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Uniformization in dimension three. (Uniformisation en dimension trois.) (French) Zbl 0942.57013

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque 266, 113-137, Exp. No. 855 (2000).
This is a survey on several results which have been obtained by various people in the last 20 years, concerning the geometrization of three-manifolds. The main body of the paper concerns the following theorem of Thurston:
Hyperbolization theorem: Let \(M\) be a compact orientable three-manifold which contains a properly embedded orientable surface of negative curvature whose fundamental group injects in the fundamental group of \(M\). Suppose also that every embedded two-sphere in \(M\) bounds a three-ball. Then, the interior of \(M\) carries a complete hyperbolic structure if and only if every subgroup of \(\pi_1(M)\) which is isomorphic to \(Z+Z\) is conjugate to a subgroup of the fundamental group of a boundary component of \(M\).
In proving this theorem, Thurston established several new techniques, which have been used successfully by him and by others, to prove the most basic results on three-manifold geometry and topology. The author explains the main steps of the proof of this theorem. This involves, besides hyperbolic geometry, ideas of complex analysis, group actions, representation theory and topological techniques of Haken and Kneser. The author mentions, besides the work of Thurston, works of McMullen, Morgan-Shalen, Otal, Gabai and others. He discusses also special cases of the theorem, in particular the case where \(M\) fibers over the circle. The author presents also the main conjectures on the subject, in particular the following, which is due to Thurston:
Geometrization conjecture: The interior of a compact oriented irreducible three-manifold can be canonically decomposed along a finite family of essential tori into submanifolds which carry a complete homogeneous geometric structure.
The author explains how the works of Casson-Jungries and Gabai reduce the geometrization conjecture to two other conjectures:
Uniformization conjectures: Let \(M\) be an orientable irreducible closed three-manifold. Then:
1. Conjecture (Poincaré-Smith): \(M\) admits a spherical metric if and only if its fundamental group is finite.
2. Conjecture (Thurston): \(M\) admits a hyperbolic metric if and only if it is atoroidal and has infinite fundamental group.
For the entire collection see [Zbl 0939.00019].

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
30F60 Teichmüller theory for Riemann surfaces
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