an:05508776
Zbl 1282.20046
Kent, Richard P. IV; Leininger, Christopher J.
Shadows of mapping class groups: capturing convex cocompactness
EN
Geom. Funct. Anal. 18, No. 4, 1270-1325 (2008).
00243464
2008
j
20F65 30F40 30F60 57M07 57R50 37C85
convex cocompact Kleinian groups; convex cocompact subgroups; mapping class groups of surfaces
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 \textit{B. Farb} and \textit{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 Teichm??ller space of \(S\) equipped with Teichm??ller's metric.
The mapping class group \(\text{Mod}(S)\) acts on Teichm??ller 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 \(\mathbb P\mathcal{ML}(S)\). \textit{J. McCarthy} and \textit{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 \textit{U. Hamenst??dt} [Word hyperbolic extensions of surface groups, \url{arXiv:math.GT/0505244}].
Zbl 1021.20034; Zbl 0681.57002