/<P is compact. Now since A is a lattice
in H, according to Theorem 5.27, D contains all the nqn compact simple
subgroups of H. On the other hand one sees easily that U* = N*. It
follows that if K is any maximal compact subgroup of H, KD is a closed
normal subgroup of H such that H/KD is abelian. As $ <= KD, H/KD
has finite Haar measure and is thus compact. Since K and D/<P are
compact H/A is compact This completes the proof of the theorem.
This completes the proof of Theorem 13.19.

210
XIII. Fundamental Domains
13.20. Corollary. An irreducible lattice in a connected semisimple group
G with a rank-1 factor is finitely presentable.
This follows from Corollary 13.15 when G is linear. The general case
follows easily from this.
13.21. Remark. Corollary 13.20 together with Theorem 6.15 and
theorems of Kazdan [1] and S. P. Wang [1] show that any lattice in a
semisimple Lie group is finitely generated (cf. Remarks 6.18).
13.22. Corollary. An irreducible lattice r in a connected linear semi-
simple group G with a rank-1 factor has a torsion-free subgroup of finite
index.
This is a consequence of Corollary 13.16.
13.23. Theorem. Let G be a connected semisimple algebraic group
defined over Q and TcG=G2 be an arithmetic subgroup of G
(Definition 10.12). Assume that Q-rank (G)= 1. Then r is a rank-1 discrete
subgroup of G.
Let Gj be the maximal compact normal subgroup of G and G' = G/Gl.
Let 7t: G -* G' be the natural map and F = n(F). Then F is an L-subgroup
of T (Theorems 10.17 and 10.18) and G'/r' is not compact.
Suppose now that y'sF is a unipotent element not equal to the
identity. Let y eT be any element such that n(y)=y'. Let yu be the unipotent
part of y. Evidently n(yl)=y' so that yu+e. Moreover yB€GQ.It follows
now from Corollary 12.17 that yu belongs to a unique maximal unipotent
Q-subgroup U of G. LetU=UR. Clearly UnT is a maximal unipotent
subgroup of r. Suppose now that O'er' is a unipotent element such that
& and / generate a unipotent subgroup $' of F. Let OeT be chosen such
that n(0)=ff and let 0u be the unipotent part of 0. We claim that 9us U.
To see this, let G' be the connected normal subgroup of G such that
tc(G')=G' and G'nGi is finite. The map n\&.: (5'->G' has for kernel a
central subgroup of G'. It follows that $ = n~1(<P')<=(i' is a nilpotent
subgroup of G'. On the other hand since every unipotent element of G
must necessarily belong to G' (note that Gt is compact) yu, 0ue<P. It
follows that yu and 9U generate a unipotent subgroup of G. Since 0„eGQ
as well, we conclude that 9ueU. Since y'=n(yl) q>'En(U)nr'. It follows
that n{U)nr' is a unipotent subgroup of F containing any unipotent
subgroup of F which contains y'. Thus every element y'sF belongs to a
unique maximal unipotent subgroup of F. We have therefore proved
that F has Property (R1). In particular, according to Theorem 13.3 this
means that there are only finitely many T'-conjugacy classes of maximal
unipotent subgroups of F. Now a maximal unipotent subgroup of F is

