The girth alternative for mapping class groups

Abstract

The girth of a finitely generated group $G$ is defined to be the supremum of the girth of its Cayley graphs. Let $G$ be a finitely generated subgroup of the mapping class group Mod${}_{\Sigma }$, where $\Sigma$ is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that $G$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; these alternatives are mutually exclusive. The proof is based on a simple dynamical criterion for a finitely generated group to have infinite girth, which may be of independent interest.