- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:00
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2014
1
The girth alternative for mapping class groups
Kei
Nakamura
University of California, DAVIS, UNITED STATES
Mapping class groups, girth of Cayley graphs
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.
Group theory and generalizations
225
244
10.4171/GGD/223
http://www.ems-ph.org/doi/10.4171/GGD/223