Girth-Alternative for Mapping Class Groups

Kei Nakamura (Temple University)

The girth of a finitely generated group G is defined to be the supremum of the girth of Cayley graphs for G over all finite generating sets. For a finitely generated subgroup G of the mapping class group Mod(S), where S is a compact orientable surface, we show the following "girth-alternative": G is either a non-cyclic group with infinite girth or a virtually abelian group; moreover, these alternatives are mutually exclusive. We will also briefly discuss how the above statement can be related to the notion of "free-like" groups.