Hyperbolic Manifolds and Discrete Groups

"Hyperbolic Manifolds and Discrete Groups" is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout is on the 'Big Monster', i.e., on Thurston's hyperbolization theorem, which has not only completely changed the landscape of 3-dimensional topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book presents the first complete proof of Thurston's hyperbolization theorem in the 'generic case' and an outline of Otal's proof of the hyperbolization theorem for manifolds fibered over the circle. This important work contains an extended treatment of the theory of Kleinian groups and group actions on trees, including such key topics as: the Kazhdan-Margulis-Zassenhaus theorem; the Klein and Maskit combination theorems; the Mostow rigidity theorem; the Douady-Earle extension theorem for homeomorphisms of the circle; the smoothness theorem for representation varieties of Kleinian groups; the Ahlfors finiteness theorem; the Brooks deformation theorem; characterization of pseudo-Anosov homeomorphisms; and, compactification of character varieties via group actions on trees.