Cohomology of functor categories for topological groups

Friday 8th December 2017, Royal Holloway, University of London Local organiser: Brita Nucinkis

This is the third meeting of the Research Group Functor Categories for Groups (FCG). Introduced in the '70s, fusion systems are categories which model how non-conjugate subgroups in a Sylow p-subgroup of a given finite group can fuse, i.e. become conjugate, in the whole group. The study of fusion systems has led to significant advances and improvements of proofs in group theory, and also provided useful links with algebraic topology.

The focus of the meeting will be on cohomology of functor categories for topological groups.

The venue for the talks is Royal Holloway Bedford Square House at 11 Bedford Square, London, WC1B 3RF. The timetable is as follows:

• 1.00-2.00: Xiaolei Wu (University of Bonn), On the finiteness of the classifying space for the family of virtually cyclic subgroups
• 2.15-3.15: Nadia Mazza (Lancaster University), Invertible modules for discrete groups
• 3.15-3.45: tea break;
• 3.45-4.45: Ian Leary (University of Southampton), Subgroups of almost finitely presented groups

The FCG Research Group is supported by an LMS Joint Research Groups in the UK Scheme 3 grant. Limited funding is available for PhD students, allocated on a first-come-first-served basis. In addition, the LMS administers a Childcare Supplementary Grant Scheme.

Abstracts

Xiaolei Wu (University of Bonn), On the finiteness of the classifying space for the family of virtually cyclic subgroups

In this talk, I will first give an introduction to the classifying space for the family of subgroups with many examples. Then I will concentrate on the case when the family is the set of virtually cyclic subgroups. In particular, I will discuss an interesting conjecture due to Juan-Pineda and Leary. The conjecture says a group admits a finite model for the classifying space for the family of virtually cyclic subgroups if and only if it is virtually cyclic. I will talk about some recent progress on this conjecture and some ideas behind the proof. I will also discuss some connections between this conjecture and conjugacy growth. This is joint work with Timm von Puttkamer.

Nadia Mazza (Lancaster University), Invertible modules for discrete groups

(Joint with Peter Symonds.) Given a commutative noetherian ring k of finite global dimension and a discrete group G, subject to certain conditions, we introduce the concept of invertible kG-module in an attempt to generalise the endotrivial modules for finite groups. In the first part of the talk we will present the categorical framework which we (want to) use, before giving the definition and a few results.

Ian Leary (University of Southampton), Subgroups of almost finitely presented groups

Groups of type FP₂, or almost finitely presented groups, are usually thought of as being `closer' to finitely presented groups than to finitely generated groups. In contrast, I have shown that every countable group embeds in a group of type FP₂.

I will sketch this result and its history. Although the statement is similar to the Higman-Neumann-Neumann embedding theorem of 1949, the proof is closer to that of the Higman embedding theorem of 1961, which concerns subgroups of finitely presented groups.