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Pure and Applied Mathematics Quarterly Volume 4, Number 1 (Special Issue: In honor of Gregory Margulis, Part 2 of 2 ) 1—69, 2008
A Summary of the Work of Gregory Margulis
Lizhen Ji
- Introduction
Gregory Margulis is a mathematician of great depth and originality. Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems. As Tits said in [Ti1, §5]:
“Margulis has completely or almost completely solved a number of important problems in the theory of discrete subgroups of Lie groups, problems whose roots lie deep in the past and whose relevance goes far beyond that theory itself. It is not exaggerated to say that, on several occasions, he has bewildered the experts by solving questions which appeared to be completely out of reach at the time. He managed that through his mastery of a great variety of techniques used with extraodinary resources of skill and ingenuity. The new and most powerful methods he has invented have already had other important applications besides those for which they were created and, considering their generality, I have no doubt that they will have many more in the future.”
Indeed, in his solution to the Oppenheim conjecture, the approach of using dynamics on homogeneous manifolds to study number theoretic questions has had far-reaching and effective applications (see §40). This is another instance of his ability of combining Lie group theory and ergodic theory and applying them to seemly unrelated fields.^1 As Howe pointed out in a survey titled “A century of Lie theory” [Ho, p. 262] more than 10 years after the above article of Tits:
Received September 14, 2007. (^1) On more than one occasions, Borel mentioned that Margulis was the first person who caused confusion between two Borels in the Borel measure and the Borel subgroups, by using both Lie theory (or rather algebraic group theory) and ergodic theory simultaneously. He also declared that he was not related to the other Borel.
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“The influence between Lie theory and ergodic theory has been mutual. A particularly striking example of this was Margulis’s use of ergodic theory in the proof of his Superrigity Theorem [Zi1], which was then reinterpreted as being a result in ergodic theory by Zimmer. A very recent example of this mutual interaction is the Margulis proof of the Oppenheim Conjecture, followed quickly by Ratner’s broad generalization of the key ergodic-theoretic result underlying his proof [Ra1-6].”
The impact of this important work of Margulis on the Oppenheim conjecture was also confirmed by Borel [Bo3, p. 14]:
“G.A.Margulis ... proved the Oppenheim conjecture... This breakthrough opened the way to an intense activity both on refinements of the Oppenheim conjecture, on the Raghunathan conjecture and a related one by Dani.”
It is also well-known that the Margulis lemma is a basic tool in hyperbolic geometry and more generally geometry of manifolds of nonpositive curvature and convergence and collapsing of manifolds (see §6), and his result on the generalized prime number theorem on lengths of closed geodesics has generated an active theory covering several different fields (see §14). His explicit construction of expander graphs and the solution of a long-standing problem of Banach-Ruziewicz on the uniqueness of finitely additive, rotation invariant measure on the n-sphere Sn^ are two other results of great originality (see §16 and §17).
In order to convey a sense of the width and depth of his work, this article tries to summarize some major results of Margulis. For a very good summary of some earlier results of Margulis, especially his work on arithmeticity and super-rigidity of lattices of higher rank semisimple Lie groups, see the article [Ti1] by Tits, written for the occasion when Margulis was awarded a Fields medal in 1978 in Helsinki.
The current article is divided into topics which are arranged roughly according to the times they were first considered by Margulis or according to the years of publication in MathSciNet.^2
Descriptions are usually brief for many obvious reasons, for example, due to lack of knowledge of the author about many topics covered by the work of Mar- gulis. But, whenever possible and appropriate, we will try to explain a bit of
(^2) Another method to organize this artcle, which might be better, is to group papers according
to the topics under discussion. We have tried to do this to certain extent by putting several papers into one section under a common title, for example, the sections on arithmeticity and super-rigidity of higher rank lattices summarize many papers of Margulis. But the arrangement according to the publication dates in this article might give a historic sense to the work of Margulis. To help organize this paper and orient the reader, we have inserted a detailed table of contents. (^3) The paper [67] is not listed in MathSciNet for some reason.
