Convex cocompact subgroups of mapping class groups

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.