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

*(French)*Zbl 0671.57008The 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.

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

##### MSC:

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) |