#### Volume 6, issue 1 (2002)

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Convex cocompact subgroups of mapping class groups

### Benson Farb and Lee Mosher

Geometry & Topology 6 (2002) 91–152
 arXiv: math.GR/0106190
##### Abstract

We develop a theory of convex cocompact subgroups of the mapping class group $MCG$ of a closed, oriented surface $S$ of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup $G$ of $MCG$ defining an extension $1\to {\pi }_{1}\left(S\right)\to {\Gamma }_{G}\to G\to 1$, we prove that if ${\Gamma }_{G}$ is a word hyperbolic group then $G$ is a convex cocompact subgroup of $MCG$. When $G$ is free and convex cocompact, it is called a Schottky subgroup.

##### Keywords
mapping class group, Schottky subgroup, cocompact subgroup, convexity, pseudo-Anosov
##### Mathematical Subject Classification 2000
Primary: 20F67, 20F65
Secondary: 57M07, 57S25