# 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 2020-21, the meetings will tentatively be as follows:

• October 23 2020, UCL/Zoom
• December 11 2020, Southampton/Zoom
• March 26 2021, Warwick/Zoom
• June 2021, Oxford

Abstracts and titles of previous talks are available here.

# Details of our next meeting

## Location: Warwick/Zoom, March 26, 2020

### Intersection of parabolic subgroups in large-type Artin groups

#### 1-2PM

Maria Cumplido (Sevilla)

Artin groups are a natural generalisation of braid groups from an algebraic point of view: in the same way that braids are obtained from the presentation of the symmetric group, other Coxeter groups give rise to more general Artin groups. There are very few results proven for every Artin group. To study them, specialists have focused on some special kind of subgroup, called "parabolic subgroups". These groups are used to build important simplicial complexes, as the Deligne complex or the recent complex of irreducible parabolic subgroups. The question "Is the intersection of parabolic subgroups a parabolic subgroup?" is a very basic question whose answer is only known for spherical Artin groups and RAAGs. In this talk, we will see how we can answer this question in Artin groups of large type, by using the geometric realisation of the poset of parabolic subgroups, that we have named "Artin complex". In particular, we will show that this complex in the large case has a property called sistolicity (a sort of weak CAT(0) property) that allows us to apply techniques from geometric group theory. This is a joint work with Alexandre Martin and Nicolas Vaskou.

#### 2:15-3:15PM

Jean Pierre Mutanguha (Max Planck)

The study of outer automorphisms of free groups borrows a lot of tools and ideas from the study of mapping classes of closed orientable surfaces. One tool that’s still missing is the canonical decomposition of mapping classes: up to isotopy, an orientation preserving surface homeomorphism preserves a unique minimal multicurve and the restriction to (orbits of) components of the multicurve’s complement is either a pseudo-Anosov or a finite-order homeomorphism. We will translate this canonical decomposition in terms of R-trees and then describe an analogue for exponentially growing outer automorphisms of free groups.

### The virtual section problem for the Mapping Class groups.

#### 3:30-4:30

The virtual section problem for the Mapping Class group $$\mathrm{Mod}(S)$$ of a surface $$S$$ asks whether there exists a section from G to $$\mathrm{Homeo}(S)$$ (the group of homeomorphisms of $$S$$), where $$G<\mathrm{Mod}(S)$$ is a finite index subgroup. I discuss the history of this problem including the most recent results by Chen-Markovic.

# Reimbursements

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

Lars Louder
Dept of Mathematics
University College London
Gower St
London
WC1E 6BT

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