
Lectures on Hyperbolic Geometry
Springer (Publisher)
Published on 3. September 1992
Book
Paperback/Softback
XIV, 330 pages
978-3-540-55534-6 (ISBN)
Description
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.
More details
Series
Edition
1st ed. 1992. 2nd printing 2003
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Primary & secondary/elementary & high school
Research
Edition type
New edition
Illustrations
XIV, 330 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 19 mm
Weight
528 gr
ISBN-13
978-3-540-55534-6 (9783540555346)
DOI
10.1007/978-3-642-58158-8
Schweitzer Classification
Content
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.