Shadows of mapping class groups: capturing convex cocompactness.

*(English)*Zbl 1282.20046From 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].

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 \(\mathbb{C}\) |