XIII. Fundamental Domains
211
of the from n{U)r\ /"" where t/=UR and U is a maximal unipotent Q-sub-
group of G; also U/U n r is compact so that UnTis Zariski dense in U;
finally since Gt is a compact normal subgroup of G, 7t_1(7t(t/)) is the direct
product of U and Gt so that U is the set of all unipotent elements in
Gj U=n~l n(U). From these considerations we conclude that there are
only finitely many T-conjugacy classes of maximal unipotent Q-sub-
groups of G. We have therefore proved
13.24. Theorem. G be a connected semisimple algebraic group defined
and of rank 1 over Q. let G=Gj and r c G be an arithmetic subgroup. Let
Gl be the maximal compact normal subgroup ofG and n: G—> G'=G/Gl be
the natural map. Then n(r) has Property (R1). Consequently G has only
finitely many f-conjugacy classes of maximal unipotent Q-subgroups.
We continue with the notation introduced above. We will show now
that r'(=7t(r)) has Property (R2). As was observed above, if A' is a
maximal unipotent subgroup of /"", A' = n{JJ)r\V where t/=UR and U
is a maximal unipotent Q-subgroup of G. Moreover n(U)/A' is compact
so that U'=n(U) is the Zariski closure of A' in G'. Now let N(U) (resp.
N{U')) be the normaliser of U in G (resp. G'). Then n(N{U)) = N{Ur) and
N([/)=dGj. One sees easily moreover that if N°{U) (resp. N°{U'j) is the
" {geN(U)\lntg preserves the Haar measure on U}
(resp. {geN([/')|Intg preserves the Haar measure on [/'}), n(N°(U))=
N°{U'). Thus to show that N°(t/')/A/r°(t/')nr is compact it suffices to
show that N°{U)/N°{U)r\r is compact.
Now U being a maximal unipotent Q-subgroup of G, N(U) the
normaliser of U in G is of the form Z(T) • U where Z(T) is the centraliser
of a maximal Q-split torus T which normalises U. The group Z(T) is
reductive and is of the form T • M where TnMis finite and the identity
component M° of M has no non-trivial characters defined over Q and
(hence) Mq consists entirely of semisimple elements (cf. Chapter XII:
12.2-12.5). It follows that Mg/MgnT is compact. Since U/UnT is
compact Mg" • [//M|J -UnT is also compact. Since G hasQ-rank 1, one
sees that Mjj U is of codimension 1 in N(U)(=N(V)n G). On the other
hand N°(U) is also of codimension 1 in N(U) and, as is easily seen,
contains MjJ U. Thus Mj(t/). U is the identity component of N°{U).
Finally N°(U) is the R-rational points of the algebraic group
{g€N(U)|det<7(g)=±l}
where a denotes the adjoint action of N{U) on the Lie algebra u of U.
Thus N°(U) has only finitely many connected components
(Preliminaries § 2.3). It follows that N°(U)/N°(U)r\ T is compact. This completes
the proof of Theorem 13.23.

212
XIII. Fundamental Domains
13.25. Corollary. Let G be a semisimple algebraic group defined and of
rank 1 over Q and let r<=.GQ be an arithmetic subgroup. Then r is finitely
presentable and is a lattice in GR.
We end this chapter with a final result on arithmetic groups. This
theorem (Theorem 13.26 below) asserting the finiteness of certain double-
coset classes is the starting point in one approach to the construction of
fundamental domains for arithmetic subgroups of semisimple algebraic
Q-groups of higher rank. For the construction of the fundamental
domain itself with this result as the starting point we refer to Borel
[1,§16].
13.26. Theorem. Let G be a connected semisimple algebraic group
defined over Q. Let rcGQ be an arithmetic subgroup of G. Let P be a
minimal parabolic subgroup of G defined over Q. Then the set r\GQ/PQ of
double coset classes is finite.
We will argue by induction on the Q-rank of G. When Q-rank (G)=0,
G=P and there is nothing to prove. For the case Q-rank G = 1, we first
interpret r\GQ/PQ suitably. Let U be the unipotent radical of P. Then U
is a maximal unipotent Q-subgroup of G and every maximal unipotent
Q-subgroup of G is conjugate to U by an element of GQ. Moreover P is
the normaliser of U in G (Chapter XII: 12.4-12.5). Thus r\GQ/PQ can
be identified with the set of T-conjugacy classes of maximal unipotent
Q-subgroups of G; and Theorem 13.24 asserts that this set is indeed
finite.
For the general case we utilise the Bruhat decomposition. According
to 12.13 we have (in the notation introduced in Chapter XII),
GQ=UQQ^Z(S)QUQ=UQ Q^PQ.
Let/: QW^>QW° be the natural bijection. Now each element w0eQW°
can be written as a product
w0=««,■*« sap
of reflections with respect to the simple roots: {a,eQ.d, lgigp}. We
denote by /(vv0) the least integer p for which we have such an expression
for w0. For we W let l{w)=l(f{w)). For an integer I, let W{I)={weQW,
l(w)=l}. We will now prove inductively that for every / we have a finite
set S, such that
uQw(0PQ-=r-a,.pQ.
Assume this result proved for r <l. Let w0e W{f). Then
W0 = Sa-<7

