Publications

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

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 thick-thin decomposition which is of independent interest.

In this paper we prove that, under mild assumptions, a lattice in a product of semi-simple 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.

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 finite-dimensional representation with infinite image over a commutative unital ring? If $ X$ is the Bruhat-Tits 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 Bruhat-Tits 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.

Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. 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 Margulis-Zimmer super-rigidity phenomenon [2].