Homeomorphism groups of surfaces via fine curve graphs

The goal of our project is to use ideas and methods from geometric group theory to study homeomorphism groups of surfaces. More concretely, we study the fine curve graph: a Gromov hyperbolic space on which the homeomorphism group acts in an interesting way.

In this talk, I will survey recent progress in this program, including a description of (part of) the Gromov boundary — which in turn allows to certify that most relative pseudo-Anosov homeomorphisms have positive stable commutator length — and connections between geometric properties of the graph to classical invariants in surface dynamics, namely rotation sets.