##
**Hyperbolic manifolds and discrete groups.**
*(English)*
Zbl 0958.57001

Progress in Mathematics (Boston, Mass.). 183. Boston, MA: Birkhäuser. xxv, 467 p. (2001).

The main goal of the book is to present a proof of Thurston’s Hyperbolization Theorem (“The Big Monster”): Suppose that \(M\) is a compact atoroidal Haken 3-manifold that has zero Euler characteristic. Then the interior of \(M\) admits a complete hyperbolic structure of finite volume. (This is exactly the beginning of the book).

The hyperbolization theorem has been announced by W. P. Thurston around 1976. Thurston himself never published a complete proof of his theorem (see the interesting comments in the last section of his paper [Bull. Am. Math. Soc., New Ser. 30, No. 2, 161-177 (1994; Zbl 0817.01031)], but prepared the ground for its proof by explaining the general scene and the basic concepts involved in various papers and preprints, and in particular in his Princeton lecture notes which were widely distributed and which revolutionized the whole field of low-dimensional topology, hyperbolic geometry, Kleinian groups and geometric group theory (part of these notes have appeared in his book “Three-dimensional geometry and topology”, Princeton Math. Ser. 35 (1997; Zbl 0873.57001)). An early and careful account of the proof has been given by Morgan in the book [J. W. Morgan (ed.) and H. Bass (ed.), The Smith conjecture, Pure Appl. Math., Academic Press, 122 (1984; Zbl 0599.57001)] whose positive solution was a first major success of the new theory (using the still incomplete proof of the hyperbolization theorem). Then, in the next twenty years, many mathematicians worked on various aspects of the new theory, formalizing and extending the ideas of Thurston, often from different points of view. We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself which we will not repeat here in detail (there are too many different aspects and persons involved in the story).

The proof of the hyperbolization theorem naturally divides into two cases: the “generic” case when the 3-manifold does not fiber over the circle, using induction on the length of a Haken-decomposition or hierarchy of the 3-manifold, and the case of a surface bundle over the circle. For the bundle case, a complete proof has been written up by J.-P. Otal [Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996; Zbl 0855.57003)]. In the present book, a complete proof is presented for the first case. The main work of the book is to assemble and adapt the (published and unpublished) work done by many authors in the last two decades in order to amalgam it into a coherent proof of the hyperbolization theorem which is, of course, a tremendeous work for a result of the complexity of the hyperbolization theorem.

“The book contains essentially no new results. I prefered to use whatever I could find in the existing literature. Sketches of the proofs (or just the references) are given mostly in the cases when I was comfortable with the proofs that are already published. I tried to give proofs if they seemed more transparent than those in the standard references, if they are unpublished, or if they are not very long but are central for our discussion.”

The first 15 chapters of the book present various subjects and results needed for the proof of the hyperbolization theorem which is then outlined in chapter 15 and carried through in the following chapters. The headlines of the 20 chapters of the book are as follows:

Three-dimensional topology; Thurston norm; Geometry of the hyperbolic space; Kleinian groups; Teichmüller theory of Riemann surfaces; Introduction to orbifold theory (with a proof that every isomorphism between fundamental groups of certain 3-orbifolds is induced by a homeomorphism, a result needed for the inductive step); Complex projective structures; Sociology of Kleinian groups (deformations, algebraic and geometric convergence, rigidity theorems); Ultralimits of metric spaces; Introduction to group actions on trees; Laminations, foliations and trees; The Rips theory (of group actions on trees which is a main tool in the book for the proof of the Bounded Image Theorem and thus for the “Final Gluing”, the final step of the induction); Brook’s theorem and circle packings; Pleated surfaces and ends of hyperbolic manifolds; Outline of the proof of the hyperbolization theorem; Reduction to the bounded image theorem; The bounded image theorem; Hyperbolization of fibrations (the case of surface bundles over the circle for which a proof is sketched and an alternative shorter proof is given in special cases); The orbifold trick (needed to reduce the inductive step in the proof of the hyperbolization theorem to the final gluing); Beyond the hyperbolization theorem.

This is an important book which had to be written; some parts are still technical and will certainly be streamlined and shortened in the next years, but together with Otal’s work a complete published proof of the hyperbolization theorem is finally available. Apart from the proof itself, the book contains a lot of material which will be useful for various other directions of research. Based on the hyperbolization theorem, a proof of the orbifold geometrization theorem has been given in the meantime by Boileau-Leeb-Porti and by D. Cooper, C. D. Hodgson, and S. P. Kerckhoff [see their paper in “Three-dimensional orbifolds and cone manifolds”, Math. Soc. Japan Memoirs 5 (2000)]. There remains the general geometrization conjecture for 3-manifolds (including the Poincaré conjecture), and we close with a citation from Thurston’s paper mentioned above. “What mathematicians most wanted and needed from me was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds. It is unlikely that the proof of the general geometrization conjecture will consist of pushing the same proof further”.

