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

1. Introduction and basic notions
2. General questions on discrete subgroups and discrete transformation groups.
3. 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.
4. Crystallographic groups (Bieberbach theorems)
5. Nilpotent Lie groups and their lattices
6. Solvable Lie groups and their lattices (Mostow rigidity)
7. 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.

Definition 1 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".

Example 1
• 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.
Definition 2 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.
Example 2 An important subclass is given by taking to be a Lie group and be a discrete subgroup.

Here is a less trivial example.

Example 3 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 .
Example 4 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.

Definition 3 Let be a topological group and be a topological space. A topological action of on is a continuous map satisfying
1. for any ,
2. 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.

Definition 4 The space with a -action is called a -space.
Definition 5 The group with an action on is called a transformation group.

We can even put more structures on the spaces and the groups :

1. 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.
2. is a complex manifold (or complex space), is either discrete or a complex Lie group and the action is by holomorphisms.
3. 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.

Definition 6 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.
Proposition 1 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.

1. It occurs naturally (e.g., ).
2. It provides important examples in geometric group theory and combinatorial group theory.
3. Given a manifold with nontrivial fundamental group, then acts on as Deck transformations.
4. 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:

1. Mapping class groups of surfaces with the actions on the Teichmuller spaces.
2. 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.
Definition 7 A discrete subgroup of of a topological group is called a uniform lattice if is compact.
Proposition 2
1. Every discrete subgroup of is of the form where 's are linearly independent.
2. 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 .

Definition 8 A subset of is called a fundamental set for the -action on if meets every -orbit exactly once, namely and for any in .
Remark 1 We always have a fundamental set by the axiom of choice, i.e., by picking one element from each orbit. However, it may not be a Borel subset.
Proposition 3 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 .
Proof The proof goes back to [1]. First let us construct a Borel fundamental set . Since is a discrete subgroup of , there exists an open subset containing the identity such that . Since is a topological group , there exists an open neighbourhood containing such that . Then the -translates of are disjoint. Since is secondly countable, there exists a sequence of such that . Now let one can check that is a Borel fundamental set.

Let be the projection. Then is one-to-one. Define an induced measure on by where is a Borel subset of . We need to show that this induced meausre is independent on the choice of . Suppose are two Borel fundamental sets. Then it suffices to show that for any subset which is invariant under . By the invariance of and the definition of , we have This completes the proof. ¡õ

Definition 9 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 .
Remark 2 In general, lattices are not necessarily uniform (e.g., ).

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 .

Definition 10 The function is called the modular function of . If , the is called unimodular (hence any left invariant Haar measure is also right invariant).
Example 5 Let be a real semisimple Lie group, then is unimodular, since the modular function is homomorphism, is abelian and .
Proposition 4 If admits a lattice, then is unimodular.
Proof 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
Example 6 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.
Question Does every unimodular (e.g., abelian) group admit lattices?
Answer 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.
Remark 3 Any locally compact field that contains as a dense subfield is either or for some prime .

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.

Theorem 1 (Borel) If is a real semisimple Lie group with finitely many connected components, then admits lattices (both uniform and nonuniform).
Remark 4 For a -adic semisimple Lie group, any lattice is uniform.

## 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.)

Definition 11 Let be a set of generators of . Define . Define the word metric for any . This metric is left invariant under , namely .
Remark 5 is a discrete metric space, hence not connected.
Definition 12 The Cayely graph consists of the elements of as vertices. An edge between exists if and only if . Then is connected.
Remark 6 There are many generating sets, but the word metrics defined by them are quasi-isometric in the following sense.
Definition 13 Let , be two generating sets of . Then there exists constants such that for any ,
Remark 7 The Calay graph is a metric space by declaring that every edge has length 1. Then the metric space is quasi-isometric to . is connected but is not simply connected in general.
Example 7 , . Then is the infinite grid on and is not simply connected.
Example 8 , . 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 .

Example 9 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.

Theorem 2 (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.

Theorem 3 (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.

Definition 14 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 .

Definition 15 A domain in (somewhere between closed and open) is called a fundamental domain for in if and the interiors of 's are disjoint (cf. Definition 8).

Usually we require that is not too complicated, e.g., the boundary of is small: . If has a measure invariant under , we usually require that . If is a manifold, we usually require that consists of submanifolds of smaller dimensions.

Definition 16 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.

Theorem 4 is a homeomorphism if and only if is locally finite.
Proof We omit the proof here. ¡õ
Theorem 5 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.
Proof 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.

Example 10 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.
Proposition 5 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.

Definition 17 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.
Proposition 6 Let be a connected topological space. If is a rough fundamental domain (open or closed) and if finite, then is finitely generated.
Proposition 7 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 .

Definition 18 Any discrete subgroup of acting on is called a Fuchsian group.

Fuchsian groups divide into two types:

1. the first kind: if the limit set of in the boundary is the whole boundary.
2. the seconnd kind: if it is not of the first kind.
Proposition 8 If is a lattice, then is of the first kind.
Example 11 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.
Remark 8 The well-known description of the fundamental domain of appeared early in Gauss' Disquisitiones Arithmeticae when he studied the problem of representing integers by quadratic forms. The space of all positive definite quadratic forms corresponds to the space of all positive definite matrices. Note that acts on all positive definite matrices of determinant 1 by transitively and the stabilizer of is , so the space of all positive definite matrices of determinant 1 corresponds to . For , the quadratic forms and take the same set of values on integers . So this problem boils down to finding the simplest quadratic form in each orbit. The reduction process goes by starting with any quadratic form and reducing it to the simplest one.

An important part of Minkowskii's geometry of numbers consists of reduction theory of finding fundamental domains of on or on .

Proposition 9 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.

Lemma 1 (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.

Remark 9 A Lie group is called linear if it can be embedded into some . So a finitely generated subgroup of a linear Lie group admits a finite index torsion-free subgroup . However, if is not linear, this is not necessarily true.
Question When is a discrete subgroup of a Lie group finitely generated?
Theorem 6 (Siegel) If is a lattice of , then is finitely generated.

More generally, we have:

Theorem 7 If is a lattice of a semisimple Lie group , then is finitely generally.
Remark 10 is a semisimple Lie group, but has rather special properties.

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.

Definition 19 A Fuchsian group is called geometrically finite if it admits a fundamental domain that is bounded by finitely many geodesics (called a geodesic polygon).
Proposition 10 Let be a Fuchsian group. If is geometrically finite, then is finitely generated.
Proof 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.

Theorem 8 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.

Proposition 11 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

Definition 20 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.
Remark 11 It turns out that acts isometrically and transitively on and . So Kleinian groups can be also defined as discrete subgroups of .

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.

Question 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).

Theorem 9 (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.

Definition 21 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 .
Question Is every arithmetic subgroup of a congruence subgroup?
Answer 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.
Question 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.

Definition 22 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.

Definition 23 A subgroup is called an arithmetic subgroup if is commensurable with .
Remark 12 A -structure on a Lie group is essentially a linear algebraic group such that .
Theorem 10 (Borel, Harish-Chandra) If is a semisimple Lie group, then any arithmetic subgroup of is a lattice.
Proposition 12 Let be a lattice, then is not uniform if and only if contains unipotent elements.
Remark 13 The same thing holds for lattices in semisimple Lie groups. This is a conjeture of Selberg and it is proved by Kazhdan-Margulis.
Question 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:

Theorem 11 (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.