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 → π 1 (S) → Γ G → G → 1, we prove that if Γ 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 π 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. AMS Classification numbers Primary: 20F67, 20F65Secondary: 57M07, 57S25

Subgroups of mapping class groups from the geometrical viewpoint

“…B. Farb and L. Mosher have extended this notion to subgroups of Mod(S) [27]. A finitely generated subgroup of Mod(S) is said to be convex cocompact if it satisfies one of the conditions in the following theorem.…”

“…Sketch of proof from [27]. The proof of McCarthy's Tits Alternative implies that for sufficiently large m, the group ϕ m , ψ m is free.…”

Surface subgroups of mapping class groups

“…In [13] the notion of convex cocompactness of subgroups of Γ g is given, and an attempt to extend this to geometrically finite is proposed in [33]. In particular, the question of geometrical finiteness of the groups in Theorem 3.1 as well as others constructed in [23] is posed.…”

“…In particular, the question of geometrical finiteness of the groups in Theorem 3.1 as well as others constructed in [23] is posed. We refer the reader to [13] and [33] for more on the notions of convex cocompact and geometrically finite subgroups in the context of Γ g , as well as for questions concerning the geometrical finiteness of various subgroups.…”

“…To that end, we are particularly interested in the nature of surface subgroups of Γ g . For an exploration of other analogies between Γ g and Kleinian groups see [13], [19] and [33].…”