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- conjectures 11. Normal subgroups of groups of rational points and Margulis-Platonov
- Characterization of arithmetic groups
- Further developments on rigidity of lattices
- measure 14. Counting lengths of geodesics and volume entropy, and Margulis
- Long time behavior of unipotent group orbits
- Expander graphs and work in combinatorics
- Invariant measures on spheres and Rn
- Strong approximation of algebraic groups
- Maximal subgroups of lattices and linear groups
- Decomposition of lattice subgroups into amalgams
- conjecture 21. Actions of affine transformations, Milnor conjecture, Auslander
- Absence of invariant analytic hypersurfaces.
- Proportionality of covolume of lattices
- The Oppenheim conjecture on values of quadratic forms
- Quantitative version of the Oppenheim conjecture
- Oseledets multiplicative ergodic theorem and Lyapunov exponents
- Proximal linear maps
- QC-maps and Carnot-Carath´eodory spaces
- Bounded orbits and unbounded orbits of flows
- Compact quotients of homogeneous spaces
- Metric Diophantine approximation and Khintchine-type theorems
- Hyperbolic Penrose tile
- Wiener ergodicity for semisimple Lie groups
- Finite coverings of hyperbolic manifolds
- A generalized Tits alternative
- Rigidity of actions of Lie groups and lattices
- Random walks on homogeneous spaces and reduction theory
- Coarse metrics on reductive groups
A Summary of the Work of Gregory Margulis 5
- Effective ergodic theory 54
- Problem lists by Margulis 55 References to the publication of Margulis 58 References to papers by others 62
- Results proved during the undergraduate years
Margulis wrote three papers [66] [67] [68] when he was an undergraduate stu- dent at the Moscow State University. Even though it was not unusual for un- dergraduate students of Moscow State University to do research, what he has achieved was very significant. In fact, the paper [68] was reported by Borel [Bo2] at the famed Bourbaki seminar.
He wrote his first paper [66] when he was a third year undergraduate student and attended Dynkin’s seminar. At that time, structure of positive harmonic functions was one of the problems studied by Dynkin.
Starting at the end of the third year, Sinai became his advisor. Margulis’ second paper [67] proved exponential growth of the fundamental group of a 3- dimensional manifold admitting Anosov flows, and it was published as an appen- dix to a survey by Anosov and Sinai [AS]. Participation in Sinai’s seminar and discussions with him and his other students gave Margulis valuable experience in ergodic theory and dynamical systems, and it played a very important role in his mathematical career. In fact, his thesis is titled On some aspects of the theory of Anosov flows [04-1], which includes the counting of closed geodesics and lattice points in terms of the entropy and a construction of an important measure, now called the Margulis measure (or Bowen-Margulis measure) (see §14). Ergodic theory and dynamical systems are crucial in his celebrated later works such as arithmeticity and super-rigidity of irreducible lattices in higher rank semisimple Lie groups and the solution of the Oppenheim conjecture etc.
At the end of his undergraduate study, Margulis wrote a well-known paper [68] with his classmate D.Kazhdan, which solved several outstanding problems, including a conjecture of Selberg on the existence of unipotent elements in non- uniform lattices. The well-known Margulis Lemma was also motivated by a result proved in this paper (see §6).
After this paper [68], Margulis became very much involved in the theory of discrete subgroups of Lie groups. In particular he started to work on the problem of arithmeticity of nonuniform lattices. He succeeded after several years by using the approach based on the study of unipotent elements in a nonuniform lattice and on a nondivergence result for actions of unipotent groups on the space of lattices.
A Summary of the Work of Gregory Margulis 7
of exponential growth if the number of elements of Γ in a metric ball of radius R with respect to dS grows exponentially in R.
Though the word metric dS and hence the exact sizes of metric balls depend on the choice of the generating set S, the notion of exponential growth of Γ does not depend on the choice, and the exponential growth rate is an important large scale invariant of the group Γ.
The main result of the paper [67] says that if a three dimensional manifold M admits an Anosov flow, then the fundamental group of M has exponential growth.
This puts an algebraic topological restriction on three dimensional manifolds which admit Anosov flows.
It might be helpful to understand this result in the following context. It was known (see [AS]) that if M is a compact Riemannian manifold with strictly negative sectional curvature, then the geodesic flow of M is an Anosov flow.
