**Speaker:** Collin Bleak (University of St Andrews)

**Title:** On finite generation for groups of homeomorphisms of Cantor spaces

**Abstract:** Given a Cantor space , we define a broad class of subgroups of the group of homeomorphisms of so that, if a given group is a subgroup of a finitely generated subgroup of and is in our class, we can immediately conclude that is two-generated. The argument has connections with Higman and Epstein’s general arguments toward simplicity for groups of homeomorphisms, and is general enough to immediately prove two-generation, e.g., for many relatives of the Thompson groups, and many other groups as well. Joint with James Hyde.

**Speaker:** Vaibhav Gadre (University of Glasgow)

**Title:** Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

**Abstract:** Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the circle at infinity. They ask if the exponent vanishes for almost all points with respect to Lebesgue measure on the circle. We answer their question in the affirmative by considering the behavior of the word metric along typical geodesic rays. This is joint work with Joseph Maher and Giulio Tiozzo.

**Speaker:** Richard Hepworth (University of Aberdeen)

**Title:** Homological stability

**Abstract:** A sequence of groups and inclusions

is said to satisfy homological stability if, in each degree, the induced maps on homology groups become isomorphisms once you go far enough along the sequence. Stability is known for many families of groups (symmetric, general linear, mapping class, …), usually without knowing the homology itself.

I will give a tour of homological stability, and end by describing my own work on the subject.