Groups, Geometry, and Dynamics
Full-Text PDF (242 KB) | Metadata | Table of Contents | GGD summary
Published online: 2014-05-13
The girth alternative for mapping class groupsKei Nakamura (1) University of California, Davis, USA
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.
Keywords: Mapping class groups, girth of Cayley graphs
Nakamura Kei: The girth alternative for mapping class groups. Groups Geom. Dyn. 8 (2014), 225-244. doi: 10.4171/GGD/223