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Shadows of mapping class groups: capturing convex cocompactness. (English) Zbl 1282.20046
From the introduction: We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher [Geom. Topol. 6, 91-152 (2002; Zbl 1021.20034)].
A Kleinian group $$\Gamma$$ is a discrete subgroup of $$\text{PSL}_2(\mathbb C)$$. When non-elementary, such a group possesses a unique non-empty minimal closed invariant subset $$\Lambda_\Gamma$$ of the Riemann sphere, called the limit set. A Kleinian group acts properly discontinuously on the complement $$\Delta_\Gamma$$ of $$\Lambda_\Gamma$$ and so this set is called the domain of discontinuity.
Let $$S$$ denote an oriented complete hyperbolic surface of finite area, $$\text{Mod}(S)=\pi_0(\text{Homeo}^+(S))$$ its group of orientation preserving self-homeomorphisms up to isotopy, and $$\mathcal T(S)$$ the Teichmüller space of $$S$$ equipped with Teichmüller’s metric.
The mapping class group $$\text{Mod}(S)$$ acts on Teichmüller space $$\mathcal T(S)$$ by isometries, and W. Thurston discovered a $$\text{Mod}(S)$$-equivariant compactification of $$\mathcal T(S)$$ by an ideal sphere, the sphere of compactly supported projective measured laminations $$\mathbb P\mathcal{ML}(S)$$. J. McCarthy and A. Papadopoulos have shown that a subgroup $$G$$ of $$\text{Mod}(S)$$ has a well-defined limit set $$\Lambda_\Gamma$$, although it need not be unique or minimal, and that there is a certain enlargement $$Z\Lambda_\Gamma$$ of $$\Lambda_\Gamma$$ on whose complement $$G$$ acts properly discontinuously [Comment. Math. Helv. 64, No. 1, 133-166 (1989; Zbl 0681.57002)]. So such a group has a domain of discontinuity $$\Delta_G=\mathbb P\mathcal{ML}(S)-Z\Lambda_G$$.
Our purpose here is to strengthen the analogy between convex cocompact Kleinian groups and their cousins in the mapping class group. Our first main result is the following:
Theorem 1.2. Given a finitely generated subgroup $$G$$ of $$\text{Mod}(S)$$, the following statements are equivalent:
$$\bullet$$ $$G$$ is convex cocompact.
$$\bullet$$ The weak hull $$\mathfrak H_G$$ is defined and $$G$$ acts cocompactly on $$HG$$.
$$\bullet$$ Every limit point of $$G$$ is conical.
$$\bullet$$ $$G$$ acts cocompactly on $$\mathcal T(S)\cup\Delta_G$$.
Theorem 3.9 provides much stronger information than is needed to prove Theorem 1.2. We state it here as it may be of independent interest.
Theorem 3.9. Let $$G$$ be a subgroup of $$\text{Mod}(S)$$. If $$\Delta_G\neq\emptyset$$ and $$G$$ acts cocompactly on $$\Delta_G$$, then every lamination in $$\Lambda_G$$ is uniquely ergodic, $$Z\Lambda_G=\Lambda_G$$, and $$\mathcal H_G$$ is defined and cobounded. Furthermore, $$G$$ has a finite index subgroup all of whose non-identity elements are pseudo-Anosov.
Our second main theorem is the following:
Theorem 1.3. A finitely generated subgroup $$G$$ of $$\text{Mod}(S)$$ is convex cocompact if and only if sending $$G$$ to an orbit in the complex of curves defines a quasi-isometric embedding $$G\to\mathcal C(S)$$.
Remark. This theorem was independently discovered by U. Hamenstädt [Word hyperbolic extensions of surface groups, arXiv:math.GT/0505244].

##### MSC:
 20F65 Geometric group theory 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 30F60 Teichmüller theory for Riemann surfaces 57M07 Topological methods in group theory 57R50 Differential topological aspects of diffeomorphisms 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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