On the other hand, shortly after the paper [67], Milnor [Mi] proved a well- known result that if M is a compact Riemannian manifold with strictly negative sectional curvature, then the fundamental group of M has exponential growth.
- Volume of locally symmetric spaces and existence of unipotent elements
If the volume of a Riemannian manifold (or orbifold) is finite, then it provides an important basic geometric invariant of the manifold. It is a multiplicative in the following sense. Let X is a Riemannian manifold, and Γ 1 , Γ 2 two discrete groups acting isometrically on X with finite volume quotients Γ 1 \X and Γ 2 \X. Assume that Γ 1 ⊂ Γ 2. Then
vol(Γ 1 \X)/vol(Γ 2 \X) = [Γ 2 : Γ 1 ].
This implies that if Γ 1 is small, the volume vol(Γ 1 \X) is big.
If X is a Riemannian symmetric space of noncompact type and Γ is a discrete isometry group with vol(Γ\X) < +∞, i.e., Γ is a lattice, then in many cases, it is easy to construct subgroups Γ′^ with arbitrary large index [Γ : Γ′]. In other words, there is no finite upper bound on volumes of locally symmetric spaces Γ\X modeled on each given symmetric space X.
The first striking result in the joint paper of Margulis with Kazhdan [68] is that there is a uniform positive lower bound for the volumes of all quotients Γ\X, which only depends on the invariant Riemannian metric on X.
When X is equal to the Poincare upper half plane H^2 = {x+iy | x ∈ R, y > 0 }, this result was known before and is classical (see [Si] [Bea, Chap. 10]).
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This result on the lower bound is a consequence of the following result: Let G be a connected linear semisimple Lie group without compact factor. Let ρ( , ) be a right invariant metric on G. There exist a neighborhood V˜ 1 of e, and constants C 1 > 0 and c > 1 such that for any discrete subgroup Γ of G, one can find some g ∈ G satisfying the two following conditions:
(1) ρ(e, g) ≤ C 1 , (2) for any γ ∈ Γ ∩ V˜ 1 , γ 6 = e, the following inequality holds:
ρ(e, gγg−^1 ) ≥ cρ(e, γ).
The proof of this result is based on the following result: For every connected Lie group G, there exists a neighborhood U of the identity element such that for every discrete subgroup Γ in G, the elements in the intersection Γ ∩ U generate a nilpotent subgroup.
This latter result was the starting point of the Margulis lemma (see §6) and also used to prove the second major result of [68]. In fact, it was used by Kazhdan and Margulis [68] to prove a conjecture of Selberg [Se2, p. 180]: If G is a connected linear semisimple Lie group, and Γ ⊂ G is a non-uniform lattice, i.e., Γ\G has finite measure with respect to any Haar measure of G but is not compact, then Γ contains a nontrivial unipotent element.
When G = SL(2, R), this conjecture was known earlier and unipotent ele- ments correspond (or fix) the cuspidal (or infinite, or ideal) points of a Dirichlet fundamental domain for the Γ-action on H^2. This conjecture is also known for arithmetic subgroups; in fact, it was the content of a conjecture of Godement for arithmetic subgroups, proved independently by Borel and Harish-Chandra [BoHC], and Mostow and Tamagawa [MoT]. (Non-uniform arithmetic subgroups contain many unipotent elements. Their intersection with Q-parabolic subgroups contains lattices of the real locus of the unipotent radical of the parabolic sub- groups.)
Among all discrete groups Γ of G, lattices are particularly important for var- ious reasons. First, arithmetic subgroups are lattices. Second, for the study of automorphic functions, this assumption is often needed (see [Se1, p. 101]).
The presence of unipotent elements in non-uniform lattices has played an im- portant role in the study of rigidity and arithmeticity of Γ. See [Se1, §7.4] for one of the original motivations. Another motivation is that to study the Selberg trace formula, a precise description of the cusps (i.e., noncompact parts) of a good fundamental domain, for example a Dirichlet fundamental domain, is im- portant [Se2]. If Γ is either an arithmetic subgroup or a lattice acting on H^2 , the cusps are described by unipotent elements in Γ, and the above conjecture is the first step towards such goals. As Borel explained in [Bo3, p. 8], “This latter statement [on existence of unipotent elements] was also not unexpected, but it was
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constant ε > 0 depending only on n such that if V n^ is a compact Riemannian manifold with sectional curvature strictly bounded between 0 and − 1 , and two elements α, β ∈ π 1 (V, v 0 ) can be represented by loops of length less than or equal to ε, then there is a natural number m such that αm^ and βm^ generate a nilpotent subgroup of π 1 (V, v 0 ).
