Robert J. Zimmer

Subrelations of ergodic equivalence relations

We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II 1 . Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II 1 . We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II 1 -action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

Lattices in Semisimple Groups and Invariant Geometric Structures on Compact Manifolds

The aim of this paper is to describe a geometrization of the Mostow-Margulis theory of rigidity and representations of discrete subgroups of semisimple groups. More precisely let H be a connected semisimple Lie group and Γ ⊂ Halattice subgroup. Let G be another Lie group. The general problem considered by Mostow and Margulis was to study the homomorphisms π: Γ → G. The Mostow rigidity theorem of course deals with the case in which G is semisimple and π(Γ) is a lattice in G, and the Margulis superrigidity theorem deals with the more general case in which π(Γ) is merely assumed to be Zariski dense in G. While we shall recall the precise results later, we simply remark here that the ultimate conclusion is that one can essentially understand all such homomorphisms. Roughly speaking, π either extends to a smooth homomorphism of H or π(Γ) has compact closure (in which case one also has information on the closure), or is a combination of these cases. A geometric generalization of the notion of a homomorphism Γ → G is of course the notion of an action of Γ by automorphisms of a principal G-bundle.

Ergodic theory, semisimple lie groups, and foliations by manifolds of negative curvature

© Publications mathématiques de l’I.H.É.S., 1982, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

VARIANTS OF KAZHDAN'S PROPERTY FOR SUBGROUPS OF SEMISIMPLE GROUPS

Some variants of Kazhdan’s property (T) for discrete groups are presented. It is shown that some groups (e.g. SLn (Q),n≧3) which do not have property (T) still have some of these weaker properties. Applications to cohomology and infinitesimal rigidity for certain actions on manifolds are derived.

Representations of fundamental groups of manifolds with a semisimple transformation group

group ? 1 (M) , and the Lie groups that can act on M. More precisely, let G be a connected semisimple Lie group of higher real rank, and suppose G acts continuously on a (topological) manifold M, preserving a finite measure. The main theme of this paper is that the representation theory of 7r I(M) in low dimensions is to a large extent controlled by that of G (the latter of course being well understood). In particular, under natural hypotheses (e.g., that the action of G on M is engaging, i.e., there is no loss of ergodicity in passing to finite covers; see Definition 3.1 below), we prove that if G has no nontrivial

Fundamental groups of negatively curved manifolds and actions of semisimple groups

THIS PAPFR is part of the investigation of the fundamental groups of manifolds on which a non-compact semisimple Lie group may act. Earlier work in this direction has dealt largely with the linear representation theory of the fundumcntal group. In this paper we consider homomorphisms into isomctry groups of negatively curved manifolds. and in particular the question as to when a scmisimplc Lie group may act on a manifold whose fundamcnt;ll group is isomorphic to th;tt of a ncgativcly curved manifold.

A structure theorem for actions of semisimple Lie groups

We consider a connected semisimple Lie group G with finite center, an admissible probability measure , on G, and an ergodic (G, ,u)-space (X, v). We first note (Lemma 0.1) that (X, v) has a unique maximal projective factor of the form (G/Q, vo): where Q is a parabolic subgroup of G, and then prove: 1. Theorem 1. If every noncompact simple factor of G has real rank at least two, then the maximal projective factor is nontrivial, unless v is a G

Ergodic theory, group representations, and rigidity

In these lectures we discuss some topics concerning the relationship of ergodic theory, representation theory, and the structure of Lie groups and their discrete subgroups.

Rigidity of Furstenberg entropy for semisimple Lie group actions

Abstract We consider the action of a semi-simple Lie group G on a compact manifold (and more generally a Borel space) X , with a measure ν stationary under a probability measure μ on G . We first establish some properties of the fundamental invariant associated with a (G,μ) -space (X,ν) , namely the Furstenberg entropy [? ] , given by h μ (X,ν)= ∫ G ∫ X − log dg −1 ν dν (x) dν(x) dμ(g). We then prove that when (X,ν) is a P -mixing (G,μ) -space [? ] , and R -rank (G)=r≥2 , the value of the Furstenberg entropy must coincide with one of the 2r values hμ(G/Q,ν0) , where Q⊂G is a parabolic subgroup. We also construct counterexamples to show that this conclusion fails for both non- P -mixing actions and actions of groups with R -rank 1 . We also characterize amenable actions with a stationary measure as the actions having the maximal possible value of the Furstenberg entropy. We give applications to geometric rigidity for actions with low Furstenberg entropy, to orbit equivalence and to the cohomology of actions with stationary measure.

Volume preserving actions of lattices in semisimple groups on compact manifolds

© Publications mathématiques de l’I.H.É.S., 1984, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.