  The girth of a finitely generated group G is the supremum of the girth of
Cayley graphs for G over all finite generating sets. Let G be a finitely
generated subgroup of the mapping class group Mod(S), where S is a compact
orientable surface. Then, either G is virtually abelian or it has infinite
girth; moreover, if we assume that G is not infinite cyclic, these alternatives
are mutually exclusive.