In [Gr1], Gromov called this result Margulis Lemma. There are also various other versions of Margulis Lemma based on similar ideas. For applications to group actions on Riemannian manifolds, one version of the Margulis Lemma can be stated as follows [BaGS, p. 101, p. 107]: Let X be a simply connected and nonpositively curved complete Riemannian manifold with sectional curvarure bounded from below by − 1 and Γ is a discrete group acting isometrically and properly on X. For any μ > 0 and x ∈ X, define a subgroup
Γμ,x = 〈{γ ∈ Γ | d(x, γx) ≤ μ}〉.
Then there exist a constant μ 0 and an integer I 0 only depending on n such that for every μ ≤ μ 0 and x ∈ X, the subgroup Γμ,x is virtually nilpotent; in fact, it contains a nilpotent subgroup of index bounded uniformly by I 0.
As a consequence, if the sectional curvature of X is strictly negative, then the thin part (Γ\X)≤μ of such a quotient Γ\X is basically either a cusp or a neck, when the μ-thin part is by definition the set of points ¯x ∈ Γ\X where the injectivity radius is less than μ. The reason is that for every point x ∈ X, the injectivity radius of the image ¯x of x in Γ\X is equal to infγ∈Γ d(x, γx), and its small neighborhoods are described by quotients of nilpotent subgroups (see [Fu, Theorem 4.6, p. 206]).
As mentioned earlier, another type of applications of the Margulis Lemma con- cern convergence and collapsing of families of Riemannian manifolds. They were initiated by Gromov and carried out extensively by Cheeger, Fukaya and Gro- mov et al. Basically, the presence of nilpotent structure is related to collapsing of Riemannian manifolds to lower dimensional ones.^6 See [Fu] for a survey on generalizations of the Margulis Lemma in geometry and applications on conver- gence of Riemannian manifolds under bounds on curvatures and diameters, or other geometric conditions.
- Arithmeticity of lattices
An important consequence of the uniformization theorem for Riemann surfaces is that there are in general positive dimensional families of non-conjugate lattices in SL(2, R) (or rather P SL(2, R)). For example, if Γ ⊂ SL(2, R) is a lattice
(^6) This is closely related to the horospherical decomposition of symmetric spaces with respect to parabolic subgroups, and the collapsing of the Borel-Serre compactification of arithmetic locally symmetric spaces to the reductive Borel-Serre compactification.
A Summary of the Work of Gregory Margulis 11
such that Γ\H^2 is a closed surface of genus g ≥ 2, then Γ belongs to a (6g − 6)- dimensional family of non-conjugate lattices, or equivalently their corresponding hyperbolic surfaces are not isometric.
This deformation, or equivalently non-rigidity, of such lattices implies immedi- ately that most of lattices in SL(2, R) are not arithmetic subgroups, since there are only countably many arithmetic subgroups.
On the other hand, for other semisimple linear Lie groups, it is difficult to construct non-arithmetic lattices. One reason is that there is no uniformization theorem for higher dimensional manifolds which we can use as above. (To see lack of any direct generalization of the uniformization theorem in one complex variable, note that all simply connected domains in C are biholomorphic to each other. But this is completely false for Cn, n ≥ 2; in fact, there are uncountably infinitely many of non-biholomorphic simply connected (or even contractible) domains in Cn.)
After success with several special cases, Selberg made the following conjectures [Se1, p. 119] [Se3, §5]:
(1) Let G be a linear semisimple Lie group, and Γ ⊂ G an irreducible lattice. Then Γ can be deformed into a group whose matrix representation has entries from some number field, and the denominators of these entries are uniformly bounded. (2) If the real rank of G is at least 2 and Γ is a non-uniform lattice, then Γ is an arithmetic subgroup of G with respect to a suitable Q-structure on G.
