The Hyperbolization Theorem for Fibered 3-manifolds

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American Mathematical Soc., Jan 1, 2001 - Mathematics - 126 pages
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A 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of the hyperbolization theorem for 3-manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way.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
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