Surface Homeomorphisms through the lens of geometric group theory

Closed surfaces, and their self-homeomorphisms, are central throughout modern geometry, topology and dynamics. Nevertheless, our understanding still shows large gaps, and in particular the question of which groups can act on surfaces and how is wide open. Similarly, the dynamics of homeomorphisms acting on surfaces is very rich classical area that is still full of open questions.In this talk we will present results from a recent research programme that aims to approach questions about homeomorphism groups using coarse geometry, and in particular the action on a suitable negatively curved graph: the fine curve graph. We will present an overview of these methods, and applications in dynamics, geometric topology, and geometric group theory.