Groups and Geometry in South England

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 Bristol, Cambridge, London, Oxford, Warwick, and Southampton, and organised by Martin Bridson, Mark Hagen, Peter Kropholler, Robert Kropholler, Lars Louder, John Mackay, Ashot Minasyan, Saul Schleimer, and Henry Wilton. We have been awarded LMS Scheme 3 funding, as well as support from the Isaac Newton Institute for Mathematical Sciences.

In 2023-24, the meetings will tentatively be as follows:

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 15 March 2024

In Oculus OC1.01:

1:30 Relatively Hyperbolic Groups, JSJ decompositions and the Farrell--Jones conjecture

Naomi Andrew (Oxford)

The Farrell--Jones conjecture predicts that the K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, a positive answer for a family of groups allows one to calculate these K-groups, as well as classifying the manifolds admitting a given fundamental group, and implying a positive answer to the Borel conjecture. I will talk about recent work with Yassine Guerch and Sam Hughes where we show that it holds for extensions of many relatively hyperbolic groups, as well as for automorphism groups in the one-ended case. The proof uses a result of Knopf for groups acting acylindrically on trees, so I will also discuss the tools from JSJ theory that we worked with to arrange for these actions.

2:30 First-order logic and acylindrically hyperbolic groups

Jonathan Fruchter (Bonn)

In 1945, Alfred Tarski posed the following question: do all non-abelian free groups have the same first-order theory? It took over 60 years until Sela gave a positive answer to this question, and the path towards this final answer is paved with many interesting (and sometimes lesser-known) results and techniques. I will talk about joint work with Simon Andre, in which we generalize some of these techniques to the acylindrically hyperbolic setting. These results also allow us to draw conclusions which stretch beyond the realm of first-order logic.

3:30-4:00 TEA Somewhere

In B3.02 Zeeman Building:

5:00 Approximating hyperbolic lattices by cubulations

Eduardo Reyes (Bonn/Yale)

The fundamental group of an \(n\)-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space \(\mathbb H^n\). If \(n\) is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on CAT\((0)\) cube complexes. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on \(\mathbb H^n\) by actions on these complexes, solving a conjecture of Futer and Wise. The main tool is a codimension-1 generalization of the space of geodesic currents introduced by Bonahon.

Reimbursements

UPDATE: There is travel money available, prioritising speakers, PhD students, and ECRs. DO NOT FILL OUT ANY FORMS! Just send me an email at l.louder AT ucl.ac.uk and I will set things in motion.

We are on less of a shoestring budget than we were, but still try to minimize your travel costs!