The grand arc graph

Abstract

In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and δ-hyperbolic. We also show that the mapping class group acts on the grand arc graph by isometries and that the action is quasi-continuous, which is a coarse relaxation of a continuous action. When the surface has stable maximal ends, we also show that this action has finitely many orbits.

Two results on end spaces of infinite type surfaces

“…See Section 2 for the definition of the partial order on ends types. This notion has turned out to be a useful one and it been appeared many times in the literature, see for instance [1,2,3,7]. A partial converse to this statement was proved in [6,Prop.…”