These are my live-TeXed notes (reorganized according to my tastes) for the course Discrete subgroups of Lie groups and discrete transformation groups by Professor Lizhen Ji, July 19 — August 9, at Math Science Center of Tsinghua University.
The plan of this course
- Introduction and basic notions
- General questions on discrete subgroups and discrete transformation groups.
- Fuchsian groups, namely discrete subgroups of
(
). In particular, the modular group
and its congruence subgroups. Kleinian groups, namely discrete subgroups of
. They provide special, important examples of lattices of semisimple Lie groups and also nonlattices.
- Crystallographic groups (Bieberbach theorems)
- Nilpotent Lie groups and their lattices
- Solvable Lie groups and their lattices (Mostow rigidity)
- Semisimple Lie groups and their lattices
The main purpose of this course is to study lattices of semisimple Lie groups. But we need to understand lattices of non-semisimple Lie groups too. (Unfortunately, only the first three topics were covered in this course).
Introduction
The topic of this course is a central subject of modern mathematics. It is closely related to many areas, e.g., group theory, geometry (differential geometry, algebraic geometry, arithmetic geometry), analysis, number theory (automorphic forms) and algebra (algebraic
-theory).
Major figures in this subject includes Poincare, Klein, Fricke, and recently C. L. Siegel, A. Weil, A. Selberg, A. Borel, Harish-Chandra, Piatetski-Shapiro, G. Margulis, D. Kazhdan, Maltsev, Auslander, Hsien-Chung Wang, R. Langlands. Two important books are Raghunathan's and Zimmer's.
Now let us explain the "what, why and how" of the words in the course title.
A discrete group is a group with the discrete topology.
This definition is not very interesting at first sight since the discrete topology may not appear "naturally".
with the natural topology.
with the discrete topology. But the usual topology is more natural since it is a connected Lie group. The inclusion of
into
induces the usual discrete topology on
.
with the discrete topology. The induced topology of
from
is not discrete. Nevertheless, The discrete topology on
is natural and important since it can be realized as a discrete subgroup of an important locally compact topological space, i.e., the ring of adeles.
Let

be a topological group (usually we assume

is locally compact). A subgroup

is called a
discrete subgroup if the induced topological on

is discrete.
An important subclass is given by taking

to be a Lie group and

be a discrete subgroup.
Here is a less trivial example.
Let

. Then

is a Lie group since it is the inverse image of the regular value 1 under

(or, it is a closed subgroup of a Lie group). The
modular group 
is a discrete subgroup of

since

is discrete in

. Also,

is a discrete subgroup of

and

. The first one of the three embeddings is the most natural one. Firstly,

is a lattice, namely

is finite, but

is infinite. Secondly, from the point of view of algebraic groups,

is an arithmetic subgroup of the semisimple linear algebraic group

and

is the

-locus of

.
Let

, for any nonzero vector

, we have an embedding

Also, given two linear independent vectors

, we have an embedding

which is more natural since

is finite. In other words, we are interested in discrete subgroups

of

that are not "too small" in

.
Now let us come to the second key word in the course title — transformation groups, which first appeared in Lie's book Theorie der transformations gruppe I, II, II.
Let

be a topological group and

be a topological space. A
topological action of

on

is a continuous map

satisfying
for any
,
- for any
and any
,
.
For any
, the
-action is a homeomorphism. Let
be the group of homeomorphisms of
, then we get a homomorphism
. The following two concepts are closely related.
The space

with a

-action is called a
-space.
The group

with an action on

is called a
transformation group.
We can even put more structures on the spaces
and the groups
:
is a manifold,
is either a discrete group or a Lie group and the action is by diffeomorphisms. This is the subject of the classical transformation group theory.
is a complex manifold (or complex space),
is either discrete or a complex Lie group and the action is by holomorphisms.
is an algebraic variety,
is an algebraic group and the action is by morphisms.
We are interested in the case that
is an infinite discrete group and
is a topological space or manifold, which is important for, e.g., geometric group theory.
Let