XIII. Fundamental Domains
213
where ueQA and as W{1 — 1). Consider any element of the form
u-w0-p, ueUQ, pePQ.
Now let G(ot) be the subgroup generated by
{expgta|fceZfc=,=0}.
One sees easily that this is a semisimple Q-group of Q-rank 1. Moreover
we can assume sa so chosen that it belongs to G(ot). Now let U(ot)=
G (a) n U and let U' be the subgroup generated by {exp gp|/?eQ^+, /? not
a multiple of a}. Then it is known that u can be written in the form u=ua-u'
with HaeU(ot), u'eU'. Moreover s~1U'stcU. We thus find
u-w0-p=ua-sa-vap
where veU' and uasaeG(a). Since the result is true for Q-rank-1 groups
we have a finite set Sf for each /?eQ.d such that
G(/?)Q=(G(/?)nr)S/,(G(«nPQ).
We have thus setting P(/8)=G(/8)nP
u • w0 • pe(G(«)nr)2aP(a)Q • vap.
Clearly P(a)Q • p • a ■ p cz UQ • W{1 — 1) PQ since P(a) <=P. Now by induction
hypothesis
UQm'-l)PQ^2.-iPQ.
Thus we find
Now for each xeS^xTx-'nT has finite index in r. We can thus find
for each x e 2a a finite set Fx a such that xr<=r-Fxa and setting 5^ = (J Fx a
we see that „ ^ ^ „,
7. a
Itfollows that uw0 peT S^-Sj_ j PQ. If we set [j S'0LSi_1=3i we have
This proves the theorem.
13.27. Remark. A. Borel ([1], § 16) shows how one can describe
fundamental domains for arithmetic subgroups of semisimple algebraic
group with Theorem 13.26 as the starting point. This construction yields
as a consequence among numerous other things the result below
(Theorem 13.28). We will make use of this result in Chapter XIV. For the proof
we refer to Borel (loc. cit.).

214
XIII. Fundamental Domains
13.28. Theorem. Let G be a semisimple algebraic group defined over Q
and GR the set ofR-rational points of G. Let r<=GR be an arithmetic
subgroup ofG. Then T is a lattice in GR.
As has been proved in Chapter X, when GQ has no unipotent element
in it or equivalently when Q-rank G=0, Gjr is compact. When G has
Q-rank 1, we have established that r is a lattice such that G^jr is non-
compact. Borel's construction of the fundamental domain takes care of
all Q-ranks.
13.29. Remark. Theorems 13.19 and 13.12 together generalise the
work of Garland-Raghunathan [1] where fundamental domains of the
type described in Theorem 13.12 are exhibited for lattices in rank 1
groups. The approach in this book, however is somewhat different from
that of Garland-Raghunathan. Also using the techniques of this paper,
one can prove that irreducible lattices in semisimple groups with a
rank 1 factor and without compact factors are "almost always" locally
rigid in the sense of Chapter VI.

Chapter XIV
Existence of Lattices
Our aim in this chapter is to prove the following result due to Borel [4].
(The proof given here is practically a reproduction.)
14.1. Theorem. Let G be a connected non-compact semisimple Lie
group. Then G has both uniform and non-uniform lattices.
One sees easily that the general case is easily deduced from the case
when G is simple and is in addition the adjoint group (of its Lie algebra).
In view of our results in Chapter X and Theorem 13.28, Theorem 14.1 is
a direct consequence of the following theorem.
14.2. Theorem. Let G be a non-compact connected simple Lie group
isomorphic to its adjoint group. Then we can find a (connected) semisimple
algebraic group G (resp. H) defined over Q with the following properties.
1) There is a surjective homomorphism q>: G|j—>G (resp. i//: H|j—>G)
with compact kernel (here G{| (resp. H{j) is the identity component of the
R-rational points GR (resp. HR) of G (resp. H)).
2) IfT is an arithmetic subgroup ofG (resp. H) GJT (resp. HJT) is
non-compact (resp. compact).
This theorem in turn is a fairly simple consequence of the following
algebraic result.
14.3. Proposition. Let g be a real semisimple Lie algebra. Then there
exists a Lie algebra gQ over Q, an involution 6Q: gQ—>gQ and a (Q-linear)
injective Lie algebra homomorphism
/' 9q->9
such that (i) the induced map
gQ®QR->g
is an isomorphism and (ii) the natural extension 8 of Qq to q is a Carton
involution.
Moreover if q is non-compact the Q-structure may be so chosen that
there exists ^(+0) in Qq such that adX is nilpotent.

