Past Algebra Seminar

Informally speaking, a finitely generated group G is said to be {\it Golod-Shafarevich} (with respect to a prime p) if it has a presentation with a ``small'' set of relators, where relators are counted with different weights depending on how deep they lie in the Zassenhaus p-filtration. Golod-Shafarevich groups are known to behave like (non-abelian) free groups in many ways: for instance, every Golod-Shafarevich group G has an infinite torsion quotient, and the pro-p completion of G contains a non-abelian free pro-p group. In this talk I will extend the list of known ``largeness'' properties of Golod-Shafarevich groups by showing that they always have an infinite quotient with Kazhdan's property (T). An important consequence of this result is a positive answer to a well-known question on non-amenability of Golod-Shafarevich groups.

Many classical results and conjectures in representation theory of finite groups (such as

theorems of Thompson, Ito, Michler, the McKay conjecture, ...) address the influence of global properties of representations of a finite group G on its p-local structure. It turns out that several of them also admit real, resp. rational, versions, where one replaces the set of all complex representations of G by the much smaller subset of real, resp. rational, representations. In this talk we will discuss some of these results, recently obtained by the speaker and his collaborators. We will also discuss recent progress on the Brauer height zero conjecture for 2-blocks of maximal defect.

The study of representation growth of infinite groups asks how the

numbers of (suitable equivalence classes of) irreducible n-dimensional

representations of a given group behave as n tends to infinity. Recent

works in this young subject area have exhibited interesting arithmetic

and analytical properties of these sequences, often in the context of

semi-simple arithmetic groups.

In my talk I will present results on the representation growth of some

classes of finitely generated nilpotent groups. They draw on methods

from the theory of zeta functions of groups, the (Kirillov-Howe)

coadjoint orbit formalism for nilpotent groups, and the combinatorics

of (finite) Coxeter groups.

We classify certain linear representations of finite groups with a large orbit. This is motivated by a question on the number of conjugacy classes of a finite group.

To any Coxeter group (W,S) together with an appropriate representation on V one may associate various categories of "singular Soergel bimodules", which are certain bimodules over invariant subrings of

regular functions on V. I will discuss their definition, basic properties and explain how they categorify the associated Hecke algebras and their parabolic modules. I will also outline a motivation coming from geometry and (if time permits) an application in knot theory.

In a joint project with Christopher Voll, I have investigated the representation zeta functions of compact p-adic Lie groups. In my talk I will explain some of our results, e.g. the existence of functional equations in a suitable global setting, and discuss open problems. In particular, I will indicate how piecing together information about local zeta functions allows us to determine the precise abscissa of convergence for the representation zeta function of the arithmetic group SL3(Z).

I will discuss the following

Conjecture B: Finitely generated abstract images of profinite groups are finite.

I will explain how it relates to the width of words and conjugacy classes in finite groups. I will indicate a proof in the special case of 'non-universal' profinite groups and propose several directions for future work.

This conjecture arose in my discussions with various participants of a workshop in Blaubeuren in May 2007 for which I am grateful. (You know who you are!)