Conference Program - Thursday 09

13:30 - 14:00 Registration
14:00 - 14:50 Normal subgroups of Chevalley groups
Benjamin Klopsch (Heinrich‐Heine‐Universität Düsseldorf)
In my talk I will report on joint work with Ilir Snopce regarding the normal subgroups of universal Chevalley groups over compact discrete valuation rings. Apart from inevitable exceptions in Chevalley type A and possible exceptions in residue characteristics \(2\) and \(3\), we establish that normal subgroups are squeezed in between principal congruence subgroups in a surprisingly strong uniform way. The proof comes down to a Lie-theoretic result of independent interest: Apart from expected exceptions in type A and possible exceptions in characteristics \(2\) and \(3\), we show that for every Chevalley Lie algebra \(L\) over a field \(F\) and every non-zero element \(z\) of \(L\), the Lie algebra \(L\) is the \(F\)-linear span of commutators \([[[z,u],v,]w]\) with \(u\), \(v\), \(w\) in \(L\). The result is motivated by past, current and future applications to the study of normal subgroup zeta functions of Chevalley groups over pro-p rings.
15:00 - 15:50 Probabilistic zeta functions of profinite groups
Ged Corob Cook (University of Lincoln)
A profinite group \(G\) is said to have UBERG if, for some \(n\), the probability of generating the completed group algebra (over \(\hat{\mathbb{Z}}\)) is positive. One can calculate this probability explicitly in terms of the number of absolutely irreducible representations over finite fields, and this calculation leads to a Dirichlet series associated to \(G\), which converges in a half-plane when \(G\) has UBERG. It is possible to calculate the abscissae of these zeta functions for many groups of interest, in particular for free pro-poly-\(S\) groups, where \(S\) is a set of finite simple groups. This includes free pro-\(p\) and free prosoluble groups. For more general sets \(S\), I will sketch how we can asymptotically count absolutely irreducible representations, using ideas from the work of Jaikin-Zapirain and Pyber on permutation groups. I will also describe the connection between the free pro-\(p\) case, the search for primes in arithmetic progressions, and the Mersenne primes.
15:50 - 16:30 Coffee & Tea!
16:30 - 17:20 Double-coset zeta functions for groups acting on trees and buildings
Bianca Marchionna (Universität Bielefeld and Universitá di Milano‐Bicocca)
The double-coset zeta functions introduced by I. Castellano, G. Chinello and T. Weigel admit a geometric and favorable description in the case of locally compact groups acting properly, cocompactly and "sufficiently transitively" either on trees or on buildings. We will sketch a method for finding the meromorphic continuation of such zetas, and then give a criterion for connecting their value in \(-1\) and the Euler characteristic of the group. We will eventually discuss how this result (in several cases) relates to the simplicity of the group itself.
17:30 - 18:20 Arithmetic Representation Growth of Virtually Free Groups
Fabian Korthauer (Heinrich‐Heine‐Universität Düsseldorf)
Arithmetic representation growth deals with counting representations of finitely generated groups over finite fields. For virtually free groups various types of representations are counted by evaluating certain counting polynomials in the cardinality of the ground field. We will sketch how these polynomials arise and how they can be practically computed. If time permits, we also mention some intriguing applications to the complex representation theory of virtually free groups in terms of the cohomology of their moduli spaces of representations.

Conference Program - Friday 10

09:30 - 10:20 Representation growth of \(p\)-adic analytic groups
Margherita Piccolo (Heinrich‐Heine‐Universität Düsseldorf)
The representation growth of a group \(G\) measures the asymptotic distribution of its irreducible representations. Whenever the growth is polynomial, a suitable vehicle for studying it is a Dirichlet generating series called the representation zeta function of \(G\). One of the key invariants in this context is the abscissa of convergence of the representation zeta function. The spectrum of all abscissae arising across a given class of groups is of considerable interest and has been studied in some cases. In the realm of \(p\)-adic analytic groups (with perfect Lie algebra), the abscissae of convergence are explicitly known only for groups of small dimensions. But there are interesting asymptotic results for "simple" \(p\)-adic analytic groups of increasing dimension. In this talk, I will give an overview of the main tools and ingredients in this area and I will report on recent work joint with Moritz Petschick to enlarge the class of groups.
10:20 - 11:00 Coffee & Tea!
11:00 - 11:50 Reidemeister Spectrum and Zeta functions
Paula Macedo Lins de Araujo (University of Lincoln)
A group automorphism \(\varphi: G \to G\) induces the action \(g\cdot x= gx\varphi(g)^{-1}\) on \(G\). The orbits of such action are called twisted conjugacy classes (also known as Reidemeister classes). In the past few years, the sizes of such classes have been intensively investigated. One of the main goals in the area is to classify groups where all classes are infinite. For groups not having such property, one is interested in the possible sizes of the classes. In this talk, we will discuss how zeta functions of groups can be used to determine these sizes for certain nilpotent groups.
12:00 - 12:50 B-telescopes and profinite completions
Eduard Schesler (FernUniversität in Hagen)
Since the early 90's, infinite products of finite groups keep appearing in the study of growth functions in group theory such as subgroup growth, representation growth, and residual finiteness growth. However, it took until 2006 when Kassabov and Nikolov discovered that many infinite products of finite groups arise as profinite completions of finitely generated groups. In this talk I will introduce the notion of a B-telescope, which gives rise to rather elementary examples of finitely generated groups whose profinite completion is an infinite product of finite groups. Moreover, I will discuss some applications of B-telescopes in the study of amenability and profinite rigidity of groups. This talk is based on a joint work with Steffen Kionke.