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The
shape of hyperbolic Dehn surgery space
Craig D Hodgson and Steven P Kerckhoff |
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Geometry & Topology 12 (2008) 1033–1090
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Abstract |
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In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic –manifolds with “tubular boundary”. In particular, this applies to complements of tubes of radius at least around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston’s hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of “uniform” size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (eg volumes and core geodesic lengths) during hyperbolic Dehn filling. This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups. |
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Dedicated to Bill Thurston on his 60th birthday |
Keywords
hyperbolic Dehn surgery, harmonic deformation
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Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57N10
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References |
Forward citations |
Publication
Received: 22 September 2007
Accepted: 20 February 2008
Published: 12 May 2008
Proposed: Dave Gabai
Seconded: Cameron Gordon, Walter Neumann
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Authors |
| Craig D Hodgson | |
| Department of Mathematics and
Statistics University of Melbourne Victoria 3010 Australia |
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| Steven P Kerckhoff | |
| Department of Mathematics Stanford University Stanford, CA 94305 USA |
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