Kleinian Groups and Hyperbolic 3-Manifolds
Proceedings of the Warwick Workshop, September 11–14, 2001
Edited by Y. Komori
Edited by V. Markovic
Edited by C. Series
Publisher: Cambridge University Press
Print Publication Year: 2003
Online Publication Date:September 2009
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511542817.003
This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angles) can be turned into effective control on the deformations that do exist. This leads to precise analytic and geometric versions of the idea that hyperbolic structures with short geodesics are close to hyperbolic structures with cusps. The paper also outlines a new harmonic deformation theory which applies whenever there is a sufficiently large embedded tube around the singular locus, removing the previous restriction to cone angles at most 2π.
The local rigidity theorem of Weil [Wei60] and Garland [Gar67] for complete, finite volume hyperbolic manifolds states that there is no non-trivial deformation of such a structure through complete hyperbolic structures if the manifold has dimension at least 3. If the manifold is closed, the condition that the structures be complete is automatically satisfied. However, if the manifold is non-compact, there may be deformations through incomplete structures. This cannot happen in dimensions greater than 3 (Garland-Raghunathan [GRa63]); but there are always non-trivial deformations in dimension 3 (Thurston [Thu79]) in the non-compact case.
In [HK98] this rigidity theory is extended to a class of finite volume, orientable 3-dimensional hyperbolic cone-manifolds, i.e. hyperbolic structures on 3-manifolds with cone-like singularities along a knot or link.