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A covering theorem for hyperbolic 3-manifolds and its applications. (English) Zbl 0863.57010
This paper is dealing with a well-known conjecture on hyperbolic 3-manifolds, of whether a hyperbolic 3-manifold \(N\) with finitely generated fundamental group is topologically tame, i.e. \(N\) is homeomorphic to the interior of a compact 3-manifold. Actually, the main theorem describes how geometrically infinite ends of topologically tame hyperbolic 3-manifolds cover. This result generalizes a theorem of W. P. Thurston [The geometry and topology of 3-manifolds, Princeton Lect. Notes] proved for geometrically tame manifolds whose compact cores have incompressible boundary, and has many applications [the author, in ‘Low dimensional topology’, Conf. Proc. Lect. Notes Geom. Topol. 3, 21-30 (1994; Zbl 0849.57014)]. Much of the paper addresses various aspects of the properties of geometrical finiteness and topological (geometrical) tameness in the context of Thurston’s work on hyperbolic structures. In particular, it gives a complete characterization of geometrically finite covers of an infinite volume topologically tame hyperbolic 3-manifold. The author also discusses the applications of his theorem to understanding algebraic and geometric limits of sequences of Kleinian groups.

MSC:
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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