Convex cocompact subgroups of mapping class groups
نویسندگان
چکیده
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 de ning an extension 1 ! 1(S) ! ΓG ! G ! 1, we prove that if ΓG 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: su ciently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup. AMS Classi cation numbers Primary: 20F67, 20F65 Secondary: 57M07, 57S25
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