Bouts des variétés hyperboliques de dimension 3. (Ends of hyperbolic 3-dimensional manifolds). (French) Zbl 0671.57008

The author proves a number of powerful theorems which settle in the affirmative a conjecture of W. Thurston [The geometry and topology of 3-manifolds, Lecture Notes, Princeton Univ. 1976-1979] that the ends of a closed surface are geometrically tame.
One consequence of the author’s result is a partial proof of A. Marden’s conjecture [Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014)] that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact manifold. This consequence was noted by Thurston; namely that his conjecture implies Marden’s conjecture if the fundamental group of the manifold is not a (non-trivial) free product.
Another consequence (also noted by Thurston) of the Thurston conjecture settles a question of L. V. Ahlfors [Am. J. Math. 86, 413-429 (1964; Zbl 0133.042)] concerning the limit set of a finitely generated Kleinian group.
Reviewer: L.P.Neuwirth


57N10 Topology of general \(3\)-manifolds (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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