Arc and curve graphs for infinite-type surfaces

TL;DR

This paper investigates the properties of arc and curve graphs on infinite-type surfaces, demonstrating hyperbolicity in some cases and non-hyperbolicity in others, with implications for understanding their geometric structures.

Contribution

It extends known hyperbolicity results to a broader class of punctured surfaces and analyzes the geometric rank of specific curve graph subgraphs.

Findings

Arc graphs are connected, uniformly hyperbolic, and have infinite diameter.

Subgraphs of curve graphs intersecting a fixed separating curve have infinite diameter.

Certain curve graph subgraphs are not hyperbolic and have geometric rank 3.

Abstract

We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured surfaces. Second, we study the subgraph of the curve graph spanned by those elements which intersect a fixed separating curve on the surface. We show that this graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.