# A note on the curve complex of the 3-holed projective plane

## TL;DR

This paper studies the curve complex of a 3-holed projective plane, showing it can be exhausted by finite rigid sets, its automorphism group matches the mapping class group, and it is quasi-isometric to a tree.

## Contribution

It establishes a finite rigid set exhaustion of the curve complex and proves the automorphism group equals the mapping class group, also showing the complex's quasi-isometry to a tree.

## Key findings

- Curve complex admits an exhaustion by finite rigid sets
- Automorphism group is isomorphic to the mapping class group
- Curve complex is quasi-isometric to a simplicial tree

## Abstract

Let $S$ be a projective plane with $3$ holes. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets. As a corollary, we obtain that the group of simplicial automorphisms of $\mathcal{C}(S)$ is isomorphic to the mapping class group $\mathrm{Mod}(S)$. We also prove that $\mathcal{C}(S)$ is quasi-isometric to a simplicial tree.