216
XIV. Existence of Lattices
We will now deduce the Theorem 14.2 from Proposition 14.3.
Let G be a connected semisimple Lie group and g its Lie algebra. Let
g=I © p be a Cartan decomposition of g with I as the algebra. According
to Proposition 14.3, we can find a basis {XJl^A^n} of g consisting of
elements belonging to I or p and such that for 1 g A, n g n,
v=l
where c^eQ. Let g=g ® RC. We identify GL(g) with GL(n, C) by means
of the basis {Xx\ 1 ^xgn}. Let G denote the group of Lie algebra
automorphisms of g. Since c^eQ, one sees immediately that G = Autg is an
algebraic subgroup of GL(w, C) defined over Q. GR is then evidently the
automorphism group of the Lie algebra g. Now it is well-known that the
identity component Gj of GR is canonically isomorphic to the group G.
Now let r<=GR be an arithmetic subgroup. We assume the Q-form gQ to
be so chosen that there exists X(+0) in gQ such that adX is nilpotent.
Then one sees that if xeGR is the (inner) automorphism exp(adX),
xfeT for some integer n. Thus T contains a unipotent element and hence
GR/r is not compact. This completes the proof for the non-uniform case.
To construct a uniform subgroup as in Theorem 14.2 we argue as
follows. Consider the Lie algebra gQ and the involution 6 constructed
as in Proposition 14.3. Let u="|/2 and v be the unique positive square
root of u. Let {XA|l_U^n} be a basis of gQ (over Q) chosen such that
0(Xa)=Xa for N<a^n and 0(X,.)=-X,. for lgigN. Let K=Q{]/2)
and let m (resp. mj be the vector space over Q("|/2) spanned by
{Aa|N<a_n} and {vX^l^i^N} (resp. spanned by {XJN< a gn} and
(V^l • v X; 11 ^ i g N}). Now we have g ^ m ®Q(vT) R. Let g„=m„® Q(/2)R.
It is easy to see that the Killing form of gu is negative definite. Thus the
adjoint group of gu is compact. Consider now the Lie algebra h = g ©gu.
Let bQc:h be the Q-linear span of the vectors {Aa={Xa, XJ, N <a^n},
{B,.=(t>X,,]/^Ti>X,), l^igN}, L4;=(uXa)-uXa),N<a^n} and
{B'(=(urX;, — uvXt), l^i^N} in h. One sees easily that hQ is a Q-Lie-
subalgebra of h and that hQ ® QR^b. Using the basis
we identify h=h®RC with C" and GL(h) with GL(n,C). Then the
automorphism group H of the Lie algebra h is an algebraic Q-subgroup
GL(n, C). Let r be an arithmetic subgroup of HR. Then we claim that
HR/r is compact. To prove this it suffices to show that Tn Hj contains
no nontrivial unipotent elements. By passing to a subgroup of finite index
we can assume that r stabilises the lattice
L= £ ZA.Q £ ZB(.