The first conjecture implies that if a lattice Γ is (locally) rigid, i.e., does not admit nontrivial deformations, then Γ has a matrix realization with entries given by algebraic numbers [Se3, p. 159, p. 164].
Earlier, the assumption that the rank of G being at least 2 was not made for Conjecture (2), but only the group SL(2, R) and its lattices were excluded. After some non-arithmetic lattices acting on the real hyperbolic spaces Hn, n = 3, 4, were discovered, the conjecture of Selberg was modified (see [PS1, p.3]).
The importance of these conjectures, in particular the second one, is that the reduction theory for arithmetic subgroups can be used to understand the structure of neighborhoods at infinity of Γ\G (or Γ\X), which is fundamental to the theory of automorphic forms for Γ. Indeed, as Selberg pointed out in [Se2, p. 180]:
“The most serious obstacle to carrying out of the idea sketched above is that, with the exception of the case of a hyperbolic plane (and of course for the product of a hyperbolic plane and euclidean or compact symmetric spaces) it is not known
A Summary of the Work of Gregory Margulis 13
proof of (ii) is based on the phenomenon of polynomial divergence (applied to actions of {u(t)} on the exterior products of Rn) and on a rather sophisticated induction argument related to the geometry of numbers. The technique used in the proof of (ii) was refined and quantified first by S.G.Dani in [Da1] (in the proof of the quantitative recurrence of unipotent orbits to compact sets) and later in a joint work of Margulis with Kleinbock [98-2] and by Eskin, Mozes and Shah in their work on counting of integral points on homogeneous varieties [EMS].
Margulis’ work on the above result (ii) ([71] or [75-3]) provided him with some intuition which played an important role in his much later work on unipotent flows (this intuition was based on the understanding of the importance of the polynomial divergence in the theory of unipotent flows).
The detailed proof of the main result announced in [69-3] was given in [74-2] and [75-2] (actually the paper [75-2] was submitted for publication much earlier than [74-2]). Another important ingredient in the proof is a construction from representation theory. This construction was used later by Oh [Oh4] in her work on discrete groups generated by lattices in horospherical subgroups. It should be mentioned that Raghunathan obtained an independent proof of the main result announced in [69-3].^7
In [75-2] Margulis also proved the strong rigidity for non-uniform lattices in higher rank groups using a totally different method (see the next section for more detail).
Reduction of the proof of arithmeticity of non-uniform lattices in higher rank groups to the main result announced in [69-3] was started in [74-2] and completed in [75-1].
In [PS1, p. 3] (see also [PS2, p. 189]), Piatetski-Shapiro conjectured that irreducible uniform lattices Γ in G of rank at least 2 are also arithmetic. This is a big step from the conjecture of Selberg, since unipotent elements of non-uniform lattices Γ play crucial roles in the proof of arithmeticity of Γ in the above papers. He defined arithmetic subgroups of Lie groups which are not linear Lie groups. Briefly, a lattice Γ in a semisimple Lie group G is called arithmetic if there is a linear semisimple Lie group G′^ defined over Q and a surjective homomorphism ϕ : G′(R) → G with a compact kernel and an arithmetic subgroup Γ′^ of G′(Q) such that ϕ(Γ′) is commensurable with Γ.
In [PS1, pp. 4-5], Piatetski-Shapiro also introduced arithmetic subgroups of linear p-adic Lie groups and products of both real Lie groups and linear p-adic Lie groups, and formulated questions on rigidity and arithmeticity of lattices in such locally compact groups.
(^7) See the article [Bo3, pp. 7-10, §10] for a very good summary of Raghunathan’s contributions to strong rigidity and arithmeticity of lattices and a history of related results.
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The paper [74-1] [75-4] outlined a proof of this conjecture of arithmeticity of Piatetski-Shapiro on uniform irreducible lattices of semisimple Lie groups G when the rank of the symmetric space associated with G is at least two. The full details appeared in [77-2] [84-1].
A comprehensive discussion of these and other results is in the book [91-1]. Another account together with some generalizations on rigidity of group actions is given in the book [Zi1]. For a vivid description of the proof of arithmeticity theorem, see [Ti1] and also [Bo3, §7, pp. 7-10].
