JournalsggdVol. 8, No. 1pp. 225–244

The girth alternative for mapping class groups

  • Kei Nakamura

    University of California, Davis, USA
The girth alternative for mapping class groups cover

Abstract

The girth of a finitely generated group GG is defined to be the supremum of the girth of its Cayley graphs. Let GG be a finitely generated subgroup of the mapping class group ModΣ_\Sigma, where Σ\Sigma is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that GG 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. 225–244

DOI 10.4171/GGD/223