Connectedness of the Gromov boundary of fine curve graphs

Abstract

The fine curve graph was introduced to study homeomorphism group of surfaces. In this paper we study the topology of the Gromov boundary of this graph for closed surfaces with higher genus. We first prove a bounded geodesic image theorem for the fine curve graph, a consequence of which is the non-compactness of the Gromov boundary. Using this theorem, we are able to show that the Gromov boundary is linearly connected with respect to some visual metric.