Lectures on Hyperbolic Geometry

by Benedetti, R.

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Title Statements

Full title :Lectures on Hyperbolic Geometry / by Riccardo Benedetti, Carlo Petronio.

Summary

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.

Edition Statements

Publication Date:1992.

Publication Statements

Publisher(s):Springer Berlin Heidelberg
Place of Publication:Berlin, Heidelberg
Language:eng
Publication Date:1992.

The Physical Object

Description:1 online resource (xiv, 332 pages 175 illustrations)

Edition Identifiers

ISBN:9783642581588, 3642581587
oclc:ocn851820175
LC Classification:QA641-670
Dewey Decimal:516.36

Ch. A. Hyperbolic Space. A.1. Models for Hyperbolic Space. A.2. Isometries of Hyperbolic Space: hyperboloid Model. A.3. Conformal Geometry. A.4. Isometries of Hyperbolic Space: Disc and Half-space Models. A.5. Geodesics, Hyperbolic Subspaces and Miscellaneous Facts. A.6. Curvature of Hyperbolic Space -- Ch. B. Hyperbolic Manifolds and the Compact Two-dimensional Case. B.1. Hyperbolic, Elliptic and Flat Manifolds. B.2. topology of Compact Oriented Surfaces. B.3. Hyperbolic, Elliptic and Flat Surfaces. B.4. Teichmuller Space -- Ch. C. The Rigidity Theorem (Compact Case). C.1. First Step of the Proof: Extension of Pseudo-isometries. C.2. Second Step of the Proof: Volume of Ideal Simplices. C.3. Gromov Norm of a Compact Manifold. C.4. Third Step of the Proof: the Gromov Norm and the Volume Are Proportional. C.5. Conclusion of the Proof, Corollaries and Generalizations -- Ch. D. Margulis' Lemma and its Applications. D.1. Margulis' Lemma. D.2. Local Geometry of a Hyperbolic Manifold.

D.3. Ends of a Hyperbolic Manifold -- Ch. E. The Space of Hyperbolic Manifolds and the Volume Function. E.1. The Chabauty and the Geometric Topology. E.2. Convergence in the Geometric Topology: Opening Cus…

Subjects

applied mathematics Géométrie différentielle globale Géométrie hyperbolique Global differential geometry Hyperbolische Geometrie Hyperbolische meetkunde mathematics Mathématiques