Curvature and rank of Teichmüller space

By J. B. Brock, Benson Farb. 2001

95 citations recorded by the source corpus.

Let 5 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 P(S) to prove that the Weil-Petersson metric on Teichmiiller space Teich (S) is Gromov-hyperbolic if and only if d(S) 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of Rk,k > 2); when d(S) 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) > 2. 1. Introduction. The Weil-Petersson metric on Teichmiiller space Teich (5) has many curious properties. It is a Riemannian metric with negative sectional curvature, but its curvatures are not bounded away from zero or negative infinity. It is geodesically convex, but it is not complete. In this paper we show that in spite of exhibiting negative curvature behavior, the Weil-Petersson metric is not coarsely negatively curved except for topologically simple surfaces S. Our main theorem answers a question of Bowditch (Be, Question 11.4).

Subfields: Applied Mathematics, Geometry and Topology, Oncology

View Paper: https://www.semanticscholar.org/paper/8ae464dcd0d866e5b6e3b1f921ead48893d6ee64

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Citations in the corpus

Contracting boundaries of CAT(0) spaces (2013, background): “These notions have proved fruitful in analyzing right angled Artin groups [3], Teichmüller space [2, 7, 8, 9], the mapping class group [2], CAT(0) spaces [4, 22, 29], and Out(Fn) [1] among others (see also [13, 14, 19, 20]).”
Dimension and rank for mapping class groups (2005, methodology): “) We also note that the lower bound on the geometric rank of Teichmüller space is obtained in [BF].”
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space (2005, unknown): “20F67; 30F60, 57M07, 20F65”
Relatively hyperbolic groups: geometry and quasi-isometric invariance (2006, result): “Other similar notions can be found in [BF01], and in [HK05] in the context of CAT(0) metric spaces.”
Quasiflats in hierarchically hyperbolic spaces (2017, background): “esolves Brock’s question in the armative. The only previously known cases of this question were: in the rank one cases, where QUASIFLATS IN HIERARCHICALLY HYPERBOLIC SPACES 3 the space is hyperbolic [BF06]; and, in the three rank two cases, where the space is relatively hyperbolic, [BM08, Theorem 3]”