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29 March 2024
 
  » arxiv » math.GT/0109048

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The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
Jeffrey F. Brock ;
Date 7 Sep 2001
Subject Geometric Topology; Dynamical Systems MSC-class: 2000 MSC. Primary 30F40; Secondary 30F60, 37F30 | math.GT math.DS
AbstractWe introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the Weil-Petersson distance with the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian as well as a new finiteness criterion for geometric limits.
Source arXiv, math.GT/0109048
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