On the geometry of graphs associated to infinite-type surfaces

Abstract

Consider a connected orientable surface S of infinite topological type, i.e. with infinitely-generated fundamental group. Our main purpose is to give a description of the geometric structure of an arbitrary subgraph of the arc graph of S, subject to some rather general conditions. As special cases, we recover results of Bavard (Geom Topol 20, 2016) and Aramayona–Fossas–Parlier (Arc and curve graphs for infinite-type surfaces. Preprint, 2015). In the second part of the paper, we obtain a number of results on the geometry of connected, $$\mathrm{Mod}(S)$$ -invariant subgraphs of the curve graph of S, in the case when the space of ends of S is homeomorphic to a Cantor set.