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The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

Author: Jeffrey F. Brock
Journal: J. Amer. Math. Soc. 16 (2003), 495-535
MSC (2000): Primary 30F40; Secondary 30F60, 37F30
Published electronically: March 4, 2003
MathSciNet review: 1969203
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Abstract: We present a coarse interpretation of the Weil-Petersson distance $d_{\mathrm {WP}}(X,Y)$ between two finite area hyperbolic Riemann surfaces $X$ and $Y$ using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph 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 conformal boundary is comparable to the Weil-Petersson distance $d_{\mathrm {WP}}(X,Y)$. In applications, we relate the Weil-Petersson distance to the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian for $Q(X,Y)$, and give a new finiteness criterion for geometric limits.

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Additional Information

Jeffrey F. Brock
Affiliation: Mathematics Department, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Keywords: Hyperbolic manifold, Kleinian group, pants decomposition, Teichmüller space, Weil-Petersson metric, limit set
Received by editor(s): October 30, 2001
Published electronically: March 4, 2003
Additional Notes: Research partially supported by NSF grant DMS-0072133 and an NSF postdoctoral fellowship.
Article copyright: © Copyright 2003 American Mathematical Society