# 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.

## Key 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.