XIV. Existence of Lattices
217
Let 7i (resp. 7t„) denote the projection of h on g (resp. gu). Then one
sees easily that n and nu are injective when restricted to L. Moreover
if ysr is any element nu(L) and n(L) are stable under y. It is easily
seen moreover that nu(L)<=mu while n(L)<=m, and that y acts uni-
potently on nu(L) if and only if it acts unipotently on n{L) (and hence
on L as well). Suppose now that yEH^nT. The group H{J is the direct
product of G and Gu (= adjoint group of gj in a natural fashion:
H=GxGu. If ysr, y is of the form (6,6J with 6u=e if and only if y = e.
Now Gu being compact 6U is not unipotent if y+e. Thus T contains no
unipotents. Hence Hr/H,, n T is compact. We clearly have a projection
H|J—> G with a compact group viz. Gu, as kernel. This completes the proof
of Theorem 14.2.
We will now take up the proof of Proposition 14.3.
14.4. We adopt the following notation for the rest of this chapter.
g, a real semisimple Lie algebra,
i}=I©p, a Cartan decomposition with I as the algebra,
6, the associated Cartan involution,
t, a Cartan subalgebra of t,
h, a Cartan subalgebra of g containing t,
hp = hnp,
9=9 ® rC, the complexification of g.
A{ , ), the Killing form on g.
For a Q- or R-linear subspace Keg, V denotes the C-linear span
of Kin g. In particular h is a Cartan subalgebra of g.
gB=I-(-|/ —lp, the compact form of g associated to g and 0,
I)*, the dual Homc(l), C) of !),
//fl(aeh*), the element of h such that A(H, Ha) = a{H) for all //eh,
g"(aeh*), the space {xeg|ad H{x)=a(H)x for //eh},
$, the system of roots of g with respect to h i.e., the set
{aeh*|gfl=,=0,a*0},
h°, the Q-linear span of {Ha\ae<P},
h° = h°np.
Then h0 = t° + hJJ, a direct sum. Also, the inclusions t°^->t and h0<-->l)
induce isomorphisms t° ® qC-=i-* t and h° ® QC —-» h. We now choose
once for all an ordered basis {//,|l^i^/{ of h° (over Q) such that
{//(|l_i^rj span t° and {//,|r<ig/} span h°. We order the vector space
h0* = HomQ(h°,Q) ( = {c.eh*|fl(h°)c:Q() as follows. For a, />eb0*, a + b,
let i(«,fc)=inf{i|(a-fc)(//,)+0, lgig/j; then a^b if {a-b)(Hl(aJ)))^0.

218
XIV. Existence of Lattices
For ae«P, a(h°)c:Q so that «Pch°*. Let
«P+ = {ae$|a>0}.
Let A be the sytem of simple roots i.e. A = {ae$+\a+b+c for any
b,ce<P+}.
We will now establish a simple lemma needed later. We use the
notation introduced above.
14.5. Lemma. 6 leaves h stable; the induced map 'Oi^O'^ofthe dual b *
maps <P into itself and A into itself.
That 6 leaves h stable is evident from our choice of h. Since 0 is an
automorphism '8 will permute the roots of g. Hence '0(<i&)=$. We will
now show that if «£$, a is non-zero on t Let H°el$ and Esq". Now E
can be written in the form X+Y with Xel, Yep. Since [11] el and
p,p]cp, we see that if [/J,E]=0 for /Jet, then \H,X] = \H, 7]=0.
It follows that if a(t)=0, \H°, 7] commutes with t and belongs to I,
hence belongs to t. Thus for any Het, we have
A{\H°, Y], H)= -A{Y, [ff°, ff])=0.
Since A( , ) restricted to t is non-degenerate [//°, 7]=0. Now we have
a (H°) ■ E=[ff0, X + 7] = [ff0, X] € p.
Thus if a{H°)+0, E=Y, a contradiction since [/J°, 7]=0. We have
therefore proved that if ae$, a is non-zero on t. Now 6 is trivial on t
so that a'='0(a)=a on t It follows from our definition of the order that
a'>0 if and only if a>0 i.e. '0(#+)=4>+. Hence '0(/d)=/d. This proves
the lemma.
We now state a theorem due to Chevalley [3] without proof.
14.6. Theorem. For each ae$, we can choose a non-zero element
EaeQa satisfying the following conditions
(1) lEa>E_J=-H* = 2HJa{Ha),
(2) \Ea,E^=0 if a,be<P, a + b+0, a+b$<P,
(3) \Ea,E^ = NBtbEa+b if a,b, a+be<P, where
(4) A^=AU_»=±(P+1),
and p^O is the greatest integer such that a—pbe$. Moreover we have
+ j;R/^I(£,-£J.
He*

