Curvature and rank of Teichmuller space

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, decomposidom P((S)) to prove that the Weil-Petersson metic 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);wften d(S) < 2, we combine the hyperbolicity of the comprex of curves and the reative 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.