On the arc and curve complex of a surface

TL;DR

This paper investigates the automorphism group and geometric properties of the arc and curve complex of a surface, establishing its relation to the mapping class group and hyperbolicity, with implications for understanding surface topology.

Contribution

It proves that the automorphism group of the arc and curve complex matches the extended mapping class group for most surfaces and shows the complex is Gromov-hyperbolic, extending known results.

Findings

Automorphism group of $AC(S)$ equals the extended mapping class group for most surfaces.

The link of each vertex characterizes the type of curve or arc.

$AC(S)$ is Gromov-hyperbolic under certain conditions.

Abstract

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of $AC(S)$ coincides with the natural image of the extended mapping class group of $S$ in that group. We also show that for any vertex of $AC(S)$, the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in $S$ that represents that vertex. We also give a proof of the fact if $S$ is not a sphere with at most three punctures, then the natural embedding of the curve complex of $S$ in $AC(S)$ is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on $S$, was already known. As a corollary, $AC(S)$ is Gromov-hyperbolic.