Discontinuous Groups of Isometries in the Hyperbolic Plane

Werner Fenchel, Jakob Nielsen

Walter de Gruyter, May 12, 2011 - Mathematics - 385 pages

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.

Contents

• I Möbius transformations and noneuclidean geometry: 1
• 2 Crossratio: 4
• 3 Möbius transformations direct and reversed: 6
• 4 Invariant points and classification of Möbius transformations: 8
• 5 Complex distance of two pairs of points: 14
• 6 Noneuclidean metric: 18
• 7 Isometric transformations: 23
• 8 Noneuclidean trigonometry: 27
• 9 Products and commutators of motions: 43
• II Discontinuous groups of motions and reversions: 58
• 11 Groups with invariant points or lines: 70
• 12 A discontinuity theorem: 78
• 13 Fgroups Fundamental set and limit set: 82
• 14 The convex domain of an Fgroup Characteristic and isometric neighbourhood: 95
• 15 Quasicompactness modulo F and finite generation of F: 115
• III Surfaces associated with discontinuous groups: 127
• 17 Area and type numbers: 135
• IV Decompositions of groups: 153
• 19 Decomposition of groups: 174
• 20 Decompositions of Fgroups containing reflections: 196
• 21 Elementary groups and elementary surfaces: 213
• 22 Complete decomposition and normal form in the case of quasicompactness: 242
• 23 Exhaustion in the case of nonquasicompactness: 270
• V Isomorphism and homeomorphism: 283
• 25 Topological and geometrical homeomorphism: 308
• 26 Construction of gmappings Metric parameters Congruent groups: 318
• Symbols and definitions: 349
• Alphabets: 353
• Bibliography: 355
• Index: 361

About the author

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

Bibliographic information

• Title: Discontinuous Groups of Isometries in the Hyperbolic Plane; Volume 29 of De Gruyter Studies in Mathematics
• Authors: Werner Fenchel, Jakob Nielsen
• Editor: Asmus L. Schmidt
• Edition: illustrated
• Publisher: Walter de Gruyter, 2011
• ISBN: 3110891352, 9783110891355
• Length: 385 pages
• Subjects: Mathematics / Calculus; Mathematics / General; Mathematics / Geometry / General; Mathematics / Mathematical Analysis