be a Hausdorff space with

-action on

. We say the action of

is
discontinuous, if for any

, the orbit

of

is a discrete subset of

. We say the action of

is
properly discontinuous, if for any compact subset

, the set

is finite.
If the

-action on

is properly discontinuous, then the quotient

with the quotient topology is Hausdorff.
Here are a few reasons why we study discrete subgroups.
- It occurs naturally (e.g.,
).
- It provides important examples in geometric group theory and combinatorial group theory.
- Given a manifold
with nontrivial fundamental group, then
acts on
as Deck transformations.
- Many natural spaces arises from quotients
(e.g., locally symmetric space).
Groups are interesting in themselves, to understand their properties, the "only" effective way is to use their actions on suitable spaces. So groups that admit good actions on nice spaces are interesting and special. Besides discrete subgroups of Lie groups, two other very important discrete transformation groups are:
- Mapping class groups of surfaces with the actions on the Teichmuller spaces.
- Let
be the free group on
generators, the outer automorphism group
is the most important group in combinatorial group theory. In 1980, Culler-Vogtmann discovered the Outer space
, a contractible space (which is a simplicial complex) where
acts simplically and properly.
is the space of marked metrics, which is related to the hot topic of tropical geometry.
A discrete subgroup of

of a topological group

is called a
uniform lattice if

is compact.
- Every discrete subgroup
of
is of the form
where
's are linearly independent.
is a uniform lattice if and only if
is finite.
We can define the more general notion of volume on the quotient
using a fundamental set of the
-action on
.
A subset

of

is called a
fundamental set for the

-action on

if

meets every

-orbit exactly once, namely

and

for any

in

.
Let

be a secondly countable locally compact topological group and

be a left invariant Haar measure on

, then for any discrete subgroup

of

, there exists an induced measure on

.
A discrete subgroup

of a locally compact topological group

is called a
lattice (or lattice subgroup) if

is finite. In other words, let

be a left invariant measure on

, a discrete subgroup

is called a lattice if

is finite, where

is the induced meausre on

.
Let
be a locally compact topological group. Then
admits a left invariant Haar measure unique up to multiple. Let
be a left invariant Haar measure and let
be the right action, then
induces a measure
and
is also left invariant. By the uniqueness of left invariant Haar measure, there exists a constant
such that
.
The function

is called the
modular function of

. If

, the

is called
unimodular (hence any left invariant Haar measure is also right invariant).
Let

be a real semisimple Lie group, then

is unimodular, since the modular function

is homomorphism,

is abelian and
![$G=[G,G]$](./latex/latex2png-DiscreteSubgroups_13762369_-5.gif)
.
If

admits a lattice, then

is unimodular.
There exists a Borel fundamental set

such that

. One can check that for any

,

is also a fundamental set. So

(finite) and

is unimodular.
¡õ
The affine group

is not unimodular, as the measure

is left invariant but not right invariant. Hence it doesn't admit lattices. Similarly the

is not unimodular, hence it doesn't admit lattices either.
Does every unimodular (e.g., abelian) group admit lattices?
No,

is abelian but doesn't have any notrivial discrete subgroups. If

is a discrete subgroup containing

, then

but

, contradicting the fact that

is discrete.
It is still an open problem to decide when a unimodular Lie group group admits lattices. Nevertheless, the anwser is affirmative for semisimple Lie groups.
(Borel)
If

is a real semisimple Lie group with finitely many connected components, then

admits lattices (both uniform and nonuniform).
Questions and problems about discrete subgroups
We start by listing several problems concerning a discrete subgroup
to motivate our discussion.
Finite generation Namely, find finitely many elements
such that every element of
can be written as
where
. The existence of the generation is relatively easy (e.g., Kazhdan property (T)). However, finding the explicit generators are generally much harder, e.g., how about the generators of
for
? The finite generation is useful for geometric group theory (word metric, Cayely graph, etc.)
Let

be a set of generators of

. Define

. Define the
word metric 
for any

. This metric is left invariant under

, namely

.
The
Cayely graph 
consists of the elements of

as vertices. An edge between

exists if and only if

. Then

is connected.
Let

,

be two generating sets of

. Then there exists constants

such that for any

,

,

. Then

is the infinite grid on

and is not simply connected.

