# Trimester Seminar

## Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall

## Thursday, December 13th, 2:30 p.m.

### Conjugacy classes and centralisers in classical groups

Speaker: Giovanni de Franceschi (Auckland)

#### Abstract

We discuss conjugacy classes and associated centralisers in classical groups, giving descriptions which underpin algorithms to list these explicitly.

## Thursday, December 13th, 2 p.m.

### PFG, PRF and probabilistic finiteness properties of profinite groups

Speaker: Matteo Vannacci (Düsseldorf)

#### Abstract

A profinite group G equipped with its Haar measure is a probability space and one can talk about "random elements" in G. A profinite group G is said to be **positively finitely generated** (PFG) if there is an integer k such that k Haar-random elements generate G with positive probability. I will talk about a variation of PFG, called "positive finite relatedness" (PFR) for profinite groups. Finally I will survey some recent work-in-progress defining higher probabilistic homological finiteness properties (PFP_n), building on PFG and PFR. This is joint work with Ged Corob Cook and Steffen Kionke.

## Thursday, December 13th, 11 a.m.

### Strong Approximation for Markoff Surfaces and Product Replacement Graphs

Speaker: Alex Gamburd (Graduate Centre, CUNY)

#### Abstract

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, in particular the intimate relation with product replacement graphs, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.

## Wednesday, December 12th, 11 a.m.

### McKay graphs for simple groups

Speaker: Martin Liebeck (Imperial College)

#### Abstract

Let V be a faithful module for a finite group G over a field k, and let Irr(kG) denote the set of irreducible kG-modules. The McKay graph M(G,V) is a directed graph with vertex set Irr(kG), having edges from any irreducible X to the composition factors of the tensor product of X and V. These graphs were first defined by McKay in connection with the well-known McKay correspondence. I shall discuss McKay graphs for simple groups.

## Tuesday, December 11th, 11 a.m.

### Groups, words and probability

Speaker: Aner Shalev (Hebrew University of Jerusalem)

#### Abstract

I will discuss probabilistic aspects of word maps on finite and infinite groups. I will focus on solutions of some probabilistic Waring problems for finite simple groups, obtained in a recent work with Larsen and Tiep. Various applications will also be given.

## Thursday, December 6th, 1:30 p.m.

### Conjugacy growth in groups

Speaker: Alex Evetts

## Thursday, December 6th, 2:10 p.m.

### Zeta functions of groups: theory and computations

Speaker: Tobias Rossmann

#### Abstract

I will give a brief introduction to the theory of zeta functions of (infinite) groups and algebras. I will describe some of the techniques used to investigate these functions and give an overview of recent work on practical methods for computing them.

## Thursday, December 6th, 3:10 p.m.

### Zeta functions of groups and model theory

Speaker: Michele Zordan

#### Abstract

In this talk we shall explore the connections between rationality questions regarding zeta functions of groups and model theory of valued fields.

## Wednesday, December 5th, 11 a.m.

### Towards Short Presentations for Ree Groups

Speaker: Alexander Hulpke (Colorado State University, Fort Collins)

#### Abstract

The Ree groups 2G2(32m+1) are the only class of groups for which no short presentations (that is of length polynomial in log(q) are known. I will report on work of Ákos Seress of myself that found a likely candidate for such a short presentation, as well as the obstacles that lie in the way of proving that it is a presentation for the group.

## Tuesday, December 4th, 11 a.m.

### Finding involution centralisers efficiently in classical groups of odd characteristic

Speaker: Cheryl Praeger (University of Western Australia)

#### Abstract

Bray's involution centraliser algorithm plays a key role in recognition algorithms for classical groups over finite fields of odd order. It has always performed faster than the time guaranteed/justified by complexity analyses. Work of Dixon, Seress and I published this year give a satisfactory analysis for SL(n,q). And we are slowly making progress with the other classical groups. The "we" are Colva Roney-Dougal, Stephen Glasby and me - and we have conquered the unitary groups so far.

