×

zbMATH — the first resource for mathematics

The hyperbolization theorem for fibre manifolds of dimension 3. (Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3.) (French) Zbl 0855.57003
Astérisque. 235. Paris: Société Mathématique de France, x, 159 p. (1996).
This book is devoted to proving the Thurston hyperbolization theorem for the case of compact, atoroidal, irreducible, sufficiently large 3-manifolds which are fibered over the circle.
W. P. Thurston’s more general theorem asserts the existence of a hyperbolic structure on the interior of any compact, atoroidal, irreducible, sufficiently large 3-manifold [Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982; Zbl 0496.57005)]. The author’s proof follows the lines of an unpublished 1986 preprint of Thurston’s except at “l’étape essentielle – ‘le théorème de la limite double’ ” which is done in a new way. The degree of detail of the proof is suggested by the range of the contents:
Chapter 1: Teichmüller spaces and Kleinian groups (18 pp.); Chapter 2: Degeneracies of hyperbolic structures and real trees (16 pp.); Chapter 3: Geodesic laminations and real trees (7 pp.); Chapter 4: Geodesic laminations and Gromov convergence (11 pp.); Chapter 5: The theorem of the double limit (4 pp.); Chapter 6: The hyperbolization theorem for fibered manifolds (16 pp.); Chapter 7: Sullivan’s theorem (9 pp.); Chapter 8: Actions of surface groups on real trees (19 pp.); Chapter 9: Two examples of hyperbolic manifolds fibered over the circle (4 pp.); Appendix: Geodesic laminations (8 pp.).

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M50 General geometric structures on low-dimensional manifolds
51M10 Hyperbolic and elliptic geometries (general) and generalizations
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDF BibTeX XML Cite