Besides non-arithmetic lattices acting on the real hyperbolic spaces [GPS], there are also non-arithmetic lattices acting on the complex hyperbolic spaces of low dimensions [DeM]. On the other hand, lattices acting on the two types of other rank one symmetric spaces of noncompact type are arithmetic [Co] [GP]. Non-arithmetic lattices in rank 1 p-adic semi-simple Lie groups always exist [Lu1].
In the above discussions, we concentrated on lattices in semisimple Lie groups. In fact, Margulis proved that if S is a finite set of primes and Gp is a semisimple p- adic Lie group with trivial center, any irreducible lattice in the product
p∈S Gp is arithmetic if the sum of the ranks of Gp over p ∈ S is at least 2 (see [91-1]).
In his lecture at Yale in 1992, Selberg suggested that some of his work should imply that any discrete subgroup containing lattices from the opposite horospher- ical subgroups of a product of SL(2, R) is an arithmetic subgroup (or close to it). This is stronger than the arithmeticity of such discrete subgroups, since they may not be a priori lattices. Margulis raised this question for discrete subgroups Γ of general higher rank semisimple Lie groups G such that Γ contain lattices from opposite horospherical subgroups and intersect any normal subgroups trivially, i.e., asked whether such groups Γ are arithmetic under the above assumption. If G is a split higher rank simple Lie group and not equal to SL(3, R), then this propblem of Selberg and Margulis was solved positively by Oh [Oh4]. As men- tioned above, a construction from representation theory by Margulis in [74-2] and [75-2] was used crucially here in [Oh4].
- Local and strong rigidity of lattices and locally symmetric spaces
Arithmeticity of lattices is closely related to rigidity of lattices. In fact, one of the main motivations for Selberg is to use deformation rigidity to prove arith- meticity of lattices. On the other hand, rigidity of lattices is also natural from the point of view of geometry of locally symmetric spaces.
Besides [Se2] [Se3], the first papers on rigidity of lattices and locally symmetric spaces include [Ca] [CaV] [We]. The motivation of [CaV] is to apply the general
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This formulation is closely related to the Borel conjecture in geometric topol- ogy: If two closed aspherical manifolds M and N are homotopic, then they are homeomorphic.
By definition, M is an aspherical manifold if πi(M ) = { 1 } for all i ≥ 2. Locally symmetric spaces of noncompact type are aspherical manifolds, and the Mostow strong rigidity settles this conjecture when both M and N are locally symmetric spaces.
The Borel conjecture is related to classification of manifolds and was directly motivated by an earlier result of Mostow on solvmanifolds. Though many im- portant cases have been proved, it is not completely solved yet and is a major unsolved problem in geometric topology. See [Fa] and references there (also [Ji1] for more references).
In the above version of the Mostow strong rigidity, the lattices Γ are assumed to be co-compact. A natural problem is to remove this condition. This was achieved by Margulis in [75-2] when the rank of G is at least two, and the proof was totally different from the remarkable proof of Mostow of strong rigidity of uniform lattices [Mos1] and was more in the style of the theory of automorphisms of classical groups. The rank one case was proved by Prasad [Pr1] by generalizing the method of [Mos1]. (See [Kos] and the introduction of [Pr1] for the history of the Mostow strong rigidity for non-uniform lattices). Therefore the strong rigidity holds for all irreducible lattices in semisimple Lie groups with trivial centers which are not locally equal to SL(2, R) and have no compact factors.
In an earlier paper [Mos2, §12], Mostow proved the following result: Let G be the group of isometries of a real hyperbolic space X of dimension n ≥ 3 , and Γ, Γ′ be two lattice subgroups of G. Let θ : Γ → Γ′^ be an isomorphism and ϕ : X → X a quasi-conformal homeomorphism such that
ϕ(γx) = θ(γ)ϕ(x)
for all γ ∈ Γ and x ∈ X. Then θ extends to an inner automorphism of G. In particular, Γ\X and Γ′\X are isometric.
A corollary of this result is the following: Assume that M and N are two com- pact Riemannian n-manifolds with constant negative curvature and n > 2. If M and N are diffeomorphic, then they conformally equivalent and hence isometric.