## Thursday, November 29th, 2 p.m.

### Density of small cancellation presentations

Speaker: Michal Ferov

## Thursday, November 29th, 2 p.m.

### Constructing Grushko and JSJ decompositions: a combinatorial approach

Speaker: Suraj Krishna

#### Abstract

The class of graphs of free groups with cyclic edge groups constitutes an important source of examples in geometric group theory, particularly of hyperbolic groups. In this talk, I will focus on groups of this class that arise as fundamental groups of certain nonpositively curved square complexes. The square complexes in question, called tubular graphs of graphs, are obtained by attaching tubes (a tube is a Cartesian product of a circle with the unit interval) to a finite collection of finite graphs. I will explain how to obtain two canonical decompositions, the Grushko decomposition and the JSJ decomposition, for the fundamental groups of tubular graphs of graphs. While our algorithm to obtain the Grushko decomposition is of polynomial time-complexity, the algorithm for the JSJ decomposition is of double exponential time-complexity and is the first such algorithm with a bound on its time-complexity.

## Thursday, November 29th, 2 p.m.

### On the Burnside variety of groups

Speaker:Rémi Coulon

## Thursday, November 22th, 3 p.m.

### From the Principle conjecture towards the Algebraicity conjecture

Speaker:Ulla Karhumäki

#### Abstract

It was proven by Hrushovski that, if true, the Algebraicity conjecture implies that if an infinite simple group of finite Morley rank has a generic automorphism, then the fixed point subgroup of this automorphism is pseudofinite. I will state some results suggesting that the converse is true as well, and further, present a possible strategy for proving that the Principle conjecture and the Algebraicity conjecture are actually equivalent.

## Thursday, November 15th, 3 p.m.

### Separating cyclic subgroups in the pro-p topology

Speaker: Michal Ferov

## Thursday, November 15th, 3 p.m.

### Refinements and filters for groups

Speaker: Josh Maglione

## Thursday, November 8th, 2 p.m.

### On spaces of Lipschitz functions on finitely generated and Carnot groups

Speaker: Michal Doucha

#### Abstract

The motivation for this work comes from functional analysis, namely to study the normed spaces of Lipschitz functions defined on metric spaces, however certain natural restrictions lead us to focus on finitely generated and Lie groups as metric spaces in question. We show that whenever $\Gamma$ is a finitely generated nilpotent torsion-free group and $G$ is its Mal'cev closure which is Carnot, then the spaces of Lipschitz functions defined on $\Gamma$ and $G$ are isomorphic as Banach spaces. This applies e.g. to the pairs $(\mathbb{Z}^d, \mathbb{R}^d)$ or $(H_3(\mathbb{Z}), H_3(\mathbb{R}))$. I will focus on the group-theoretic content of the results and on the relations between finitely generated nilpotent torsion-free groups and their asymptotic cones (which are Carnot groups) and Mal'cev closures. Based on joint work with Leandro Candido and Marek Cuth.

## Thursday, November 8th, 2 p.m.

### The Probability Distribution of Word Maps on Finite Groups

Speaker: Turbo Ho

## Thursday, November 8th, 2 p.m.

### How to construct short laws for finite groups

Speaker: Henry Bradford

## Friday, November 2th, 2 - 4 p.m.

### Groups, boundaries and Cannon--Thurston maps

Speaker: Giles Gardam

### On the isomorphism problem for one-relator groups

Speaker: Alan Logan

### String C-group representations for symmetric and alternating groups

Speaker: Dimitri Leemans

#### Abstract

A string C-group representation of a group G is a pair (G,S) where S is a set of involutions generating G and satisfying an intersection property as well as a commuting property. String C-group representations are in one-to-one correspondance with abstract regular polytopes. In this talk, we will talk about what is known on string C-group representations for the symmetric and alternating groups. We will also explain some open questions in that area that involve group theory, graph theory and combinatorics.

### Lecture Notes

## Monday, October 29th, 3:15 p.m.

### Product set growth in groups and hyperbolic geometry

Speaker: Markus Steenbock

Abstract

We discuss product theorems in groups acting on hyperbolic spaces:

for every hyperbolic group there exists a constant $a>0$ such that for every

finite subset U that is not contained in a virtually cyclic subgroup,

$|U^3|>(a|U|)^2$. We also discuss the growth of $|U^n|$ and conclude that the entropy of $U$ (the limit of $1/n log|U^n|$ as $n$ goes to infinity) exceeds $1/2\log(a|U|)$. This generalizes results of Razborov and Safin, and answers a question of Button. We discuss similar estimates for groups acting

acylindrically on trees or hyperbolic spaces. This talk is on a joint work with T. Delzant.

