# 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 2016-17, the meetings will tentatively be as follows:

- 27 October 2017, UCL
- 1 December 2017, Southampton
- February 2018, Warwick
- May 2018, 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 1 December, 2017

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

### 1:15 Horofunctions and groups of linear growth

Matthew Tointon (Neuchatel)

One can define a boundary of a metric space using certain functions called 'horofunctions'. When the metric space is a Cayley graph there is a natural action of the group on this boundary. Anders Karlsson has proposed a potential new method for proving Gromov's theorem on groups of polynomial growth using this action. In this talk I will show that such a method can be made to work in the case of groups of linear growth. Joint work with Ariel Yadin.

### 2:30 Conjugation invariant geometry of \(\operatorname{SL}(n,\mathbb{Z})\)

Jarek Kedra (Aberdeen)

It is known that the group \(\operatorname{SL}(n,\mathbb{Z})\) for \(n>2\) is boundedly generated by elementary matrices. It almost immediately follows that every conjugation invariant norm on \(\operatorname{SL}(n,\mathbb{Z})\) is bounded. In particular, the word norm associated with a conjugation invariant generating set has finite diameter. I will discuss the dependence of the diameter on the choice of a generating set and present applications to finite simple groups \(\operatorname{PSL}(n,q)\). This is a recent joint work with Assaf Libman and Ben Martin.

### 3:30 Tea

### 4:00 Exotic building lattices

Stefan Witzel (Bielefeld)

Lattices on buildings arise naturally as \(S\)-arithmetic groups. However, there are also lattices on exotic buildings, which cannot be \(S\)-arithmetic, and they will be the topic of the talk. Exotic lattices are interesting because they share certain properties with their arithmetic counterparts while they are intriguingly different in other respects. I will survey some of the recent results on exotic lattices and discuss some open questions

# 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