Schedule

The notion of weakly almost periodic representation (WAP) is natural, generalizing simultaneously unitary representations, probability measure preserving actions and WAP actions (which is a type of actions studied in topological dynamics). It provides a convenient frame work that I will present in my lecture. When such representations appear as coefficient modules, fixed point theorems could be applied in order to obtain vanishing result for cohomology. I will present easy proofs of vast generalizations of the classical vanishing theorems of Blanc and of Shalom.
Based on a joint work with Christian Rosendal and Roman Sauer.

We study the limiting behavior of the discrete spectra of congruence subgroups of an irreducible arithmetic lattice in a semisimple Lie group $G$. Assuming that the subgroups in question do not contain any non-trivial elements of finite order, one expects their spectra to converge to the Plancherel measure of $G$ (the limit multiplicity property). We are able to prove this property for the lattices ${\rm SL} (n, \mathfrak{o}_F)$, where $F$ is a number field, and obtain conditional results in general. The focus lies on
the case of non-compact quotients, where the spectra have a continuous part.

There are two main parts of the proof, which is based on Arthur's trace formula. First, we prove some general results on congruence subgroups of arithmetic lattices and their intersections with conjugacy classes. Second, we reduce the control of the continuous spectrum to two conjectural properties of intertwining operators, one global and one local, which we can verify for the groups ${\rm GL} (n)$ and ${\rm SL} (n)$. This is joint work with Erez Lapid (Jerusalem) and Werner Müller (Bonn).

I will discuss the representation theory of countable linear groups into two specific polish targets. Sym(Z) the full symmetric group of a countable set and Aut(X,B,m) the automorphism group of a standard Borel probability space. The structure of linear groups, helps us establish similarities between their representation theory and that of finite or algebraic groups.
The talk will be based on 4 papers, two joint with Tsachik Gelander and one joint with Dennis Gulko.

Given measure-preserving map of an infinite (sigma-finite) measure space, the appropriate version of the ergodic theorem is Hopf's ratio theorem, which gives a.e. convergence of the ratios of the orbital sums of two L^1 functions. This theorem is classical for actions of the integers but only recently has progress been made for other groups. It turns out that the theorem can be extended to Z^d, but fails for many (maybe most) other groups, including infinite rank abelian groups and the discrete Heisenberg group. The validity of the theorem is intimately relatd to the combinatorics and geometry of the group. I will discuss these old, recent and new results, and some future directions.

Higher-dimensional Dehn functions are quasi-isometry invariants which measure the difficulty to fill k-spheres by (k+1)-balls. I will determine these invariants for uniform S-arithmetic groups (over global fields). I will also adress the mostly conjectural
picture for non-uniform S-arithmetic groups.

This talk is about the entropy of group actions of amenable groups. I will present recent progress on questions asked by Christopher Deninger about the entropy of certain principal algebraic dynamical systems. I will show that the entropy of an algebraic dynamical system agrees with the L2-torsion of the dual module over the integral group ring of the group acting. As a by-product we prove vanishing of the L2-torsion of amenable groups, which was conjectured by Wolfgang Lueck. This is joint work with Hanfeng Li.

The Mordell constant of a d-dimensional unimodular lattice is defined to be $2^{-d} sup\ vol(V)$, where V ranges over all symmetric boxes with sides perpendicular to the axes, containing no nonzero lattice points. The Mordell Gruber spectrum is the set of all possible values for this constant, over all lattices in dimension d. In joint work with Uri Shapira we give a dynamical interpretation of the Mordell constant in terms of the action of the diagonal group on the space of lattices, and use dynamical results for this action to obtain new information about the spectrum.

In this talk I will introduce Anosov representations and explain a construction of domains of discontinuity for such representations. I will then describe several application of this construction.