Lectures on Hyperbolic Geometry

Riccardo Benedetti, Carlo Petronio

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English

Springer-Verlag Berlin and Heidelberg GmbH & Co. K
03 September 1992

Mathematics & Sciences; Geometry; Topology

Series: Universitext

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 Teichmüller 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.

By: Riccardo Benedetti, Carlo Petronio
Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Country of Publication: Germany
Dimensions: Height: 235mm, Width: 155mm, Spine: 18mm
Weight: 1.090kg
ISBN: 9783540555346
ISBN 10: 354055534X
Series: Universitext
Pages: 352
Publication Date: 03 September 1992
Audience: College/higher education , Professional and scholarly , Further / Higher Education , Undergraduate
Format: Paperback
Publisher's Status: Active

A. Hyperbolic Space.- B. Hyperbolic Manifolds and the Compact Two-dimensional Case.- C. The Rigidity Theorem (Compact Case).- D. Margulis’ Lemma and its Applications.- E. The Space of Hyperbolic Manifolds and the Volume Function.- F. Bounded Cohomology, a Rough Outline.- Notation Index.- References.