# Abstracts

### Minicourse

E. Militon (Nice)

Title: Action of the group of homeomorphisms of a surface on its fine curve graph (parts 1, 2, 3)

Abstract: The fine curve graph of a surface S is a Gromov hyperbolic graph, introduced by Jonathan Bowden, Sebastian Hensel and Richard Webb, on which the group of homeomorphisms of S acts faithfully by isometries. In this mini-course, I will first give an overview of the current results about this graph. Then, I will give a more in-depth explanation of the links between the dynamics of a homeomorphism of a surface and the isometry type of its action on the fine curve graph.

### Research Talks

Collin Bleak (St Andrews)

Title: A Smorgasbord of Thompson's Groups Results

Abstract: We give a collection of results that are the results of ongoing work on the structures of the R. Thompson groups F and T. All results are answers to questions that have been asked of us lately by other researchers interested in Thompson groups: Does TxT embed into T? Is T truly maximal in V? Is V a quotient of C2*C3? Joint with various collaborators (question dependent, including Rachel Skipper, Martyn Quick, James Belk, Raad Al Kohli, Casey Donoven, and Scott Harper).

Lei Chen (Maryland)

Title: Mapping class groups of circle bundles over a surface

Abstract: In this talk, we study the algebraic structure of mapping class group Mod(M) of 3-manifolds M that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku and Alina Beaini.

Thomas Koberda (Virginia)

Title: Things that homeomorphism groups of manifolds know

Abstract: I will discuss a result, joint with J. de la Nuez Gonzalez, which shows that homeomorphism groups of compact manifolds interpret the full second order theory of their countable subgroups. As one application, I will show that there is a single first order group theoretic sentence that is true in the homeomorphism group of a compact 2-manifold if and only if the manifold's mapping class group is linear. I will also discuss some self-reference paradoxes that arise among homeomorphism groups of manifolds.

André Nies (Auckland)

Title: Describing a group by a first-order sentence (slides here)

Abstract: A group G is finitely axiomatisable (FA) within a class C if there is a finite axiom system (in an appropriate first-order language) such that, up to isomorphism, G is the only group in C satisfying it. We survey results starting from 21 years back, when the speaker introduced this notion for the class C of finitely generated groups (and the usual first-order language of groups). First examples included the Baumslag-Solitaer groups B(1,n) for n>1. Lasserre (2014) showed that the Thompson groups are FA within the f.g. groups. With Segal and Tent (2021) we address FA in classes of profinite groups. We show in particular that within the pro-p groups, each group G of finite rank is FA using the language of group theory expanded by function symbols for exponentiation with p-adic numbers.

Javier de la Nuez-Gonzalez (KIAS)

Title: A regularity gap for groups of bi-absolutely continuous homeomorphisms

Abstract: I will discuss recent results of mine which fit into a long-term project of understanding whether all sensible notions of regularity of manifold transformations, taking the lattice of group topologies using the lattice of group topologies on the diffeomorphism group below the Whitney topology as a proxy. In this talk, I will focus on the group of those homeomorphisms of a manifold endowed with a suitable measure for which both the homeomorphism and their inverse are absolutely continuous in the sense of Banach. These groups admit a natural Polish group topology finer than the C^0 topology. I will discuss a result of mine according to which there is a gap between these two topologies, which can be interpreted as saying that absolute continuity is a minimal reasonable non-trivial notion of regularity.

Yash Lodha (Hawaii)*

Title: Finitely presented simple torsion free groups of homeomorphisms of R.

Abstract: I will present a construction of finitely presented simple torsion free groups that act by piecewise linear homeomorphisms of R. These are the first such examples, and only the second torsion-free construction of finitely presented simple groups since the Burger-Mozes construction from 1999. Moreover, our examples are also of type F_{\infty} and admit a nontrivial homogeneous quasimorphism, providing the first torsion-free solution to Problem 14.13.(b) in the Kourovka notebook. This is joint work with James Hyde.