On the geometry of graphs associated to infinite-type surfaces

TL;DR

This paper investigates the large-scale geometry of subgraphs of the arc and curve complexes of infinite-type surfaces, revealing new insights into their structure and symmetry properties.

Contribution

It characterizes the geometry of invariant subgraphs of arc and curve complexes for infinite-type surfaces, extending previous results to a broader class of surfaces.

Findings

Describes the large-scale geometry of invariant subgraphs

Recovers main results of Bavard and Aramayona-Fossas-Parlier

Provides new insights into the structure of infinite-type surface complexes

Abstract

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex $C(S)$ of $S$, provided they are invariant under a sufficiently big subgroup of the mapping class group $Mod(S)$. We obtain a number of consequences; in particular we recover the main results of J. Bavard and Aramayona-Fossas-Parlier .