Publications
2019

(2019). Lattices in amenable groups. Fundamenta Mathematicae. 246:(3)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). ON THE LINEARITY OF LATTICES IN AFFINE BUILDINGS AND ERGODICITY OF THE SINGULAR CARTAN FLOW. Journal of the American Mathematical Society. 32:(2)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). Coarse equivalence and topological couplings of locally compact groups. Geometriae Dedicata. 196:(1)19. Abstract
M. Gromov has shown that any two finitely generated groups and are quasiisometric if and only if they admit a topological coupling, i. e., a commuting pair of proper continuous cocompact actions X on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact secondcountable groups.
2017

(2017). Almost algebraic actions of algebraic groups and applications to algebraic representations. Groups Geometry And Dynamics. 11:(2)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). Equicontinuous actions of semisimple groups. Groups Geometry And Dynamics. 11:(3)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

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

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

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

(2016). Amenable Invariant Random Subgroups. to apear in the Israel Journal of Mathematics.
2015

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

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

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

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

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


(2014). Algebraic Representations of Ergodic Actions and Super Rigidity. Submitted, 25 pages.
2013

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

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

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

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

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

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

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

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

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

(2010). Conformal Actions on Homogeneous Lorentzian Manifolds. Journal of LieTheory 20 no. 3(2010), {469481}.
2009

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

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

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

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

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

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