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of the hyperbolization theorem for 3-manifolds that fiber over the circle, following the plan of Thurston's original (unpublished) proof, though the double limit theorem is dealt with in a different way.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 is 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.
Jean-Pierre Torrell is a Dominican priest of the Toulouse province and professor of dogmatic theology at the University of Fribourg. From 1973 to 1981, he taught at the Gregorian University in Rome and was a member of the Leonine Commission, where he contributed to research on and editing of Aquinas's works.
Teichmuller spaces and Kleinian groups Real trees and degenerations of hyperbolic structures Geodesic laminations and real trees Geodesic laminations and the Gromov topology The double limit theorem The hyperbolization theorem for fibered manifolds Sullivan's theorem Actions of surface groups on real trees Two examples of hyperbolic manifolds that fiber over the circle Geodesic laminations Bibliography Index.