## The Hyperbolization Theorem for Fibered 3-manifoldsA fundamental element of the study of 3-manifolds is Thurston's remarkable geometrization conjecture, which states that the interior of every compact 3-manifold has a canonical decomposition into pieces that have geometric structures. In most cases, these structures are complete metrics of constant negative curvature, that is to say, they are hyperbolic manifolds. The conjecture has been proved in some important cases, such as Haken manifolds and certain types of fibered manifolds. The influence of Thurston's hyperbolization theorem on the geometry and topology of 3-manifolds has been tremendous. This book presents a complete proof of the hyperbolization theorem for 3-manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way. The book should be suitable for graduate students with a background in modern techniques of low-dimensional topology and will also be of interest to researchers in geometry and topology. This is the English translation of a volume originally published in 1996 by the Societe Mathematique de France. |

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### Contents

Introduction | ix |

Teichmiiller Spaces and Kleinian Groups | 1 |

Real Trees and Degenerations of Hyperbolic Structures | 17 |

Geodesic Laminations and Real Trees | 33 |

Geodesic Laminations and the Gromov Topology | 41 |

The Double Limit Theorem | 53 |

The Hyperbolization Theorem for Fibered Manifolds | 59 |

Sullivans Theorem | 75 |

Actions of Surface Groups on Real Trees | 85 |

Two Examples of Hyperbolic Manifolds That Fiber over the Circle | 105 |

### Common terms and phrases

action of F angle atoroidal Borel set boundary Chapter closed geodesic closed set closed subset closure compact leaf compact set compact surface complement component of H2 conjugacy class conjugates connected components construction contained contradiction converges Corollary cusp defined DEFINITION deformation of F denote diffeomorphism disjoint dual tree elements g elements of F embedded endpoints equal exceptional minimal set Fact finite type fixed point Fuchsian group fundamental group group G Hence homeomorphism homotopic hyperbolic element hyperbolic manifold hyperbolic metric hyperbolic space hyperbolization theorem interior intersection number interval isometry Kleinian group lamination F leaf of F lift limit set Margulis lemma measured geodesic lamination measured lamination noncompact parabolic element preimage projection proof of Theorem Proposition pseudo-Anosov px(F quasiconformal homeomorphism quotient real tree rectangles Ri representation px resp restriction Section subgroup sufficiently large surface H2/F Teichmiiller space Thurston topology train track 72 translation distance transverse measure triangle union vertical sides