Marc Burger

Lattices in product of trees

© Publications mathématiques de l’I.H.É.S., 2000, 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/legal.php). 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.

Groups acting on trees : from local to global structure

© Publications mathématiques de l’I.H.É.S., 2000, 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/legal.php). 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.

CAT(-1)-spaces, divergence groups and their commensurators

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and their convex subsets ([B-G-S]), metric trees and piecewise hyperbolic cell complexes ([Mou],[Da],[Hag],[Be 1],[Be 2],[B-Br]). In this paper we establish certain superrigidity results for isometric actions of a group Λ on a CAT(−1)-space in the following two settings: A. The group Λ is a subgroup of a locally compact group G with Γ < Λ < ComGΓ, where Γ < G is a sufficiently large discrete subgroup and ComGΓ = {g ∈ G : g−1Γg and Γ share a subgroup of finite index} is the commensurator of Γ in G. B. The group Λ is an irreducible lattice in G := ∏n α=1Gα(kα), where each Gα is a semisimple algebraic group defined over a local field kα. The issues addressed in this paper are motivated on one hand by earlier work of G.A. Margulis ([Ma]) dealing with the linear representation theory of Λ, where in case A, G is a semisimple group and Γ < G a lattice, and on the other hand by the results of Lubotzky, Mozes and Zimmer ([L-M-Z]) concerning isometric actions of Λ on trees, where Γ < Λ < ComGΓ, G is the group of automorphisms of a regular tree and Γ < G is a lattice. Our approach to establishing superrigidity results is based on ergodic theoretic methods developed by Margulis ([Ma],[Zi 3],[A’C-B]). In this context, the following notion of boundary of a locally compact group Γ will be useful: let B be a standard Borel space on which Γ acts by Borel automorphisms preserving a σ-finite measure class μ.

Finitely presented simple groups and products of trees

Abstract We construct lattices in Aut Tn x Aut Tm which are finitely presented, torsion free, simple groups.

Horocycle flow on geometrically finite surfaces

acts on T S. It is our main goal to determine all N-invariant Radon measures on T1S. Our first remark is that if C is the cone of positive N-invariant Radon measures in the space ’(Tx S) of all Radon measures with the vague topology, then C is the closed convex hull of the union of its extremal generators [B, II No. 2]; moreover it is easily seen that a measure is on an extremal generator of C if and only if it is ergodic. This reduces the problem to the classification of all ergodic measures. To proceed further we consider the following decomposition of T S: Let S be the ideal boundary of D and A c S be the limit set of F. Using the visual map:

On Ulam stability

We study

Constructing irreducible representations of discrete groups

\epsilon

Amenable Groups and Stabilizers of Measures on the Boundary of a Hadamard Manifold

-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional

A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices

\epsilon

Isometric embeddings in bounded cohomology

-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional