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of the hyperbolization theorem for 3-manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way.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. |
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 G angle Borel set boundary Chapter closed geodesic closed set closed subset compact leaf compact set compact surface complement complete hyperbolic conjugacy class conjugates connected components constant construction contained contradiction converges Corollary curve cusp d(pi defined denote diffeomorphism disjoint dual tree elements g embedded endpoints equal exists Fact finite type fixed point Fuchsian group fundamental group group G H²/T half-leaf Hence homeomorphism homotopic hyperbolic manifold hyperbolic metric hyperbolic space hyperbolization theorem interior intersection number interval Isom(H isometry Kleinian group lamination F leaf of F lift map f map ƒ Margulis lemma measured geodesic lamination measured lamination noncompact nonzero parabolic element preimage projection Proposition pseudo-Anosov PSL2 quasi-Fuchsian quasiconformal homeomorphism quotient real tree rectangle representation resp restriction Section sequence subgroup sufficiently large Teichmüller space Thurston topology train track translation distance transverse transverse measure triangle union vertical sides