,

. Then

is a infinite tree, hence simply connected.
Finite presentation Let
be a finitely generated group, we may ask whether there are only finitely many relations between the generators. Finding the existence and explicit relations are very important (and hard) in combinatorial group theory.
Cohomology
and
can be defined as the cohomology groups and homology groups of the classifying space
. These are important invariants of
useful in topology, number theory, representation theory and differential geometry.
We are interested in knowing the conditions under which good models (closed manifolds) exist for the classifying space
. Furthermore, if
can be realized by a closed manifold, the Borel conjecture asks whether it is unique up to homeomorphism.
Large scale geometry The geometric group theory single-handedly established by Gromov asks how "big"
is, roughly speaking, the growth rate of the volumes of the balls in
. For example,
is "smaller" than the Poincare upper half plane
: the volume
in
, but
in
grows exponentially.
For
, we consider the asymptotic behaviors of the volume
.
For

,

grows polynomially, but for

, it grows exponentially.
The following striking result due to Gromov relates the large scale geometry of a group to its own algebraic structure.
(Gromov)
If

is a lattice of a nilpotent Lie group, then

has a polynomial growth. Conversely, if

has a polynomial growth, then

is
virtually nilpotent, namely there exists a finite index subgroup

of

such that

is a lattice in a nilpotent Lie group.
Rigidity and action on manifolds Mostow strong rigidity, Margulis super-rigidity, Zimmer program, etc..
Construction and classification of infinite simple groups Margulis normal subgroup theorem and the rough classification up to quasi-isometry.
(Margulis normal subgroup theorem)
If

is an irreducible lattice in a semisimple Lie group of rank

. Then every normal subgroup of

is finite or of finite index.
Now let us list some problems concerning a discrete transformation group
acting on
.
Properness If
acts on
properly, the the quotient space
is a Hausdorff sapce. Nevertheless, We are also interested in non-proper actions such as ergodic actions (chaotic actions). They are important in Mostow rigidity, Margulis' work and also number theory.
Let

be a measure on

invariant under

, the action of

is called
ergodic if any invariant subset of

is either of measure 0 or of full measure.
Orbits and quotient space Suppose
acts properly on
, we would like to understand structures (geometry, topology, analsis) of orbits of
in
and the quotient
.
- Geometry: the geometry of locally symmetic spaces
when
is a symmetic space;
- Topology: often
provides classification spaces like the classifying spaces.
- Analysis: spectual theorem of automorphic forms. The Selberg trace formular relates the geometry and analysis.
A crucial role is played by finding good fundamental domains of
.
A fundamental domain

is called
locally finite if for any

, there exists a neighbourhood

containing

such that

only meets finitely many

-translates of

. A fundamental domain

is called
globally finite if

is fintie.
Suppose
is a fundamental domain. Let
be the projection. Then
is obtained by identifying some points on the boundary
, namely we have a continuous bijection
. The significance of the local finiteness and global finiteness can be seen from the following two theorems.

is a homeomorphism if and only if

is locally finite.
We omit the proof here.
¡õ
Suppose

is a connected topological space and

acts properly on

. Let

be a fundamental domain. Assume

is open and

. Let

. If

is finite, then

is finitely generated. Briefly speaking, if

admits a globally finite fundamental domain then

is finitely generated.
Let

be the subgroup generated by

. We claim that

. If not, then by the construction of

,

decomposes as two
disjoint open subsets, which contradicts the assumption that

is connected.
¡õ
Finding fundamental domains with global finiteness for lattices in Lie groups is called reduction theory. Legendre and Gauss started this theory while studying number-theoretic problems. The finiteness property is now called Siegel finiteness.
Let

