The fine curve graph is a Gromov hyperbolic graph on which the homeomorphism group of a surface acts. It allows to apply tools from geometric group theory and the theory of mapping class groups in this setting.
In this talk, we will describe the first entries in a dictionary linking dynamical properties of homeomorphisms acting on the surface to the geometry of the action on the fine curve graph. Furthermore, we will discuss phenomena not encountered in the setting of "classical" curve graphs — namely, homeomorphisms acting as parabolic isometries. This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.
Time permitting, I will describe ongoing work with Jonathan Bowden and Richard Webb concerning the Gromov boundary of the fine curve graph.