In [Mos2], the above assumption that M and N are diffeomorphic is impor- tant. In [70-1], Margulis strengthened this latter result and proved the following result: if two compact hyperbolic manifolds M and N of dimension at least 3 are homotopic, then they are isometric.
The methods and ideas of this two pages long paper [70-1] are much more im- portant, though many people might not be aware of this fact. As it is well-known
A Summary of the Work of Gregory Margulis 17
to many people, the important notion of quasi-isometries and the method of push- ing things, for example, an equivariant homotopy equivalence, to the boundary at infinity of symmetric spaces of noncompact type are used crucially in the famous proof of the Mostow strong rigidity by Mostow in [Mos1], and have motivated a lot of recent developments in various subjects, in particular in geometric group theory. It is also well-known that Gromov revolutionized the study of finitely generated groups by putting on word metrics on them and approximating the universal covering of a compact manifold by the fundamental group with a word metric and using the idea of quasi-isometries. In this short paper [70-1], these important notions and methods were introduced for real hyperbolic spaces and discrete groups acting on them independently of [Mos1] (the methods used in [Mos1] were outlined in an ICM talk by Mostow in 1970 [Mos3]) and ahead of Gromov’s work, for example, the famous papers [Gr2] [Gr3].
- Super-rigidity of lattices
There have been a lot of extension and generalizations of the Mostow strong rigidity. One of the most significant is the super-rigidity of lattices by Margulis [75-4] [84-1] [91-1]. Many results were announced in [75-4] and some proofs were also outlined. The paper [84-1] was written in mid-1970s and appeared as an appendix to the Russian translation of [Rag1].
The history and motivations of the superrigity of irreducible lattices of higher rank semisimple Lie groups canbe best explained by a recollection of Margulis:
“In the late sixties I learnt about Mostow’s fundamental work on strong rigid- ity. Thinking about it, I at some point realized that it would be possible to prove the arithmeticity of uniform higher rank lattices if one could prove a statement which is now called superrigidity. I believe (and this was confirmed by Mostow) that the superrigidity was a new phenomenon which had not been discovered be- fore. The first proof of superrigidity was based on combination of methods from ergodic theory and algebraic group theory, and one of the important ingredients was Oseledec multiplicative ergodic theorem.^8
One of the consequences of superrigidity is the classification (in a certain sense) of finite-dimensional representations of higher rank lattices and S-arithmetic groups. The reduction of this classification to superrigidity is based on the argument which is dual (again in a certain sense) to the famous “unitary trick” of H.Weyl. Let me be a little bit more precise. Roughly speaking, the superrigidity describes rep- resentations ρ of a lattice Γ with non-compact image. But in general this image
(^8) Borel in his paper [Bo3, pp. 10, §7] wrote that “The work of Margulis [on arithmeticity of irreducible lattices of higher rank semisimple Lie groups] was based on a new principle, soon christened “superrigidity” by Mostow...”
A Summary of the Work of Gregory Margulis 19
A more important application of the Margulis super-rigidity is to prove arith- meticity of such lattices Γ. In fact, as Selberg observed in [Se3, p. 159, p. 164], if a lattice Γ in a linear semisimple Lie group G is locally rigid, then under a suitable conjugation by elements of G, the matrix entries of elements of Γ belong to a number field, hence belong to Q when embedded into a bigger general linear group by applying the functor of restriction of scalars. By the local rigidity of Weil [We] (or applying the stronger Mostow-Margulis-Prasad strong rigidity), an irreducible lattice Γ in a connected semisimple Lie group G with trivial center (hence linear) and no compact factors is locally rigid and thus admits a realiza- tion by matrices with entries contained in Q. But this does not imply that Γ is an arithmetic subgroup yet, though it is suggestive (see [PS2, p. 189, paragraph 5]). The remaining difficult step is to show that the denominators of the entries for all the elements of Γ are uniformly bounded. This is equivalent to the fact that they are bounded in the p-adic topology, which is settled by the Margulis super-rigidity when k is taken to be a locally compact, totally disconnected field such as Qp. (See [Zi, p. 120-121] [91-1, Chap. IX, §2] for details.)
