The girth of a finitely generated group is defined to be the supremum of the girth of its Cayley graphs. Let be a finitely generated subgroup of the mapping class group Mod, where is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that 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.
Cite this article
Kei Nakamura, The girth alternative for mapping class groups. Groups Geom. Dyn. 8 (2014), no. 1, pp. 225–244DOI 10.4171/GGD/223