## Thursday, October 18th, 2 p.m.

### A quick introduction to homogeneous dynamics

### Guan Lifan

## Thursday, October 18th, 2:30 p.m.

### Scale subgroups of automorphism groups of trees

### George Willis

## Thursday, October 18th, 3 p.m.

### Searching for random permutation groups

### Robert Gilman

Abstract

It is well known that two random permutations generate the symmetric or alternating group with asymptotic probability 1. In other words the collection of all other permutation groups has asymptotic density 0. This is bad news if you want to sample random two-generator permutation groups. However, there is another notion of density, defined in terms of Kolmogorov complexity, with respect to which the asymptotic density of every infinite computable set is positive. For the usual reasons the corresponding search algorithm cannot be implemented, but one may try a heuristic variation. Perhaps surprisingly, it seems to work. We present some experimental results.

## Thursday, October 11th, 3 p.m.

**Canonical conjugates of finite permutation groups**

### Robin Candy (Australian National University)

Abstract

Given a finite permutation group $G \le \operatorname{Sym}(\Omega)$ we discuss a way to find a canonical representative of the equivalence class of conjugate groups $G^{\operatorname{Sym}(\Omega)}=\left\{ s^{-1} G s \,\middle|\, s \in \operatorname{Sym}(\Omega) \right\}$. As a consequence the subgroup conjugacy and symmetric normaliser problems are introduced and addressed. The approach presented is based on an adaptation of Brendan McKay's graph isomorphism algorithm and is heavily related to Jeffrey Leon's partition backtrack algorithm.

## Thursday, October 4th, 3 p.m.

**Enumerating characters of Sylow p-subgroups of finite groups of Lie type $G(p^f)$**

### Alessandro Paolini (TU Kaiserslautern)

Abstract

Let q=p^f with p a prime. The problem of enumerating characters of subgroups of a finite group of Lie type G(q) plays an important role in various research problems, from random walks on G(q) to cross-characteristics representations of G(q). O'Brien and Voll have recently determined a formula for the generic number of irreducible characters of a fixed degree of a Sylow p-subgroup U(q) of G(q), provided p>c where c is the nilpotency class of G(q).

We discuss in this talk the situation in the case $p \le c$. In particular, we describe an algorithm for the parametrization of the irreducible characters of U(q) which replaces the Kirillov orbit used in the case p>c. Moreover, we present connections with a conjecture of Higman and we highlight a departure from the case of large p. This is based on joint works with Goodwin, Le and Magaard.

## Thursday, October 4th, 3 p.m.

**Rationality of the representation zeta function for compact FAb $p$-adic analytic groups.**

### Michele Zordan (KU Leuven)

Abstract

Let $\Gamma$ be a topological group such that the number $r_n(\Gamma)$ of its irreducible continuous complex characters of degree $n$ is finite for all $n\in\mathbb{N}$. We define the {\it representation zeta function} of $\Gamma$ to be the Dirichlet generating function \[\zeta_{\Gamma}(s) = \sum_{n\ge 1} r_n(\Gamma)n^{-s} \,\,\,(s\in\mathbb{C}).\]% One goal in studying a sequence of numbers is to show that it has some sort of regularity. Working with zeta functions, this amounts to showing that $\zeta_{\Gamma}(s)$ is rational. Rationality results for the representation zeta function of $p$-adic analytic groups have been first been obtained by Jaikin-Zapirain for almost all $p$. In this talk I shall report on a new proof (joint work with Stasinski) of Jaikin-Zapirain's result without restriction on the prime.

## Thursday, September 27th, 3:30p.m.

**Hyperbolicity is preserved under elementary equivalence**

### Simon Andre (University of Rennes)

Abstract

Zlil Sela proved that any finitely generated group which satisfies the same first-order properties as a torsion-tree hyperbolic group is itself torsion-free hyperbolic. This result is striking since hyperbolicity is defined in a purely geometric way. In fact, Sela's theorem remains true for hyperbolic groups with torsion, as well as for subgroups of hyperbolic groups, and for hyperbolic and cubulable groups. I will say a few words about these results.

## Thursday, September 6th, 3 p.m.

**Universal minimal flows of the homeomorphism groups of Ważewski dendrites**

### Aleksandra Kwiatkowska (Universität Münster)

Abstract

For each P ⊆ {3,4,...,ω}, we consider Ważewski dendrite W_{P}, which is a compact connected metric space that we can construct in the framework of the Fraïssé theory. If P is finite, we prove that the universal minimal flow of the homeomorphism group H(W_{P}) is metrizable and we compute it explicitly. This answers a question of Duchesne. If $P$ is infinite, we show that the universal minimal flow of H(W_{P}) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow with a comeager orbit.