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/CBO9780511542817.010
Subjects: Algebra, Geometry and topology
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Abstract
This work is a detailed study of the space of quasifuchsian once punctured torus groups in terms of their Ford (isometric) fundamental polyhedra. The key is a detailed analysis of how the pattern of isometric planes bounding the polyhedra change as one varies the group.
Introduction
One possible approach to “Kleinian groups” is to ask: “How do they look?” It makes sense when the groups have been associated with natural fundamental polyhedrons.
In the case of Fuchsian groups, satisfactory answers were known to Fricke [FK26], but generally the situation becomes rather complicated.
It is natural to restrict the considerations to finitely generated groups and, hence-forward, we shall do so, since, in many respects, the class of groups which cannot be generated by a finite number of Möbius transformations seems to be too extensive for general studies – see for instance the examples of Abikoff [Abi71], [Abi73].
In preparation for an intuitive treatment of Kleinian groups, Ahlfors' finiteness paper [Ahl65a] contains much of the ground material. Also, it indicates one direction in which the above question might be specified, for instance, in order to attack the problems related to the characteristics of the limit sets, namely, whether the set of limit points situated on the boundary of the Dirichlet fundamental polyhedron is finite. Ahlfors proved that it has zero area [Ahl65b].
Contributing techniques from 3-dimensional topology, Marden [Mar74] described those groups which have finite sided polyhedrons and observed that they are stable in the sense of Bers [Ber70b].
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