Geometry of the complex of curves. I: Hyperbolicity.

*(English)*Zbl 0941.32012In 1981, Harvey associated to a surface \(S\) a finite-dimensional simplicial complex \(C(S)\) called the complex of curves, which was intended to capture some of the combinatorial structure, and in particular to encode the asymptotic geometry of Teichmüller space in analogy with Tits buildings for symmetric spaces. The vertices of Harvey’s complex are homotopy classes of simple closed curves in \(S\), and the simplices are collections of curves that can be realized disjointly. By making each simplex a regular Euclidean simplex of sidelength 1, \(C(S)\) can be made into a complete geodesic metric space. This paper mainly studies the intrinsic geometry.

The main results are the following.

Theorem 1.1 (Hyperbolicity). Let \(S\) be an oriented surface of finite type. The curve complex \(C(S)\) is a \(\delta\)-hyperbolic metric space, where \(\delta\) depends on \(S\). Except when \(S\) is a sphere with 3 or fewer punctures, \(C(S)\) has infinite diameter.

Let \(T(S)\) be the Teichmüller space of \(S\) and \(\text{Ext}_x(\alpha)\) be the extremal length of simple closed curve \(\alpha\) on \(S\). For a small \(\varepsilon_0>0\), let \(H_{\alpha}= \{x\in T(S): \text{Ext}_x (\alpha)\leq \varepsilon_0\}\). Theorem 1.2 (relative Hyperbolicity 1). The Teichmüller space \(T(S)\) is relatively hyperbolic with respect to the family of regions \(\{H_\alpha\}\).

This theorem implies that the regions \(\#\{H_\alpha\}\) are the only obstructions to hyperbolicity of Teichmüller space.

Theorem 1.3 (Relatively Hyperbolicity 2). The group \(\text{Mod}(S)\) is relatively hyperbolic with respect to left-coset of a finite collection of stabilizers of curves.

The proof of the above results uses Teichmüller geometry in an essential way.

The main results are the following.

Theorem 1.1 (Hyperbolicity). Let \(S\) be an oriented surface of finite type. The curve complex \(C(S)\) is a \(\delta\)-hyperbolic metric space, where \(\delta\) depends on \(S\). Except when \(S\) is a sphere with 3 or fewer punctures, \(C(S)\) has infinite diameter.

Let \(T(S)\) be the Teichmüller space of \(S\) and \(\text{Ext}_x(\alpha)\) be the extremal length of simple closed curve \(\alpha\) on \(S\). For a small \(\varepsilon_0>0\), let \(H_{\alpha}= \{x\in T(S): \text{Ext}_x (\alpha)\leq \varepsilon_0\}\). Theorem 1.2 (relative Hyperbolicity 1). The Teichmüller space \(T(S)\) is relatively hyperbolic with respect to the family of regions \(\{H_\alpha\}\).

This theorem implies that the regions \(\#\{H_\alpha\}\) are the only obstructions to hyperbolicity of Teichmüller space.

Theorem 1.3 (Relatively Hyperbolicity 2). The group \(\text{Mod}(S)\) is relatively hyperbolic with respect to left-coset of a finite collection of stabilizers of curves.

The proof of the above results uses Teichmüller geometry in an essential way.

Reviewer: Lixin Liu (Guangzhou)