Publications
2021

(2021) Annals of Mathematics. 193, 3, p. 837861 Abstract
Let Γ be a lattice in SO0 (n,1). We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2, then Γ is arithmetic. This answers a question of Reid for hyperbolicmanifolds and, independently, McMullen for hyperbolic 3manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics, and our main result also admits a formulation in that language.
2020

(2020) Journal of the European Mathematical Society. 22, 8, p. 25372571 Abstract
A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We prove an analog of this theorem for the torsion part of the homology in all dimensions d not equal 3. Thus the total homology is controlled by the volume. This applies in particular to the classical case of hyperbolic manifolds. In dimension 3 the size of torsion homology cannot be bounded in terms of the volume. As a byproduct, in dimension d >= 4 we give a fairly precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d >= 5 up to homeomorphism. These results are based on an effective simplicial thickthin decomposition which is of independent interest.

(2020) Duke Mathematical Journal. 169, 2, p. 213278 Abstract
We introduce a class of countable groups by some abstract grouptheoretic conditions. This class includes linear groups with finite amenable radical and finitely generated residually finite groups with some nonvanishing l(2)Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Gamma in this class, we determine the general structure of the possible lattice embeddings of Gamma, that is, of all compactly generated, locally compact groups that contain Gamma as a lattice. This leads to a precise description of possible nonuniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

(2020) Compositio Mathematica. 156, 1, p. 158178 Abstract
We prove a superrigidity result for algebraic representations over complete fields of irreducible lattices in products of groups and lattices with dense commensurator groups. We derive criteria for the nonlinearity of such groups.

(2020) Geometriae Dedicata. 1, p. 113127 Abstract
We consider a finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary. We will show this for the Heisenberg group. The main tool will be a discrete version of the isoperimetric inequality.
2019

(2019) Mathematische Zeitschrift. 293, 34, p. 11811199 Abstract
In this paper we prove that, under mild assumptions, a lattice in a product of semisimple Lie group and a totally disconnected locally compact group is, in a certain sense, arithmetic. We do not assume the lattice to be finitely generated or the ambient group to be compactly generated.

(2019) Fundamenta Mathematicae. 246, 3, p. 217255 Abstract
Let G be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a nonArchimedean extension of Mostow's theorem by showing that amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a nonuniform lattice. We show that compactly generated metabelian groups, and more generally nilpotentbynilpotent groups, do have property (M). Finally, we highlight a connection of property (M) with the subtle relation between the analytic notions of strong ergodicity and the spectral gap.
2018

(2018) Journal of the American Mathematical Society. 32, 2, p. 491562 Abstract
Let $ X$ be a locally finite irreducible affine building of dimension $ \geq 2$, and let $ \Gamma \leq \operatorname {Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $ \Gamma $ linear? More generally, when does $ \Gamma $ admit a finitedimensional representation with infinite image over a commutative unital ring? If $ X$ is the BruhatTits building of a simple algebraic group over a local field and if $ \Gamma $ is an arithmetic lattice, then $ \Gamma $ is clearly linear. We prove that if $ X$ is of type $ \widetilde {A}_2$, then the converse holds. In particular, cocompact lattices in exotic $ \widetilde {A}_2$buildings are nonlinear. As an application, we obtain the first infinite family of lattices in exotic $ \widetilde {A}_2$buildings of arbitrarily large thickness, providing a partial answer to a question of W. Kantor from 1986. We also show that if $ X$ is BruhatTits of arbitrary type, then the linearity of $ \Gamma $ implies that $ \Gamma $ is virtually contained in the linear part of the automorphism group of $ X$; in particular, $ \Gamma $ is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic $ \Gamma $space attached to the the building $ X$, which we call the singular Cartan flow.

(2018) Geometriae Dedicata. 196, 1, p. 19 Abstract
M. Gromov has shown that any two finitely generated groups Γ and Λ are quasiisometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions Γ ↷ X↶ Λ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact secondcountable groups.
2017

(2017) Groups Geometry And Dynamics. 11, 2, p. 705738 Abstract
Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the Gaction on spaces of probability measures on algebraic Gvarieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of MargulisZimmer superrigidity phenomenon [2].

(2017) Groups Geometry And Dynamics. 11, 3, p. 10031039 Abstract
We study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We derive various applications, both old and new, including closedness of continuous homomorphisms, nonexistence of weaker topologies, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive ( more generally: WAP) representations.
2016

Equicontinuous actions of semisimple groups(2016) to appear in Geometry Groups and Dymnamics..

Amenable Invariant Random Subgroups(2016) to apear in the Israel Journal of Mathematics..

Almost algebraic actions of algebraic groups and applications to algebraic representations(2016) to appear in Geometry Groups and Dymnamics.

Boundary unitary representations  right angled hyperbolic buildings(2016) to apear in Journal of Modern Dynamics..
2015

Rigidity of group actions on homogeneous spaces III(2015) Duke Math. J. 164 (2015), no. 1, 115155..

On the structure and arithmetic ity of lattice envelopes(2015) Math. Acad. Sci. Paris 353 (2015), no. 5, 409413..
2014

Furstenberg Maps For CAT(0)Targets Of Finite Telescopic Dimension(2014) Submitted, 16 pages.


Algebraic Representations of Ergodic Actions and Super Rigidity(2014) Submitted, 25 pages.

On the cohomology of weaklyalmost periodic group representations(2014) Journal of Topology and Analysis Vol. 6, No. 2, 153–165.

Boundaries, rigidity of representations,and Lyapunov exponents(2014) submitted to the Proceedings of the ICM 2014, 27 pages.
2013

Weak notions of normality and vanishing up to rank in L2cohomology(2013) to appear at the IMRN, 10 pages.
2012

On some geometric representations of GLn(O)(2012) Communications in Algebra Volume 40, Issue 9, 31693191.

A fixed point theorem for L1(2012) Inventiones Math. 189 No. 1, 143148.

Integrable measure equivalence and rigidity of hyperbolic lattices(2012) to appear at the Invent. Math. DOI 10.1007/s0022201204459, 67 pages.

Boundaries Weyl groups, and Superrigidity(2012) Electronic Research Announcements in Mathematical Sciences, 19, 4148..

Rigidity of group actions on homogeneous spaces, III(2012) Duke Math Journal, under revision.
2011

Boundary Unitary Representations – irreduciblity and rigidity(2011) Journal of Modern Dynamics 5, no.1, 4969.

Simple groups without lattices(2011) Bulletin of the London Mathematical Society, 113..
2010

Conformal Actions on Homogeneous Lorentzian Manifolds(2010) Journal of LieTheory 20 no. 3(2010), {469481}..

Effcient subdivision in hyperbolic groups and applications(2010) Groups, Geometry and Dynamics, to appear, 24 pages.
2009

An embedding theorem for automorphism groups of Cartan geometries(2009) GAFA 192, 333355.
2007

Property (T) and rigidity for actions on Banach spaces(2007) Acta Math. 198, no. 1, 57{105}..

Geometric representations of GL(n;R), cellular Hecke algebras and the embedding problem(2007) Pure Appl. Algebra 208, no. 3, 905{922}.
2006

Factor and normal subgroup theorems for lattices in products of groups(2006) Invent. Math. 163 (2006), no. 2, 415{454}..
2004

Thesis: Conformal actions of simple Liegroups on Pseudo Riemannian manifolds(2004)
Supervisor: Amos Nevo
2002

Conformal actions of simple Liegroups on compact pseudo Riemannian manifolds(2002) Journal of Differential Geometry 60, 355387..