Past Algebra Seminar

A classic problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated commutative algebra by algebra automorphisms, whether the ring of invariants is still finitely generated. I shall present joint work with W. van der Kallen treating the problem for a Chevalley group over an arbitrary base. Progress on the corresponding problem of finite generation for rational cohomology will be discussed.

I'll explain how the `Auslander philosophy' from finite dimensional algebras gives new methods to tackle problems in higher-dimensional birational geometry. The geometry tells us what we want to be true in the algebra and conversely the algebra gives us methods of establishing derived equivalences (and other phenomenon) in geometry. Algebraically two of the main consequences are a version of AR duality that covers non-isolated singularities and also a theory of mutation which applies to quivers that have both loops and two-cycles.

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!)

As a small step towards an understanding of the relationship of the two fields in the title, I will present a uniformness result for embeddings of finitely generated, virtually free groups as cocompact, discrete subgroups in totally disconnected, locally compact groups.

A group is called boundedly generated if it is the product of a finite sequence of its cyclic subgroups. Bounded generation is a property possessed by finitely generated abelian groups and by some other linear groups.

Apparently it was not known before whether all boundedly generated groups are linear. Another question about such groups has also been open for a while: If a torsion-free group $G$ has a finite sequence of generators $a_1,\dotsc,a_n$ such that every element of $G$ can be written in a unique way as $a_1^{k_1}\dotsm a_n^{k_n}$, where $k_i\in\mathbb Z$, is it true then that $G$ is virtually polycyclic? (Vasiliy Bludov, Kourovka Notebook, 1995.)

Counterexamples to resolve these two questions have been constructed using small-cancellation method of combinatorial group theory. In particular boundedly generated simple groups have been constructed.

I shall explain two ways of embedding families of rescaled graphs into Cayley graphs of groups. The first one allows to construct finitely generated groups with continuously many non-homeomorphic asymptotic cones (joint work with M. Sapir). Note that by a result of Shelah, Kramer, Tent and Thomas, under the Continuum Hypothesis, a finitely generated group can have at most continuously many non-isometric asymptotic cones.

The second way is less general, but it works for instance for families of Cayley graphs of finite groups that are expanders. It allows to construct finitely generated groups with (uniformly convex Banach space)-compression taking any value in [0,1], and with asymptotic dimension 2. In particular it gives the first example of a group uniformly embeddable in a Hilbert space with (uniformly convex Banach space)-compression zero. This is a joint work with G. Arzhantseva and M.Sapir.

Cherednik algebras (always of type A in this talk) are an intriguing class of algebras that have been used to answer questions in a range of different areas, including integrable systems, combinatorics and the (non)existence of crepant resolutions. A couple of years ago Iain Gordon and I proved that they form a non-commutative deformation of the Hilbert scheme of points in the plane. This can be used to obtain detailed information about the representation theory of these algebras.

In the first part of the talk I will survey some of these results. In the second part of the talk I will discuss recent work with Gordon and Victor Ginzburg. This shows that the approach of Gordon and myself is closely related to Gan and Ginzburg's quantum Hamiltonian reduction. This again has applications to representation theory; for example it can be used to prove the equidimensionality of characteristic varieties.