Two groups are elementarily equivalent if they have the same sets of true firstorder group theoretic sentences. We prove that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker's theorem on isomorphic homeomorphism groups (1963) without relying on it. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS).

The fine curve graph of a closed surface is a graph on which the group of homeomorphisms of the surface acts faithfully by isometries. This graph is Gromov-hyperbolic. In this talk, we will explore the links between the dynamics of a surface homeomorphism and the type of isometry of its action on the fine curve graph. Joint works with Jonathan Bowden, Sebastian Hensel, Kathryn Mann, and Richard Webb and with Pierre-Antoine Guihéneuf.

As a counterpart to Deroin's minicourse, we discuss actions of groups on the circle in the C0 setting. Here, many dynamical properties of an action can be encoded by the algebraic data of a left-invariant circular order on the group. I will highlight rigidity and flexibility phenomena among group actions, and discuss new work with C. Rivas relating these to the natural topology on the space of circular orders on a group.

This is an introductory survey course on antomorphisms of free groups and the spaces they act on.