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 Teichmller space. Given a subgroup G of MCG defining an extension 1 1 (S) G G 1, we prove that if G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky subgroup of MCG, the converse is true as well; a semidirect product of 1 (S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G = Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.