Special Year 2005-06: Lie Groups, Representations and Discrete Mathematics

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely Generated residually finite group. The same holds for Cartesian products of other simple groups under...

This talk is about joint work with A. Lubotzky. Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. We have contructed new linear representations of $\A(F_n)$ arising through the action of finite index...

We will discuss the following very recent result: Theorem. Let R be any finitely generated associative (not necessarily commutative) ring, with 1. Then for any n > stable range rank of the ring R, the group EL_n(R) has Kazhdan's property (T). The...

A finitely generated group is called a Golod-Shafarevich group if it has a presentation with the following property: There exists a prime number p and a real number 0

We provide calculations of growth and spectra of Cayley and Schreier graphs related to some branch groups. Among the examples, we present a class of groups of intermediate growth defined by primitive polynomials over finite fields (the original...

Let $\alpha$ be an automorphism of a totally disconnected locally compact group $G$. There is a canonical form for $\alpha$ that partially fills the role played by the Jordan canonical form of $\mathrm{ad}( \alpha )$ in the case when $G$ is a Lie...

Studying the covolume of lattices goes back to the work of Siegel in the forties where he shows that $(2,3,7)$-triangular group is a lattice of minimum covolume in $G = \mathrm{SL}_2(\mathbb R)$. The case of $\mathrm{SL}_2(\mathbb C)$ has been open...

The main purpose of the talk is to describe an approach for the estimate of averages over orbits of certain functions on the space of lattices. These estimates play a crucial role in the study of the distribution of values at integral points of...