---
abstract: 'The *girth* of a finitely generated group $\Gamma$ is the supremum of the girth of Cayley graphs for $\Gamma$ over all finite generating sets. Let $\Gamma$ be a finitely generated subgroup of the mapping class group $\Mod({\Sigma})$, where ${\Sigma}$ is a compact orientable surface. Then, either $\Gamma$ is virtually abelian or it has infinite girth; moreover, if we assume that $\Gamma$ is not infinite cyclic, these alternatives are mutually exclusive.'
address: |
    Department of Mathematics\
    Temple University\
    Philadelphia, PA 19122
author:
- Kei Nakamura
bibliography:
- '../../../MyTeX/MyMaster.bib'
title: |
    The girth alternative for\
    mapping class groups
---

Introduction {#Introduction}
============

Let ${\Sigma}$ be a compact orientable surface, which is possibly disconnected. The *mapping class group* of ${\Sigma}$, denoted by $\Mod({\Sigma})$, is the group of isotopy classes of $\Homeo^+({\Sigma})$, i.e. the group of orientation preserving homeomorphisms of ${\Sigma}$. Mapping class groups have been studied extensively in complex analysis, low-dimensional topology, and geometric group theory for more than a century.

A fascinating aspect of the mapping class groups is that they share many properties with lattices in semi-simple Lie groups. One analogy between linear groups and mapping class groups can be seen in the following famous dichotomy regarding their subgroups, 