On limits of tame hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: On limits of tame hyperbolic 3-manifolds
zbMATH Open document ID: 0856.57011
author: Yair N. Minsky
author: Richard D. Canary
published in: Journal of Differential Geometry
publication date: 1996-08-05T00:00:00Z
review text: A hyperbolic 3-manifold is said to be topologically tame if it is homeomorphic to the interior of a compact 3-manifold. The main conjecture in the theory of Kleinian groups is probably the one that states that all hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame. One approach to this conjecture consists in trying to prove that limits in various senses of manifolds which are known to be topologically tame are topologically tame. In particular, this was the approach to Thurston who proved in the 1970's that if \(M\) is a compact 3-manifold with incompressible boundary, then any type-preserving algebraic limit of a sequence of geometrically finite hyperbolic 3-manifolds homeomorphic to the interior of \(M\) is topologically tame, and is itself homeomorphic to the interior of \(M\). In this paper, the authors prove, in the same line of ideas, that in the absence of parabolics, strong limits of topologically tame hyperbolic 3-manifolds are topologically tame. This strong convergence here means that the corresponding sequence of representations in the isometry group of the 3-dimensional hyperbolic space converges algebraically and that the images of the representations converge geometrically. As the authors say, a similar result has been proved recently, using different arguments, by K. Ohshika. The authors use then Ohshika's work to prove that again in the absence of parabolics, any algebraic limit of a sequence of topologically tame hyperbolic 3-manifolds, whose fundamental group is not a free product of surface groups and free groups, is topologically tame.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 895016
zbMATH Keywords: geometric convergence
zbMATH Keywords: algebraic convergence
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: topologically tame
zbMATH Keywords: Kleinian groups
zbMATH Keywords: fundamental groups
Wikidata QID: Q115177843
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214457896
OpenAlex ID: W1488092040
DOI: 10.4310/JDG/1214457896
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description: scientific article; zbMATH DE number 895016