In recent years hyperbolic geometry has been the object and the
preparation for extensive study that has produced important and
often amazing results and also opened up new questions. The book
concerns the geometry of manifolds and in particular hyperbolic
manifolds; its aim is to provide an exposition of some fundamental
results, and to be as far as possible self-contained, complete,
detailed and unified. Since it starts from the basics and it
reaches recent developments of the theory, the book is mainly
addressed to graduate-level students approaching research, but it
will also be a helpful and ready-to-use tool to the mature
researcher. After collecting some classical material about the
geometry of the hyperbolic space and the Teichmuller space, the
book centers on the two fundamental results: Mostow's rigidity
theorem (of which a complete proof is given following Gromov and
Thurston) and Margulis' lemma. These results form the basis for the
study of the space of the hyperbolic manifolds in all dimensions
(Chabauty and geometric topology); a unified exposition is given of
Wang's theorem and the Jorgensen-Thurston theory. A large part is
devoted to the three-dimensional case: a complete and elementary
proof of the hyperbolic surgery theorem is given based on the
possibility of representing three manifolds as glued ideal
tetrahedra. The last chapter deals with some related ideas and
generalizations (bounded cohomology, flat fiber bundles, amenable
groups). This is the first book to collect this material together
from numerous scattered sources to give a detailed presentation at
a unified level accessible to novice readers."
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