The hyperbolization theorem has been announced by W. P. Thurston around 1976. Thurston himself never published a complete proof of his theorem (see the interesting comments in the last section of his paper [Bull. Am. Math. Soc., New Ser. 30, No. 2, 161-177 (1994; Zbl 0817.01031)], but prepared the ground for its proof by explaining the general scene and the basic concepts involved in various papers and preprints, and in particular in his Princeton lecture notes which were widely distributed and which revolutionized the whole field of low-dimensional topology, hyperbolic geometry, Kleinian groups and geometric group theory (part of these notes have appeared in his book “Three-dimensional geometry and topology”, Princeton Math. Ser. 35 (1997; Zbl 0873.57001)). An early and careful account of the proof has been given by Morgan in the book [J. W. Morgan (ed.) and H. Bass (ed.), The Smith conjecture, Pure Appl. Math., Academic Press, 122 (1984; Zbl 0599.57001)] whose positive solution was a first major success of the new theory (using the still incomplete proof of the hyperbolization theorem). Then, in the next twenty years, many mathematicians worked on various aspects of the new theory, formalizing and extending the ideas of Thurston, often from different points of view. We recommend the excellent introduction of the present book for the history of the various contributions, and also for a sketch of the proof itself which we will not repeat here in detail (there are too many different aspects and persons involved in the story).

The proof of the hyperbolization theorem naturally divides into two cases: the “generic” case when the 3-manifold does not fiber over the circle, using induction on the length of a Haken-decomposition or hierarchy of the 3-manifold, and the case of a surface bundle over the circle. For the bundle case, a complete proof has been written up by J.-P. Otal [Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996; Zbl 0855.57003)]. In the present book, a complete proof is presented for the first case. The main work of the book is to assemble and adapt the (published and unpublished) work done by many authors in the last two decades in order to amalgam it into a coherent proof of the hyperbolization theorem which is, of course, a tremendeous work for a result of the complexity of the hyperbolization theorem.

“The book contains essentially no new results. I prefered to use whatever I could find in the existing literature. Sketches of the proofs (or just the references) are given mostly in the cases when I was comfortable with the proofs that are already published. I tried to give proofs if they seemed more transparent than those in the standard references, if they are unpublished, or if they are not very long but are central for our discussion.”

The first 15 chapters of the book present various subjects and results needed for the proof of the hyperbolization theorem which is then outlined in chapter 15 and carried through in the following chapters. The headlines of the 20 chapters of the book are as follows:

Three-dimensional topology; Thurston norm; Geometry of the hyperbolic space; Kleinian groups; Teichmüller theory of Riemann surfaces; Introduction to orbifold theory (with a proof that every isomorphism between fundamental groups of certain 3-orbifolds is induced by a homeomorphism, a result needed for the inductive step); Complex projective structures; Sociology of Kleinian groups (deformations, algebraic and geometric convergence, rigidity theorems); Ultralimits of metric spaces; Introduction to group actions on trees; Laminations, foliations and trees; The Rips theory (of group actions on trees which is a main tool in the book for the proof of the Bounded Image Theorem and thus for the “Final Gluing”, the final step of the induction); Brook’s theorem and circle packings; Pleated surfaces and ends of hyperbolic manifolds; Outline of the proof of the hyperbolization theorem; Reduction to the bounded image theorem; The bounded image theorem; Hyperbolization of fibrations (the case of surface bundles over the circle for which a proof is sketched and an alternative shorter proof is given in special cases); The orbifold trick (needed to reduce the inductive step in the proof of the hyperbolization theorem to the final gluing); Beyond the hyperbolization theorem.

This is an important book which had to be written; some parts are still technical and will certainly be streamlined and shortened in the next years, but together with Otal’s work a complete published proof of the hyperbolization theorem is finally available. Apart from the proof itself, the book contains a lot of material which will be useful for various other directions of research. Based on the hyperbolization theorem, a proof of the orbifold geometrization theorem has been given in the meantime by Boileau-Leeb-Porti and by D. Cooper, C. D. Hodgson, and S. P. Kerckhoff [see their paper in “Three-dimensional orbifolds and cone manifolds”, Math. Soc. Japan Memoirs 5 (2000)]. There remains the general geometrization conjecture for 3-manifolds (including the Poincaré conjecture), and we close with a citation from Thurston’s paper mentioned above. “What mathematicians most wanted and needed from me was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds. It is unlikely that the proof of the general geometrization conjecture will consist of pushing the same proof further”.

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

22E40 | Discrete subgroups of Lie groups |

53C20 | Global Riemannian geometry, including pinching |

57M50 | General geometric structures on low-dimensional manifolds |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

20F65 | Geometric group theory |