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**Thurston’s hyperbolization of Haken manifolds.**
*(English)*
Zbl 0997.57001

Hsiung, C. C. (ed.) et al., Surveys in differential geometry. Vol. III. A supplement to the Journal of Differential Geometry. Lectures on geometry and topology in honor of the 80th birthday of Chuan-Chih Hsiung, Harvard University, Cambridge, MA, USA, May 3-5, 1996. Boston, MA: International Press. 77-194 (1998).

This book-length paper comprises a complete proof of Thurston’s Hyperbolization Theorem, which says that if \(M\) is a closed irreducible, atoroidal Haken \(3\)-manifold then \(M\) is hyperbolic, i.e. it admits a complete hyperbolic metric. This Hyperbolization Theorem and Thurston’s Geometrization Conjecture announced by W. Thurston in the 1970s reveal the existence of a strong link between \(3\)-dimensional topology and the theory of Kleinian groups. Thurston has explained how to prove it in a series of papers and preprints. In the present work the author presents a complete proof of the hyperbolization theorem in the case of non-fibered Haken 3-manifolds whose fundamental group does not contain \(\mathbb Z + \mathbb Z\)-subgroups. The extension to the general case of atoroidal manifolds does not encounter any difficulties that are unfamiliar or deep, and it is covered by notes of the author and F. Paulin, Geometrie Hyperbolique et Groupes Kleiniens. For the proof in the remaining case of 3-manifolds which fiber over the circle see [J.-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996; Zbl 0855.57003)]. The proofs of the two (fibered and non-fibered) halves of the Hyperbolization Theorem may nevertheless overlap, as Thurston himself observed. For example, there is a still unsettled question: does every compact 3-manifold fibered over the circle have a finite cover that contains an incompressible surface which is not a fiber of a fibration over the circle? If it were true, the results of the present work alone would suffice to completely prove the Hyperbolization Theorem.

This very well written paper has eight chapters. In the first seven of them: Kleinian groups and Teichmüller theory, The fixed-point problem, Holomorphic quadratic differentials, The volume form on open Riemann surfaces, Contraction properties of the Theta operator, McMullen’s proof of the Fixed-point theorem, Manifolds with corners, the author gives complete accounts of several results used. The last chapter deals with the actual proof of Thurston’s Hyperbolization Theorem, from a viewpoint different from Thurston’s original approach which is covered by J. W. Morgan [On Thurston’s uniformization theorem for three-dimensional manifolds, Pure Appl. Math. 112, 37-126 (1984; Zbl 0599.57002)] and M. Kapovich [Hyperbolic manifolds and discrete groups, Prog. Math. 183 (2001; Zbl 0958.57001)]. The crucial part of the present proof relies more on Teichmüller theory than Thurston’s original proof. It is due to C. McMullen [Iteration on Teichmüller space, Invent. Math. 99, No. 2, 425-454 (1990; Zbl 0695.57012)].

For the entire collection see [Zbl 0918.00028].

This very well written paper has eight chapters. In the first seven of them: Kleinian groups and Teichmüller theory, The fixed-point problem, Holomorphic quadratic differentials, The volume form on open Riemann surfaces, Contraction properties of the Theta operator, McMullen’s proof of the Fixed-point theorem, Manifolds with corners, the author gives complete accounts of several results used. The last chapter deals with the actual proof of Thurston’s Hyperbolization Theorem, from a viewpoint different from Thurston’s original approach which is covered by J. W. Morgan [On Thurston’s uniformization theorem for three-dimensional manifolds, Pure Appl. Math. 112, 37-126 (1984; Zbl 0599.57002)] and M. Kapovich [Hyperbolic manifolds and discrete groups, Prog. Math. 183 (2001; Zbl 0958.57001)]. The crucial part of the present proof relies more on Teichmüller theory than Thurston’s original proof. It is due to C. McMullen [Iteration on Teichmüller space, Invent. Math. 99, No. 2, 425-454 (1990; Zbl 0695.57012)].

For the entire collection see [Zbl 0918.00028].

Reviewer: Boris N.Apanasov (Norman)