XIV. Existence of Lattices
219
(Chevalley's starting point is the existence of a basis satisfying somewhat
weaker conditions due to Weyl [1]. For a proof of Weyl's theorem see
for instance (Helgason [1, Theorem 5.5, Chapter III]).)
14.7. We next establish that we can in fact choose the Ea to have
additional "nice" properties with respect to 6. We observe first that if
we set a'= 6(a) for ae«P we have 6{E^=ca-Ea.; the fact that 0 is a Lie
algebra automorphism leads to the following relations among the c„, ae <P:
(6) cacbNa.j,. = ca+bNab (a,b,a+be<P),
(7) cac_a=l {ae<P).
Moreover Na, b, = + Na b since '0(<P)=<P so that we have in fact
(8) cacb=±ca+b (a,b,a+be<P).
Finally since 6 is an involution, Q2 = Identity leads to the equations
(9) c0c„, = l (ae#).
It is moreover evident that 6 leaves g„ stable. Thus 6 (£„+£_„)=
caEa. + c_a,E_a for ae«P, leads to
(10) ca=c_a {ae*).
Combining this with (9) we have
(U) ca-ca=l.
14.8. Now let /: A^>j/^T- RcC be a map such that f(a)+f(a')=0
for all as A (note that aeA if and onlyif a'e.d)and fora with a =1= a',e~ 2-f(o)=c„.
Such a choice of/ is possible since c„ has modulus 1 according to (11).
Let h'=j/^T■ hp +1(= £ R j/--l"■ *0 (&' is precisely the real subspace
of I) on which all the roots take purely imaginary values). Let HeJ) be the
unique element such that a(H)=f(a) for all aeA. Then Heh'. Let T: g—> g
be the (Lie algebra) automorphism exp(ad/J). T acts trivially on h.
For aeJ, we have 0={a+a'){H)=a(H) + a(6{H)) = a(H+d{H)). Since J
spans all of b we conclude that 6(H) = — H. From this one sees immediately
that we have
(12) /Jej/^Thpc:gli and T6=dT~l.
Now let Fb=T(Eb)(=efib)Eb). Then the {Fb\be<P} again satisfy the
conditions (l)-(5) of Theorem 14.6 (with Eb replaced by Fb). We have in
addition the following conditions:
(13) e{Fb)=±Fb if b=b' (be<P),
(14) e{Fb)=Fb if b+b' {be<P).

220 XIV. Existence of Lattices
The first of these equations is a consequence of (9). Eq. (14) is a
consequence of our choice of H: For ±beA,
6(Fb)=0(T(Eb))=^c„-Eb. = Fbl
since cb=e~2m and f(b)+f(b')=0. Now any be<P can be expressed
in the form + £ ma ■ a, ma^0; if we set \b\ = £ ma, then if \b\ > 1 we can
aeA aeA
find a eJ such that ±b = a+c' with c'e«P+,|c'| = |b| = l. A simple induction
on |i>| now proves (14).
14.9. Now let
A = {ae<P + \a' = a,d{Fa) = Fa}, B={aE<P+\a'=a,e{Fa)=-FJ,
C = {aE<P+a'*a}.
Then the subspace l(cg) is spanned as a vector space over R by the set
S(I)={/^TH* {aeAuB), \T^l{H*+H?) (ceC),
Fa + F_a(aeA), ir^l{Fa-F_J{aEA),
Fc + F_c + Fc,+F_c,(ceC) and /=l(F0-F_c + Fc.-F_c.) (ceQ) .
The space p is spanned by the set
S{p) = {H*-H*.{ceC),
V^i(Fb + F_b) {beB), Fb-F_b {beB),
/=l(Fc+F_c-Fc,-F_c.)(ceC) and (Fc-F_c-Fc. + F_c) (ceC)}.
Let Iq (resp. pQ) denote the Q-linear span of S(l) (resp. S(p)). Then it is
easily checked that Iq (resp. pQ) spans I (resp. p) as a vector space over R
and that IQ and 9q=Iq + Pq are Lie subalgebras over Q of g. Thus gQ
is the required Q-Lie algebra and 0Q=restriction of 6 to gQ, the required
involution.
14.10. We have now only to show that gQ contains an element X
such that ad X is nilpotent provided that g is non-compact.
We will consider two cases separately: Case 1: C + 0, Case 2: C=0.
14.11. Casel (C=j=0). Let ceC; then H=H*-H*.ep. On the other
hand the eigen-values of H* and H*., in the adjoint representation
belong to Q. It follows that we can find Xeqq such that
[H,X]=rX
with reQ. Lie's theorem applied to the solvable Lie subalgebra CH+CX
of g now shows that ad X is nilpotent.

