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Convex cocompact subgroups of mapping class groups. (English) Zbl 1021.20034
We recall first a classical definition. Let $$H^n$$ denote the $$n$$-dimensional hyperbolic space. A ‘convex cocompact subgroup’ of $$\text{Isom}(H^n)$$ is a discrete subgroup $$G$$ of $$\text{Isom}(H^n)$$ which acts cocompactly on the convex hull in $$H^n$$ of its limit set $$\Lambda_G$$. A ‘Schottky group’ is a convex cocompact subgroup of $$\text{Isom}(H^n)$$ which is free. The study of convex cocompact groups of isometries and of Schottky groups is an important subject in three-dimensional hyperbolic geometry. Let $$S$$ be a closed oriented surface of genus $$\geq 2$$. In the paper under review, the authors develop an “analogous” theory of convex cocompact subgroups and Schottky subgroups for the mapping class group $$MCG$$ of $$S$$, in terms of the action of $$MCG$$ on the Teichmüller space $${\mathcal T}$$ of $$S$$. Here, Teichmüller space is equipped with its Thurston boundary $$\mathcal{PMF}$$. The first result that the authors give is a theorem which gives several characterizations of convex cocompact subgroups:
Theorem 1: Let $$G$$ be a finitely generated subgroup of $$MCG$$. Then, the following are equivalent: – Some orbit of $$G$$ is quasiconvex in $$\mathcal T$$. – Every orbit of $$G$$ is quasiconvex in $$\mathcal T$$. – $$G$$ is word hyperbolic, and there is a $$G$$-equivariant embedding $$\partial f\colon\partial G\to\mathcal{PMF}$$ with image $$\Lambda_G$$ such that the following two properties hold: (i) Any two distinct points $$\xi$$ and $$\eta\in\Lambda_G$$ are the endpoints of a unique geodesic $$[\xi,\eta]$$ in $$\mathcal T$$ and (ii) If $$\text{WH}_G$$ is the “weak hull” of $$G$$, that is, the union of geodesics in $$\mathcal T$$ with distinct endpoints in $$\Lambda_G$$, then the action of $$G$$ on $$\text{WH}_G$$ is cocompact, and if $$f\colon G\to\text{WH}_G$$ is any $$G$$-equivariant map, then $$f$$ is a quasi-isometry and the following map is continuous: ${\overline f}=f\cup\partial f\colon G\cup\partial G\to{\overline{\mathcal T}}={\mathcal T}\cup\mathcal{PMF}.$ A group $$G$$ satisfying one of the equivalent three conditions of Theorem 1 is said to be a ‘convex cocompact subgroup’ of $$MCG$$. The authors show that such a group satisfies several properties which are analogous to properties of convex cocompact subgroups of $$\text{Isom}(H^n)$$. For instance, every infinite order element of a convex cocompact subgroup of $$MCG$$ is pseudo-Anosov. The limit set $$\Lambda_G$$ is the smallest nontrivial set of $$\overline{\mathcal T}$$ which is invariant under the action of $$G$$, and the action of $$G$$ on $${\mathcal{PMF}}\setminus\Lambda_G$$ is properly discontinuous. The authors define then a ‘Schottky subgroup’ of $$MCG$$ to be a convex cocompact group which is free of finite rank. They show that Schottky subgroups exists in abundance. They prove that if $$\phi_1,\dots,\phi_n$$ are pseudo-Anosov elements of $$MCG$$ whose axes have pairwise disjoint endpoints in $$\mathcal{PMF}$$, then for all sufficiently large positive integers $$a_1,\dots,a_n$$, the mapping classes $$\phi_1^{a_1},\dots,\phi_n^{a_n}$$ freely generate a Schottky subgroup of $$MCG$$. The authors apply this theory to relate convex cocompactness of subgroups of $$MCG$$ with the large scale geometry of extensions of surface groups by subgroups of $$MCG$$. In particular, they prove that if $$G$$ is a subgroup of $$MCG$$ defining an extension $$1\to\pi_1(S)\to\Gamma_G\to G\to 1$$ and if $$G$$ is a word hyperbolic group, then $$G$$ is a convex cocompact subgroup of $$MCG$$. In the case where $$G$$ is a Schottky group, then the converse is also true, that is, a semidirect product of $$\pi_1(S)$$ by a free group $$G$$ is word hyperbolic if and only if $$G$$ is a Schottky subgroup of $$MCG$$. The paper contains several interesting open questions in that theory.

MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 57M07 Topological methods in group theory 57S25 Groups acting on specific manifolds 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth
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