Since there are non-arithmetic lattices acting on the real hyperbolic spaces of all dimensions and complex hyperbolic spaces of dimensions less than or equal to 3, the Margulis super-rigidity can not be true for these lattices in P SO(n, 1) and P SU (n, 1). For the other two rank 1 symmetric spaces, the quaternionic hyperbolic spaces and the hyperbolic Cayley plane, lattices acting on them do satisfy super-rigidity properties. Over Archimedean fields R and C, it was proved by Corlette [Co], and the super-rigidity over p-adic fields was proved by Gromov and Schoen [GS]. In both cases, harmonic maps and Bochner type arguments are used. Other references on geometric super-rigidity include [JoY1-2] [MSY], where the super-rigidity is proved for uniform lattices and some non-uniform lattices.
See [Ji1-2] for more references on and some expositions of differential geometric proofs of Mostow-Margulis rigidity of locally symmetric spaces and generaliza- tions to K¨ahler manifolds, in particular, the Siu-Yau method of proving rigidity results using harmonic maps.
- Normal subgroups of lattices
As mentioned before, one of the motivations for Selberg to make the conjecture on arithmeticity of lattices is to get a better understanding of the fundamental domains of such lattices on symmetric spaces, in order to develop the spectral theory of automorphic forms and the Selberg trace formula. Another is to obtain group structures of such lattices.
The results of Margulis super-rigidity and arithmeticity of lattices and methods developed to prove them have many other striking applications on understanding intrinsic structures of such lattices.
20 Lizhen Ji
Let Γ be an irreducible lattice in a connected semisimple Lie group G with finite center and no compact factors. Assume that the rank of G is at least 2. Then a known theorem of Kazhdan implies that the quotient Γ/[Γ, Γ] is finite (see [Zi, Corollary 7.1.10, p. 132]). Clearly, [Γ, Γ] is an infinite normal subgroup of Γ. Motivated by this and a result of Vaserstein [Vas], Margulis asked [75-4, p. Problem 3] if any infinite normal subgroup of Γ has finite index. This is completely settled by him in [78-1] [78-2] [79-1]. Briefly, the results can be summarized as follows: Let G be a connected semisimple Lie group with no compact factors and finite center and of rank at least 2, and Γ ⊂ G an irreducible lattice. For every normal subgroup N ⊂ Γ, the quotient N \Γ has property T of Kazhdan. If N \Γ is amenable, then it is finite, i.e., N is of finite index; if N \Γ is not amenable, then N is contained in the center Z(Γ) of Γ and hence is finite.
Roughly, this result says that such an irreducible lattice is simple modulo finite groups. If the irreducible lattice can be realized as an arithmetic subgroup of a linear semisimple algebraic group G defined over Q, then G is almost simple over Q. This brings out another close relation (or similarity) between the algebraic group G and its lattice subgroup Γ. It also reminds one of the famous construction of finite simple groups of Lie type from Chevalley groups.
The normal subgroup theorem is extremely elegant and can stand alone as a significant result. Naturally, there have been many applications of it. According to Margulis,
“One of main applications of the normal subgroup theorem is the statement about the finiteness of index of every non-central normal subgroup of a S-arithmetic group (under certain conditions). When S is finite, this is a direct consequence of the normal subgroup theorem (in combination with the Borel-Harish-Chandra theorem). To go from finite S to infinite S, one has to use the strong approxi- mation for semi-simple groups over global fields. This reduction was first noticed by Gopal Prasad.”
Another unexpected deep result related to ideas of Margulis and this normal subgroup theorem is the construction by Burger and Mozes [BuM1] of new ex- amples of infinite simple groups satisfying the following conditions: (1) finitely presented, (2) torsion-free, (3) equal to fundamental groups of finite, locally CAT(0)-complexes, (4) of cohomological dimension 2, (5) biautomatic and (6) equal to the free amalgams of two isomorphic free groups over a common finite index subgroup.
These new groups are very different from all the known examples of finitely presented simple groups and answer positively several questions posted by various people. They arise as lattices in products of automorphism groups of regular trees, which are new classes of locally compact groups, for example, are not Lie groups. Briefly, they are given by certain lattices of Aut T 1 × Aut T 2 whose projections
