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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
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by Jeffrey F. Brock PDF
J. Amer. Math. Soc. 16 (2003), 495-535 Request permission

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
  • Email: brock@math.uchicago.edu
  • 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.
  • © Copyright 2003 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 495-535
  • MSC (2000): Primary 30F40; Secondary 30F60, 37F30
  • DOI: https://doi.org/10.1090/S0894-0347-03-00424-7
  • MathSciNet review: 1969203