On the geometry of graphs associated to infinite-type surfaces

Abstract

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 the main results of J. Bavard [2] and Aramayona-Fossas-Parlier [1].In the second part of the paper, we obtain a number of results on the geometry of connected, 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.

Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

“…However, in both of these proofs, the constant of hyperbolicity cannot be shown to be independent of the topological type of the surface. Aramayona-Valdez ask in [3] whether the graphs of nonseparating curves are uniformly hyperbolic (that is, whether the constant of hyperbolicity can be taken to be independent of the topological type of the surface). We prove in this paper that this indeed the case.…”

“…Nonetheless, many authors have recently investigated various graphs acted on by big mapping class groups, in analogy with the curve graphs of finite type surfaces: [2], [3], [4], [9], [10]. Note that the curve graph itself for an infinite type surface always has diameter two and therefore is trivial up to quasi-isometry.…”

“…The above constructions give infinite diameter hyperbolic graphs acted on by Mod(S) for certain surfaces S with infinite genus or at least one isolated puncture. Desiring to circumvent this restriction, in [3] Aramayona-Valdez studied the graph of nonseparating curves for a finite genus oriented surface, establishing that this graph is infinite diameter and that it is hyperbolic if and only if the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Therefore we have the following: Corollary 1.2.…”

Graphs of curves on infinite-type surfaces with mapping class group actions

“…Many authors have recently investigated numerous analogues of curve, arc, and pants graphs for specific classes of infinite-type surfaces [2,3,4,17] (see §1.1). The cases where the above constructions yield geometrically interesting actions of the mapping class group require the underlying surface to have a finite positive number of isolated planar ends or positive finite genus, respectively.…”

Two remarks about multicurve graphs on infinite-type surfaces

“…There has been a recent surge of interest in finding combinatorial models for mapping class groups of infinite-type surfaces, see [3,4,5,6,7,9,10,11,16].…”