Curvature and rank of Teichmüller space

Abstract

Let S be a surface with genus g and n boundary components, and let d(S) = 3 g - 3 + n denote the number of curves in any pants decomposition of S . We employ metric properties of the graph of pants decompositions P ( S ) to prove that the Weil-Petersson metric on Teichmüller space Teich ( S ) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] k , k ≥ 2); when d(S) ≤ 2, we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of P ( S ) to prove Gromov-hyperbolicity. We prove moreover that Teich ( S ) admits no geodesically complete, Mod ( S )-invariant, Gromov-hyperbolic metric of finite covolume when d(S) ≥ 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) ≥ 2.