be a proper (i.e., every bounded closed subset is compact) metric space. Let

acts isometrically. We define the
Dirichlet fundamental domain 
for

not fixed by any nontrivial element of

. In other words, we pick the elements which are closest to

in each

-orbit to form

. It is clear that

, however, the boundary may be huge. Nevertheless, we know that

is a fundamental domain in the following special case.
Let

be a Riemannian manifold and

be the Riemannian distance. Then

and

is a fundamental domain.
We can also relax the requirement on the domain in the proof of Theorem 5.
A domain

is called a
rough (coarse) fundamental domain for

, if

and the induces map

is finite-to-one, i.e., the size of the fibres are bounded.
Let

be a connected topological space. If

is a rough fundamental domain (open or closed) and

if finite, then

is finitely generated.

is compact if and only if there exists a compact rough fundamental domain.
Fuchsian groups
We shall study the Fuchsian groups in the general framework of discrete subgroups of semisimple Lie groups.
Recall that
acts on
holomorphically and isometrically by Mobius transformations. This action is transitive, i.e., for any
, there exists
such that
. The stablizer of
is
, hence
. If
is a discrete subgroup of
, then
acts properly on
.
Any discrete subgroup of

acting on

is called a
Fuchsian group.
Fuchsian groups divide into two types:
- the first kind: if the limit set of
in the boundary
is the whole boundary.
- the seconnd kind: if it is not of the first kind.
If

is a lattice, then

is of the first kind.
Let

be the modular group, then

is a lattice, therefore it is a Fuchsian group of the first kind. Note that

is finite if and only if

is finite. The latter also follows from the well-known explicit description of the fundamental domain of

:

which is also a Dirichlet fundamental domain

for

. In this case, the strip

is a rough fundamental domain. Since the integration of

is bounded on

, we know that

is a lattice. Also, Proposition
7 implies that

is not a uniform.
Assume

is a complete Riemannian manifold,

acts properly and isometrically on

, then the Dirichlet fundamental domain

is locally finite.
If
is torsion-free, then
acts freely on
and
is a covering map. However,
is not torsion-free and some elements of
fix points in
. Nevertheless,
admits finite index torsion-free subgroups. For such a torsion-free subgroup
,
is a covering map from a smooth manifold to a orbifold. Minkowskii showed that when
, the principal congruence subgroup
is torsion-free.
The following more general result is due to Selberg.
(Selberg's Lemma)
If

is a finitely generated subgroup of

, then

admits a finite index torsion-free subgroup.
In particular, every finitely generated Fuchsian group admits a finite index torsion-free subgroup.
When is a discrete subgroup

of a Lie group

finitely generated?
(Siegel)
If

is a lattice of

, then

is finitely generated.
More generally, we have:
If

is a lattice of a semisimple Lie group

, then

is finitely generally.
We shall discuss the idea of the proof of Theorem 6.
Let
be a Fuchsian group, then the Dirichlet fundamental domains
are bounded by geodesics because the bisector
is a geodesic.
A Fuchsian group

is called
geometrically finite if it admits a fundamental domain that is bounded by finitely many geodesics (called a
geodesic polygon).
Let

be a Fuchsian group. If

is geometrically finite, then

is finitely generated.
Suppose

is a fundamental domain of

bounded by finitely many geodesics in

, then there exists a pairing of sides of

by elements of

. In other words, for any geodesic side

, there exists another side

and an element

such that

. In fact, by definition, the

-translates of

cover

without overlap in the interior. So there exists a translate

such that

, which implies that there exists

such that

. Note that

, so this gives a pairing of all sides of

.
Let
be elements of
that pairs the sides of
. We claim that these elements
generates the whole group
. In fact, we can reach the translate
by a chain of neighbouring translates of
, which lies in
.
¡õ
The Siegel Theorem 6then follows from Proposition 10and the following theorem.
If

is a lattice, then every Dirichlet fundamental domain of

in

has only finitely many geodesic sides, hence

is geometrically finite.
We have proved that a geometrically finite Fuchsian group is finitely generates. The following converse is also true for Fuchsian groups.
Let

