Curvature and rank of Teichm\"uller space

Abstract

Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller 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 R^k, k >= 2); when d(S) <= 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromov-hyperbolicity. We prove moreover that Teich(S) admits no geodesically complete 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.