# 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 2022-23, the meetings will tentatively be as follows:

- 28 October 2022, UCL
- 2 December 2022, Southampton
- 3 March 2023, Warwick
- June 2023, 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

## Warwick, March 3 2023

## Room H0.58 Humanities 1:15-3:30

### 1:15-2:15 The Euler characteristic of the moduli space of graphs

Karen Vogtmann (Warwick)

The moduli space of rank n metric graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex of degree n all have the same rational cohomology. We determine the asymptotic behavior of the associated Euler characteristic, and thereby prove that the total dimension of this cohomology grows rapidly with n. This is joint work with Michi Borinsky.

### 2:30-3:30 Maximal subgroups, after Margulis and Soifer

Serge Cantat (Rennes)

Using ideas from the proof of Titsâ€™ alternative, Margulis and Soifer proved that a finitely generated group of matrices which is not virtually solvable contains uncountably many maximal subgroups; in particular, it contains maximal subgroups of infinite index. I will describe this theorem and explain how the proof can be adapted to other contexts, for example to the Cremona group in two variables.

## Go to Zeeman building

### 4:00-5:00 Warwick Colloquium: Olivia Caramello

### Tea and snacks?

## Room B3.02 Zeeman 5:15-6:30

### 5:15-6:15 One-relator groups, monoids and inverse monoids

Robert Gray (East Anglia)

It is a classical result of Magnus proved in the 1930s that the word problem is decidable for one-relator groups. In contrast, it remains a longstanding open problem whether the word problem is decidable for one-relator monoids. A natural class of algebraic structures lying between monoids and groups is that of inverse monoids. An inverse monoid is called special if it is defined by a presentation where all the defining relations are of the form w=1. There is strong motivation for studying this class coming from results of Ivanov, Margolis and Meakin (2001) who showed that if all special one-relator inverse monoids with defining relator w=1, where w is a reduced word, have decidable word problem then this would answer positively the open problem of whether all one-relator monoids have decidable word problem. In this talk I will speak about some recent results on the algebraic and algorithmic properties of special and one-relator inverse monoids. I will explain some of the methods used in this area including the theory of Schutzenberger graphs. I'll also explain the connections with some problems about one-relator groups including the submonoid membership problem, coherence, and the question of which right-angled Artin groups embed as subgroups.

# Reimbursements

There is travel money available for speakers and students. Please fill out this (please note the new form!) form and send it and your receipts to l.louder AT ucl.ac.uk.

We are on a shoestring budget, so please try to minimize your travel costs!