be a Fuchsian group. If

is finitely generated, then it is geometrically finite.
The hyperbolic plane
is a two dimensional simply connected complete Riemannian manifold of constant curvature -1. For each dimension, we have a unique hyperbolic space
with this property, which is given by
with metric 
If

is a discrete group acting isometrically and properly discontinuously on

, then

is called a
Kleinian group. If

is torsion-free, then

is a hyperbolic manifold.
We can define similar notions of Kleinian groups like Dirichlet fundamental domains (bounded by geodesic hypersurfaces) and geometrically finiteness. However, unlike the case of Fuchsian groups, a Kleinian group
is finitely generated implies neither its Dirichlet fundamental domains are bounded by finitely many geodesic sides nor
is geometrically finite. So Proposition 11 is special for Fuchsian groups.
How to construct Fuchsian groups?
The first geometric method due to Poincaré is the reverse procedure of obtaining a fundamental domain for a group
(see [2] for details).
(Poincaré polygon theorem)
Start with a connected geodesic polygon

in

and a pairing of geodesic sides of

by elements of

, then under a suitable conditions, these side-pairing elements generate a Fuchsian group such that

is a fundamental domain.
The second algebraic method depends on the notion of arithmetic subgroups.
A subgroup

of

is called an
arithmetic subgroup if

is commensurable with

. For an arithmetic subgroup

, we get a finite common cover

for

and

. The examples of arithmetic subgroups of

include congruence subgroup of

and finite index subgroup of

.
Is every arithmetic subgroup of

a congruence subgroup?
This a major problem for arithmetic groups. Fricke-Klein solved this problem and the anser is NO. There are infinitely many subgroups of finite index in

that are not congruence subgroups. However, the answer to the same question for

(

) is YES due to Bass-Milnor-Serre.

is not uniform. Fuchsian groups commensurable with

i.e., arithmetic subgroups, are not uniform as they share a common cover with

. Is there any algebraic method to construct uniform Fuchsian groups?
So far, we have restricted to subgroups of
. In order to construct unifomr Fuchsian groups algebraically, we need to find discrete subgroups that are not commensurable to
. Let
be two lattices, then
are commensurable if and only if they define the same
-structure on
, i.e.,
. So to construct uniform lattices of
, we need a different
-structure on
, namely quaternion division algebras over
. Those Riemann surfaces constructed in this way are called Shimura curves. André Weil proved that these together with the arithmetic subgroups of
exhaust all the arithmetic subgroups of
.
To define the general notion of arithmetic subgroups of
(or a real Lie group), we need to introduce the notion of algebraic groups.
A subgroup

is called an
algebraic group if

is subvariety, i.e., it is a closed subset defined by a set of polynomial equations.

is called to be
defined over 
if the coefficients of the defining polynomials are in

.
Assume
is a linear algebraic group defined over
. Weil showed that
is a Lie group with finitely many connected components.
A subgroup

is called an
arithmetic subgroup if

is commensurable with

.
(Borel, Harish-Chandra)
If

is a semisimple Lie group, then any arithmetic subgroup

of

is a lattice.
Let

be a lattice, then

is not uniform if and only if

contains unipotent elements.
Is every lattice in a semisimple lie group

an arithmetic subgroup with respect to a suitable

-structure on

?
The answer is NO for
by the Teichmuller theory. In almost all other cases, the answer is YES due to Margulis, which relies on a long line of his work on the rigidity of discrete subgroups. More precisely, we have:
(Arithmeticity theorem)
Suppose

is a semisimple Lie group without compact factor of rank

, then any irreducible lattice of

is an arithmetic subgroup.
References
[1]Siegel, C.L., Discontinuous groups, Annals of Mathematics 44 (1943), no.4, 674--689.
[2]Alan F. Beardon, The Geometry of Discrete Groups (Graduate Texts in Mathematics) (v.
91), Springer, 1983.