Curvature and rank of Teichmüller space
Abstract
Let S be a surface with genus g and n boundary components and let d(S) = 3g3+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 WeilPetersson metric on Teichmuller space Teich(S) is Gromovhyperbolic if and only if d(S) <= 2. When d(S) >= 3 the WeilPetersson metric has higher rank in the sense of Gromov (it admits a quasiisometric 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 Gromovhyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromovhyperbolic 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2001
 arXiv:
 arXiv:math/0109045
 Bibcode:
 2001math......9045B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry;
 Mathematics  Group Theory
 EPrint:
 23 pages, 1 figure