Discontinuous Groups of Isometries in the Hyperbolic Plane (inbunden)
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Format
Inbunden (Hardback)
Språk
Engelska
Antal sidor
385
Utgivningsdatum
2002-12-01
Förlag
De Gruyter
Medarbetare
Nielsen, Jakob
Illustrationer
112 Illustrations, black and white
Dimensioner
246 x 178 x 25 mm
Vikt
758 g
Antal komponenter
1
Komponenter
HC runder Rücken kaschiert
ISSN
0179-0986
ISBN
9783110175264

Discontinuous Groups of Isometries in the Hyperbolic Plane

Inbunden,  Engelska, 2002-12-01
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This is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups). The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps. This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups.
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"The Fenchel-Nielsen manuscript has been famous for a long time already and its final publication is a valuable edition to mathematical literature."EMS Newsletter "Those working in the field will be grateful to the editor Asmus Schmidt for producing this classic text; it can now be cited without the annoying reference 'Fenchel an Nielsen (to appear)'."David Singerman in: Bulletin of the London Mathematical Society 36/2004

Övrig information

Asmus L. Schmidt is Associate Professor at the Institute for Mathematical Sciences of the University of Copenhagen, Denmark.

Innehållsförteckning

Editor's preface - Short biography of the authors - Mobius transformations and non-euclidean geometry - Pencils of circles - Inversive geometry - Cross-ratio - Mobius tranformations, direct and reversed - Invariant points and classification of Mobius transformations - Complex distance of two pairs of points - Non-Euclidian metric - Geometric transformations - Non-Euclidean trigonometry - Products and commutators of motions - Discontinuous groups of motions and reversions - The concept of discontinuity - Groups with invariant points or lines - A discontinuity theorem - F-groups. Fundamental set and limit set - The Convex domain of F-group. Characteristic and isometric neighbourhood - Quasi-compactness modulo F and finite generation of F - Surfaces associated with discontinuous groups - The surfaces D modulo G and K(F) modulo F - Area and type numbers - Decompositions of groups - Composition of groups - Decomposition of groups - Decompositions of F-groups containing reflections - Elementary groups and elementary surfaces - Complete decomposition and normal form in the case of quasi-compactness - Exhaustion in the case of non-quasi-compactness - Isomorphism and homeomorphism - Topological and geometrical isomorphism - Topological and geometrical homeomorphism - Construction of g-mappings. Metric parameters. Congruent groups - Symbols and definitions - Bibliography