• 14:40-15:30 CM107 Alexandre Martin (Heriot-Watt University) On analogues of the curve complex for various groups

The action of the mapping class group of a hyperbolic surface on its curve complex has proved to be a rich source of information. This action possesses an interesting property: The action of the group extends to an action of its automorphism group. It turns out that many groups of geometric interest admit actions with a similar property. I will present such analogues of the curve complex for other types of groups, and explain how one can use the associated action to understand the automorphism group of the groups under study. In particular, I will present a construction for certain graph products. I will also highlight some similarities between these constructions. (Part of this is joint work with Anthony Genevois)

• 15:40-16:30 CM107 Ulrike Tillmann (Oxford) Homotopical group theory

We will consider some geometrically defined families of groups and study their homology and maps between them via homotopy theoretic means. In particular we will be interested in mapping class groups and braid groups.

16:30-17:00 common room Tea

• 17:00-17:50 CM107 Corinna Ulcigrai (Bristol) Hall rays for Lagrange spectra of Fuchsian groups

In this talk we will first introduce Lagrange spectra of a Riemann surfaces with cusps. Values in these spectra describe asymptotic penetration of geodesics at a cusp of the Riemann surface and also admit an interpretation in terms of Diophantine approximation in a Fuchsian group. The classical Lagrange spectrum, which correspond the case of the modular surface, has been studied for more than a century; Hall proved in 1947 that it contains a ray, i.e. an semi-infinite interval. We generalize this result to the Lagrange spectrum of any Fuchsian group which is a non-uniform lattice and to more general dynamically defined Lagrange penetration spectra. The proof uses boundary expansions, which (following Bowen-Series) encode endpoints of geodesics via a sequence of elements in the group and the study of the sum of Cantor sets in the boundary which consists of all endpoints of geodesics with bounded penetration in the cusps.

There will be a dinner in Durham. Please e-mail Anna Felikson if you plan to attend.

--------------------------------------------------------------------------------------------------

Room CM107 is on the ground floor of the maths department, to the left from the main entrance.