###
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.