Towards the boundary of the fine curve graph

Abstract

The fine curve graph was introduced as a geometric tool to study homeomorphisms of surfaces. In this paper we study the Gromov boundary of this space and the local topology near points associated with certain foliations and laminations. We then give several applications including finding dynamically explicit elements with positive stable commutator length, and proving a Tits alternative for subgroups of $\text{Homeo}(S)$ containing a pseudo-Anosov map, generalizing a result of Hurtado-Xue.

29 pages, 2 figures