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2008 The shape of hyperbolic Dehn surgery space
Craig Hodgson, Steven Kerckhoff
Geom. Topol. 12(2): 1033-1090 (2008). DOI: 10.2140/gt.2008.12.1033

Abstract

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3–manifolds with “tubular boundary”. In particular, this applies to complements of tubes of radius at least R0= arctanh(13)0.65848 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.

Citation

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Craig Hodgson. Steven Kerckhoff. "The shape of hyperbolic Dehn surgery space." Geom. Topol. 12 (2) 1033 - 1090, 2008. https://doi.org/10.2140/gt.2008.12.1033

Information

Received: 22 September 2007; Accepted: 20 February 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1144.57015
MathSciNet: MR2403805
Digital Object Identifier: 10.2140/gt.2008.12.1033

Subjects:
Primary: 57M50
Secondary: 57N10

Rights: Copyright © 2008 Mathematical Sciences Publishers

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