XIV. Existence of Lattices
221
14.12. Case 2 (C=P). In this case we necessarily have t=h and B+0.
Consider now the Lie algebra si (2, Q) of 2 x 2 matrices over Q of trace 0.
Let
1 0
0 -1
, Y=
0 1
1 0
, H=
0 1
-1 0
Then we have the bracket-relations
[X, Y] = 2tf. [H,X]=-2Y, [H, 7] = 2X.
A simple calculation shows that we have also for bsB,
[/=! ■ (F„+F_ „), Fb - F_ „]=2 j/^T • h:
[/^T • Hi, /=l(Fb+F_j] = - 2(F„ -F_„)
[/^l • /£, F,-F_ J=2/TT. (F^ + J^fc).
It follows that we have a Lie algebra homomorphism rb: si (2, Q) ->ciQ such
that rb{X)=/=l-{F„+F_b), rb{Y)={Fb-F_„) and r„(//)=/^l ■//*.
Since si (2, Q) contains nilpotent matrices the desired result follows.
14.13. Remarks. Arithmeticity of general lattices. A discrete
subgroup r in a connected semisimple Lie group G is arithmetic if it satisfies
the following condition, Let G* be the adjoint group of G and p: G—> G*
the natural map. Then there exists an algebraic group Gt defined over Q,
an arithmetic subgroup r1<=G1(i and a surjective homomorphism
/: G?R->G* such that the kernel of / is compact and /(/]) has finite
index in p(r). The examples constructed above are obviously arithmetic
in this sense. Vinberg [1] and Makarov [1] gave the first examples of
non-arithmetic lattices in semisimple groups. Vinberg [1] in fact studies
in detail Coxeter groups in SO(n, 1) and obtains criteria in order that
such groups be lattices and then gives necessary and sufficient conditions
for these lattices to be arithmetic. However these Coxeter groups (as is
seen from their very definition) are contained in the group of units of
SO(n, 1) in a number-field. All the counter examples to arithmeticity
known hitherto are in the groups SO(n, 1). In the group SL(2, R)
(=.(loc)SO(2,l)) one can find lattices which cannot be conjugated into
the Q-rational points SL(2, Q) of SL(2, C). It is not known wheter groups
other than SO(n, 1) n >0 admit a non-arithmetic lattice or not.

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Index
Algebraic groups, abelian P 2.2
— -.reductive P2.5
— —, semi-simple P2.5
— -.unipotent P2.2
Bundles, differentiable P 4,1
— .induced P4.1
—.principal P4.1
-.topological P4.1
Cartan-involution P 1.5
Cohomology of groups P 3.2
— of Lie algebras P 3.1
Criterion, Cartan's 12.1
-.Mahler's 10.9
Decomposition, Bruhat- 12,13
-, Cartan- P 1.5
—, Iwasawa- P 1.7
-, Levi- P 1,3
Groups, polycyclic 4.1
— .stronglypolycyclic 4.5
-,P/S 4,35
-, Weyl- 122
Hodge's theory 7.10
Lattice 1.8
—.uniform 1.11
—, non-uniform 1.11
-.irreducible 5.20
Lie algebras, solvable P 1.1
— -.nilpotent P 1.1
Locally rigid (homomorphism) 6.5
Neighbourhood, L-
—, Zassenhaus- 822
Nil radical, of polycyclic groups 4.8
Property, L 121
-,P 9.1
-,R1 13.2
-,R2 13.7
-,S 5.1
-,SS 5.25
Rank of nilpotent group 2.9
— of polycyclic group 4,8
Representation, extendable 4.18
—, full 4.37
-./"-admissible 7.24
—, /"-supported 7.24
Roots, k- 12.2
—, simple 12.3
Siegel set 102
Subgroups, arithmetic 10-12
—, commensurable 10.12
—, convergence of 1.20
-,£.- 1.21
—, maximum nilpotent normal P 1.2
-, net 6,10
—, parabolic 12,4
-, rank-1 discrete 13,11
—, strongly compatible 1.17
—, uniform 1.10
('P' indicates Preliminaries)