Groups, Geometry, and Dynamics

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Volume 8, Issue 1, 2014, pp. 225–244
DOI: 10.4171/GGD/223

Published online: 2014-05-13

The girth alternative for mapping class groups

Kei Nakamura[1]

(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