Geodesic
The girth alternative for mapping class groups
Kei Nakamura1
1University of California, Davis, USA
Groups, geometry, and dynamics, Tome 8 (2014) no. 1, pp. 225-244

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DOI

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 ModΣ​, where Σ 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.
DOI : 10.4171/ggd/223
Classification : 20-XX
Mots-clés : Mapping class groups, girth of Cayley graphs

Kei Nakamura  1

1 University of California, Davis, USA 
Kei Nakamura. The girth alternative for mapping class groups. Groups, geometry, and dynamics, Tome 8 (2014) no. 1, pp. 225-244. doi: 10.4171/ggd/223
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     number = {1},
     doi = {10.4171/ggd/223},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/223/}
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