# 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 2019-20, the meetings will tentatively be as follows:

• 25 October 2019, UCL
• 13 December 2019, Southampton
• 28 February 2020, Warwick
• 12 June 2020, Oxford

Abstracts and titles of previous talks are available here.

# Details of our next meeting

## Location: Oxford/Zoom, June 12

A zoom link will be sent to the list nearer the time

### Superrigidity in rank one and geometric applications

#### 2-2.45pm BST = 9-9.45am Eastern

Matthew Stover (Temple)

Margulis famously proved superrigidity of irreducible lattices in higher rank Lie groups, then used this to deduce that they are arithmetic. General superrigidity is known to fail for certain fundamental groups of hyperbolic manifolds, and there are nonarithmetic examples, but one still might wonder about the precise extent to which these fail. I will give a gentle introduction to what superrigidity means, then discuss work with Uri Bader, David Fisher, and Nicholas Miller on superrigidity of representations of fundamental groups of finite-volume real and complex hyperbolic manifolds that satisfy certain natural geometric conditions. Our main application is to prove arithmeticity finite-volume real and complex hyperbolic manifolds containing infinitely many "maximal" properly immersed totally geodesic submanifolds, thus, for example, the figure-8 knot complement is the only hyperbolic knot complement containing infinitely many properly immersed totally geodesic surfaces.

### Splitting Bianchi groups

#### 3.05-3.50pm BST = 9.05-9.50am Central

Alan Reid (Rice)

Let d be a square-free positive integer, and $$O_d$$ the ring of integers in the imaginary quadratic number field $$Q(\sqrt{-d})$$. The groups $$PSL(2,O_d)$$ are known as the Bianchi groups, and have been long studied for the connections to number theory, geometry and topology. In this talk we consider the question of whether Bianchi groups can split over a co-compact Fuchsian subgroup, and prove that for large enough $$d$$ they always do. Indeed we will sketch ideas in the proof that for large enough the Bianchi orbifold $$H^3/PSL(2,O_d)$$ contains at least (constant)$$\cdot d$$ closed orientable embedded totally geodesic 2-orbifolds. Some history, other results and applications will also be discussed time permitting. This is joint work with Junehyuk Jung.

### Filtered ends and obstructing group actions

#### 4.15-5pm BST = 9.15-10am Mountain

Emily Stark (Utah)

Studying the topology of a space at infinity offers a powerful perspective in geometric group theory. Filtered ends capture a space at infinity relative to a subspace: one considers complements of increasing neighborhoods of a subcomplex in a simplicial complex. The i-th homology groups of the complements form an inverse system, and the i-th Cech homology group is the associated inverse limit. The goal is then to compute the Cech homology groups of the filtered end and to use homological arguments to study these pairs of spaces. In this talk, I will explain how both goals are possible in the setting of coarse embeddings into coarse PD(n) spaces. Indeed, Kapovich--Kleiner proved a coarse Alexander Duality theorem to compute these deep homology groups. We extend their construction to prove a relative duality theorem, thus developing the homology theory further. As applications, one can use the deep homology groups together with homological arguments to prove that certain groups cannot act properly on a given manifold. This is joint work with Chris Hruska and Hung Cong Tran.

# 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!