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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

Volumes of hyperbolic $3$-manifolds. Notes on a paper of Gabai, Meyerhoff, and Milley


Authors: T. H. Marshall and G. J. Martin
Journal: Conform. Geom. Dyn. 7 (2003), 34-48
MSC (2000): Primary 30F40, 30D50, 57M50
Published electronically: June 17, 2003
MathSciNet review: 1992036
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Abstract: We present a new approach and improvements to the recent results of Gabai, Meyerhoff and Milley concerning tubes and short geodesics in hyperbolic $3$-manifolds. We establish the following two facts: if a hyperbolic $3$-manifold admits an embedded tubular neighbourhood of radius $r_0>1.32$ about any closed geodesic, then its volume exceeds that of the Weeks manifold. If the shortest geodesic of $M$ has length less than $\ell_0<0.1$, then its volume also exceeds that of the Weeks manifold.


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Additional Information

T. H. Marshall
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
Email: t_marshall@math.auckland.ac.nz

G. J. Martin
Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
Email: martin@math.auckland.ac.nz

DOI: http://dx.doi.org/10.1090/S1088-4173-03-00081-X
PII: S 1088-4173(03)00081-X
Received by editor(s): August 30, 2001
Received by editor(s) in revised form: April 10, 2003
Published electronically: June 17, 2003
Additional Notes: Research supported in part by the N. Z. Marsden Fund and the N. Z. Royal Society (James Cook Fellowship)
Article copyright: © Copyright 2003 American Mathematical Society