# Groups and Geometry in the South East

This is a series of meetings, with the aim of bringing together the geometric group theorists in the South East of England. The meetings are sponsored by mathematicians from the Universities of Cambridge, London, Oxford, Warwick, and Southampton, and organised by Martin Bridson, Peter Kropholler, Lars Louder, Ashot Minasyan, Saul Schleimer, and Henry Wilton. We have been awarded LMS Scheme 3 funding.

In 2018-19, the meetings will tentatively be as follows:

- 26 October 2018, UCL
- 30 November 2018, Southampton
- February 2019, Warwick
- May 2019, Oxford

To get regular updates about GGSE, please send an email to ggse-join@ucl.ac.uk.

Abstracts and titles of previous talks are available here.

# Details of our next meeting

## Friday 30 November, 2018

Location: Ketley Room, Mathematical Sciences, Southampton Uni, 54 Level 4.

### 1:30-2:30 On the abelianization of (pure) big mapping class groups.

Javier Aramayona (Madrid)

A classical theorem of Powell asserts that the mapping class group of an orientable surface of finite topological type and genus at least three has trivial abelianization. The first part of the talk will be devoted to explaining a proof of this result, as well as discussing the remaining low-genus cases.

We will then show that, in stark contrast, mapping class groups of infinite-type surfaces can have infinite abelianization. More concretely, we will explain how to construct non-trivial integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.

### 2:40-3:40 Almost finitely presented subgroups of hyperbolic groups

Robert Kropholler (Tufts)

Hyperbolic groups form a well understood class of groups. However, the subgroups of hyperbolic groups can be much wilder. One might hope that by imposing extra conditions upon subgroups they become easier to understand. A positive result of Gersten shows that if G is hyperbolic and has cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic. In particular, H is finitely presented.

I will detail work showing that this phenomenon is special to dimension 2 by constructing examples of subgroups of hyperbolic groups which are of type FP_2 but not finitely presented.

### 3:40-4:00 TEA

### 4:00-5:00 Ramanujan cubical complexes as higher-dimensional expanders.

Alina Vdovina (Newcastle)

Ramanujan graphs were first considered by Lubotzky, Phillips, Sarnak to get graphs with optimal spectral properties. In our days the theory of expander graphs and, in particular, Ramanujan graphs is well developed, but the questions is what is the best definition of a higher-dimensional expander is still wide open. There are several approaches, suggested by Gromov, Lubotzky, Alon and others, but the cubical complexes were not much investigated from this point of view. In this talk I will give new explicit examples of cubical Ramanujan complexes and discuss possible developments.

# Reimbursements

There is travel money available for speakers and students. Please fill out this form and send it, along with receipts, to Lars LouderDept of Mathematics

University College London

Gower St

London

WC1E 6BT