Bouts des variétés hyperboliques de dimension 3. (Ends of hyperbolic 3-dimensional manifolds) (Q1119931)

The author proves a number of powerful theorems which settle in the affirmative a conjecture of \textit{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 \textit{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 \textit{L. V. Ahlfors} [Am. J. Math. 86, 413-429 (1964; Zbl 0133.042)] concerning the limit set of a finitely generated Kleinian group.