The shape of hyperbolic Dehn surgery space

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 $R_{0} = arctanh (1 ∕ \sqrt{3}) \approx 0.65848$ 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.

Dedicated to Bill Thurston on his 60th birthday

Keywords

hyperbolic Dehn surgery, harmonic deformation

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 57N10

References

Forward citations

Publication

Received: 22 September 2007

Accepted: 20 February 2008

Published: 12 May 2008

Proposed: Dave Gabai

Seconded: Cameron Gordon, Walter Neumann

Authors

Craig D Hodgson
Department of Mathematics and Statistics University of Melbourne Victoria 3010 Australia

Steven P Kerckhoff
Department of Mathematics Stanford University Stanford, CA 94305 USA

Volume 12, issue 2 (2008)

Craig D Hodgson and Steven P Kerckhoff

Abstract