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Algebraic limits of geometrically finite manifolds are tame. (English) Zbl 1095.30034
An irreducible 3-manifold is called tame if it is homeomorphic to the interior of a compact 3-manifold. Marsden’s Tameness Conjecture asserts that every hyperbolic 3-manifold with a finitely generated fundamental group is tame. This conjecture has been proved recently by D. Calegari and D. Gabai [J. Am. Math. Soc. 19, No. 2, 385–446 (2006; Zbl 1090.57010)] as the culmination of the work of many authors.
The paper under review was intended as a contribution towards the proof of this conjecture. It completes the argument of [Publ. Math., Inst. Hautes Étud. Sci. 98, 145–166 (2003; Zbl 1060.30054)] of the authors in collaboration with K. Bromberg and R. Evans. Together they prove that an algebraic limit of geometrically finite hyperbolic manifolds is tame. The proof involves an intricate and delicate combination of the methods of 3-dimensional topology with those of the theory of Kleinian groups.

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
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