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
Online ISBN:9780511542817
Paperback ISBN:9780521540131
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511542817.003
Subjects: Algebra, Geometry and topology
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Abstract
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.
Introduction
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.
pp. i-iv
pp. v-vi
pp. vii-viii
Part I - Hyperbolic 3-manifolds : Read PDF
pp. 1-2
Combinatorial and geometrical aspects of hyperbolic 3-manifolds : Read PDF
pp. 3-40
Harmonic deformations of hyperbolic 3-manifolds : Read PDF
pp. 41-74
Cone-manifolds and the density conjecture : Read PDF
pp. 75-94
Les géodésiques fermées d'une variété hyperbolique en tant que nœuds : Read PDF
pp. 95-104
Ending laminations in the Masur domain : Read PDF
pp. 105-130
Quasi-arcs in the limit set of a singly degenerate group with bounded geometry : Read PDF
pp. 131-144
On hyperbolic and spherical volumes for knot and link cone-manifolds : Read PDF
pp. 145-164
Remarks on the curve complex: classification of surface homeomorphisms : Read PDF
pp. 165-180
Part II - Once-punctured tori : Read PDF
pp. 181-182
On pairs of once-punctured tori : Read PDF
pp. 183-208
Comparing two convex hull constructions for cusped hyperbolic manifolds : Read PDF
pp. 209-246
Jørgensen's picture of punctured torus groups and its refinement : Read PDF
pp. 247-274
Tetrahedral decomposition of punctured torus bundles : Read PDF
pp. 275-292
On the boundary of the Earle slice for punctured torus groups : Read PDF
pp. 293-304
Part III - Related topics : Read PDF
pp. 305-306
Variations on a theme of Horowitz : Read PDF
pp. 307-342
Complex angle scaling : Read PDF
pp. 343-362
Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds : Read PDF
pp. 363-384