The shape of hyperbolic Dehn surgery space

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 $R_0=arctanh\left(1∕\sqrt{3}\right)\approx 0.65848$ 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.