The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.
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.
Inhaltsverzeichnis
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
