# 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 2017-18, the meetings will tentatively be as follows:

• 27 October 2017, UCL
• 1 December 2017, Southampton
• 1 June 2018, Warwick
• June 2018, Oxford

Abstracts and titles of previous talks are available here.

# Details of our next meeting

## Friday 1 June, 2018

Location: University of Warwick, Zeeman Building, B3.03

Times subject to change.

### 1:15-2:15 Integrals over unitary groups, maps on surfaces, and Euler characteristics

Michael Magee (Durham)

This is joint work with Doron Puder (Tel Aviv University). For a positive integer $$r$$, fix a word w in the free group on r generators. Let $$G$$ be any group. The word $$w$$ gives a `word map' from $$G^r$$ to $$G$$: we simply replace the generators in w by the corresponding elements of $$G$$. We again call this map $$w$$. The push forward of Haar measure under $$w$$ is called the $$w$$-measure on $$G$$. We are interested in the case $$G = U(n)$$, the compact Lie group of $$n$$-dimensional unitary matrices. A motivating question is: to what extent do the $$w$$-measures on $$U(n)$$ determine algebraic properties of the word $$w$$?

For example, we have proved that one can detect the 'stable commutator length' of $$w$$ from the $$w$$-measures on $$U(n)$$. Our main tool is a formula for the Fourier coefficients of $$w$$-measures; the coefficients are rational functions of the dimension $$n$$, for reasons coming from representation theory.

We can now explain all the Laurent coefficients of these rational functions in terms of Euler characteristics of certain mapping class groups. I'll explain all this in my talk, which should be broadly accessible and of general interest. Time permitting, I'll also invite the audience to consider some remaining open questions.

### 2:45-3:45 Sphere systems at the borders of outer space

Karen Vogtmann (Warwick)

Outer space is a contractible space on which the group $$\operatorname{Out}(F_n)$$ of outer automorphisms of a free gorup acts properly but not cocompactly. Bestvina and Feighn defined a larger contractible space, called the bordification of Outer space, on which the action is both proper and cocompact. In joint work with K.-U. Bux and P. Smillie we found an equivariant deformation retract of Outer space homeomorphic to the bordification. In this talk I will interpret the boundary of this retract in terms of sphere systems in a doubled handlebody.

### 5:30-6:30 A pro-p curve complex and residual properties of the mapping class group

Gareth Wilkes (Oxford)

To study the finite quotient groups of the mapping class group it is natural to consider the outer automorphism groups of finite quotients of a surface group. Rather than study these individually, a better approach is to package the finite quotients together into a 'profinite group' which contains all the information of the finite groups in a potentially more tractable form. A more readily manipulated object is the 'pro-$$p$$ completion', where one only considers finite groups with orders a prime power. In this talk, I will discuss the ways in which the pro-p completion of a surface group may 'split over a cyclic subgroup' in a certain sense, and the techniques by which such a classification is proved. This in turn allows the construction of a pro-p analogue of a curve complex, on which the outer automorphisms of the pro-p group act. From this action we may deduce a non-trivial residual property of the mapping class group.

# 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