Kleinian Groups and Hyperbolic 3-Manifolds
Proceedings of the Warwick Workshop, September 11–14, 2001
Gero Kleineidam
Juan Souto
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/CBO9780511542817.006
Subjects: Algebra, Geometry and topology
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Abstract
We study the relationship between the geometry and the topology of the ends of a hyperbolic 3-manifold M whose fundamental group is not a free group. We prove that a compressible geometrically infinite end of M is tame if there is a Masur domain lamination which is not realized by a pleated surface. It is due to Canary that, in the absence of rank-1-cusps, this condition is also necessary.
Introduction
Marden [Mar74] proved that every geometrically finite hyperbolic 3-manifold is tame, i.e. homeomorphic to the interior of a compact manifold, and he conjectured that this holds for any hyperbolic 3-manifold M with finitely generated fundamental group. By a theorem of Scott [Sco73b], M contains a core, a compact submanifold C such that the inclusion of C into M is a homotopy equivalence. Moreover, any two cores are homeomorphic by a homeomorphism in the correct homotopy class [MMS85].
So the discussion of the tameness of M boils down to a discussion of the ends of M. The ends are in a bijective correspondence with the boundary components of the compact core C. An end E is said to be tame if it has a neighborhood homeomorphic to the product of the corresponding boundary component ∂E of C with the half-line. Hence M is tame if its ends are.
Since M is aspherical, either π1(M) = 1 or every boundary component of C is a closed surface of genus at least 1.
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