The fine curve graph is a hyperbolic graph on which the homeomorphism group of a surface acts (in an interesting way). It is motivated by, and shares many properties with, the wildly successful curve graph machinery for mapping class groups — but it also shows new behaviour not encountered in the classical setting.

In this talk, we will explore some of this new behaviour by describing (certain) Gromov boundary points and their stabilisers.

This is joint work with Jonathan Bowden and Richard Webb.