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On limits of tame hyperbolic 3-manifolds. (English) Zbl 0856.57011
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.

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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