Modern Methods in Infinite Groups

The aim of this conference is to bring together a diverse group of established experts and early career researchers working at the forefront of interactions between geometric group theory, low-dimensional topology, and algebraic topology.
A connecting theme is the work and influence of Prof. Ian J. Leary (Southampton) whose 60th birthday we wish to honour.

Venues

Wednesday, 12 June: Fry Building, LG.02
Thursday, 13 June: Canynge Hall, LG.08 
Friday, 14 June: Canynge Hall, LG.08
Conference dinner: Racks Bar & Kitchen

Abstracts

Naomi Andrew - Two Generator Subgroups of Free-by-Cyclic groups

In general, it is hard to characterise "all subgroups" of a given group. However, restricting the complexity in some way can make the problem tractable: subgroups of free groups, or of surface groups are not so bad, and cyclic subgroups don't cause too many problems. Two generators is a lot more than one, but progress can sometimes still be made: in 1979, Jaco and Shalen characterised the two-generator subgroups of fundamental groups of certain orientable three manifold.
I will talk about recent work with Edgar Bering, Ilya Kapovich and Stefano Vidussi characterising the two-generator subgroups of mapping tori of free groups, using ideas from Feighn and Handel's proof of coherence for these groups.

Macarena Arenas - Cubically presented groups, strong asphericity, and applications

We’ll explore the problem of finding effective models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, we’ll discuss the framework -- cubical small-cancellation theory -- that provides the necessary tools to do so, and we’ll explain how this viewpoint allows us to compute the homology and cohomology of various examples, including many Artin groups and other related families.

Rachael Boyd - Diffeomorphisms of reducible 3-manifolds

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space $B \mathrm{Diff}(M)$, for $M$ a compact, connected, reducible 3-manifold. We prove that when $M$ is orientable and has non-empty boundary, $B \mathrm{Diff}(M\ \mathrm{rel}\ \partial M)$ has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

Sam Fisher - Virtually free-by-cyclic RFRS groups

We give a homological characterisation of virtually free-by-cyclic groups in the class of RFRS groups; namely, we show that a RFRS group is virtually free-by-cyclic if and only if it is of cohomological dimension 2 and has vanishing second $\ell^2$-Betti number. We will also present a higher-dimensional version of the result.

Robert Kropholler - Groups of type FP_2 over fields.

I will outline a construction to create groups that are of type FP_2 over every field but not over the integers. The idea stems from Ian's construction of uncountably many groups of type FP_2. I will first give an overview of Ian's construction and then give the modifications needed to create the groups mentioned above. Time permitting I will discuss the homological filling functions of these groups. 

Monika Kudlinska - Analogues of the Thurston norm in groups

The Thurston norm is a seminorm defined on the first cohomology of a closed 3-manifold with real coefficients. Restricted to integral characters, it detects the minimal topological complexity of an embedded surface which represents the second homology class dual to the character. In this talk, we will introduce a real-valued function on the first cohomology of an arbitrary group with real coefficients which generalises the Thurston norm. We will propose a strategy for proving that such a function defines a seminorm using the theory of $L^2$-invariants. Finally, we will implement this strategy for some classes of right angled Artin groups using the recent calculations of $L^2$-Betti numbers of Artin kernels due to Fisher-Hughes-Leary.

Wolfgang Lück - Classifying spaces for families of subgroups and their applications to algebra, geometry, group theory, and K-theory

This talk is a survey on classifying spaces for families. We explain the basic properties of this notions and explain especially for the family of finite subgroups that there are often nice geometric models, such as Rips complexes or Teichmueller spaces. We will discuss finiteness properties and explain how Ian Leary and his collaborator have made substantial progress on this topic. My own iterest comes from the Farrell-Jones Conjecture and the Baum-Connes Conjecture. Moreover, I will explain how finding good models for the classifying spaces of families allows interesting computations of group cohomology and of the K- and L-theory of group rings and group $C^*$-algebras which have direct applications to the classifications of manifolds and $C^*$-algebras.

Francesco Milizia - Davis' manifolds with positive simplicial volume

The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a specific class of manifolds: those obtained from Davis’ reflection group trick. I will describe an approach that leads to the study of triangulations of spheres and simplicial maps between them. A connection of this problem with the theory of graph minors will also be presented.

Brita Nucinkis - On cohomological finiteness conditions for topological groups

In this talk I will give an introduction to analogues to the classical finiteness conditions FP_n for totally disconnected locally compact groups. I will present a construction of non-discrete tdlc groups of arbitrary finiteness length. As a bi-product we also obtain a new collection of (discrete) Thompson-like groups which contains, for all positive integers n, groups of type FP_n but not of type FP_{n+1}. This is joint work with I. Castellano, B. Marchionna, and Y. Santos-Rego.

Luis Jorge Sánchez Saldaña - Continuously many Eilenberg-Ganea type groups

The Eilenberg-Genea problem asks for the existence of a group with cohomological dimension 2 and geometric dimension 3, and it is still an open problem. Given a group $G$ and a family of subgroups $\mathcal{F}$ there are notions of cohomological and geometric dimension of $G$ relative to $\mathcal{F}$. Remarkably in this setting there are examples of group with $\mathcal{F}$-cohomological dimension 2 and $\mathcal{F}$-geoemtric dimension 3 due originally to Brady-Leary-Nucinkis. In this talk I will provide an overview of what is known about the Eilenberg-Ganea problem relative to families of subgroups.

Motiejus Valiunas - Leary--Minasyan groups and generalisations

The class of commensurating HNN-extensions of free abelian groups has been shown by Ian Leary and Ashot Minasyan to contain examples of CAT(0) groups that are not biautomatic, hence answering a long-standing open question. These groups, while generalising Baumslag--Solitar groups and thus acting as counterexamples to a variety of statements, also allow enough freedom in choosing their parameters to exhibit even more surprising behaviour.  In this talk, I will advertise Leary--Minasyan groups and discuss some results mostly concerning their (non-)biautomaticity, such as non-embeddability into biautomatic groups and a generalisation of the construction to obtain a non-biautomatic group with even stronger non-positive curvature properties.  A part of this talk is based on joint work with Sam Hughes, and a part of that part potentially also with Naomi Andrew.

Olga Varghese - Finite quotients of Coxeter groups

A standard approach to studying infinite groups is through their finite quotients. While this has limitations in general, the set of finite quotients of a Coxeter group encodes important information about it. We address the following question: For non-isomorphic Coxeter groups $W_\Gamma$ and $W_\Omega$, must $W_\Gamma$ and $W_\Omega$ have different sets of finite quotient groups?

Registration

We have reached our capacity. Unfortunately no more participants can be admitted to the conference.

The registration has closed on 28 April 2024.
The deadline to apply for financial support has passed on 31 March 2024.
Participants will be charged a registration fee of 35.00 GBP. Additionally, attending the conference dinner will cost 15.00 GBP.

Sponsors

We gratefully acknowledge the support from our funding sources.