Forthcoming events in this series
Kazhdan quotients of Golod-Shafarevich groups
Abstract
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.
Divisibility properties of character degrees and p-local structure of finite groups
Abstract
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.
Localising subcategories of the stable module category for a finite group
Representation growth of finitely generated nilpotent groups
Abstract
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.
On the number of conjugacy classes of a finite group
Abstract
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.
Hochschild and block cohomology varieties are isomorphic
Endomorphisms of tensor space and cellular algebras
Abstract
endomorphism algebras in question, both in the classical and quantum cases.
Singular Soergel Bimodules
Abstract
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.
Representation zeta functions of p-adic Lie groups
Abstract
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).
On the abstract images of profinite groups
Abstract
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!)