Shahar Mozes

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.

Tilings, substitution systems and dynamical systems generated by them

The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for one-dimensional tilings the dynamical system always contains periodic points.

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.

The word and Riemannian metrics on lattices of semisimple groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.

Divergence in lattices in semisimple Lie groups and graphs of groups

Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance 3, S is a finite set of valuations of a number field K including all infinite valuations, and O s is the corresponding ring of S-integers.

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.

On the space of ergodic invariant measures of unipotent flows

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μ n } is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G .

Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus

Let Γ be a semigroup of d × d nonsingular integer matrices, and consider the action of Γ on the torus T. We assume throughout that the action is strongly irreducible: there is no subtorus invariant under a finite index subsemigroup of Γ. The strong irreducibility assumption in particular implies that Γ acts ergodically on T (equipped with the Lebesgue measure m). Therefore the Γ-orbit of Lebesgue almost every x ∈ T is dense and in an appropriate sense even becomes equidistributed. However, when Γ is cyclic, there is a set of full Hausdorff dimension of exceptional points x for which Γ.x fails to be dense. When Γ is bigger, the distribution of individual Γ-orbits can be expected to be much more restrictive. An important result in this direction is due to Furstenberg, who showed for d = 1 (in which case Γ 1 an abelian semigroup never satisfies condition (Γ-1); indeed, the group generated by a semigroup satisfying (Γ-1) is nonamenable. Assumption (Γ-2) is a technical condition which is in particular satisfied when Γ is a Zariski dense semigroup of SLd(Z) [16].

NON-DIVERGENCE OF TRANSLATES OF CERTAIN ALGEBRAIC MEASURES

Abstract. Let G and