مشخصات کتاب

Discontinuous Groups of Isometries in the Hyperbolic Plane

ویرایش:  
نویسندگان: Werner Fenchel,  Jakob Nielsen,  Asmus L. Schmidt  
سری: De Gruyter Studies in Mathematics 
ISBN (شابک) : 3110175266, 9783110175264 
ناشر: Walter de Gruyter & Co 
سال نشر: 2002 
تعداد صفحات: 387 
زبان: English 
فرمت فایل : DJVU (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) 
حجم فایل: 6 مگابایت 

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توضیحاتی درمورد کتاب به خارجی

This book by Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had a long and complicated history. In 1938-39, Nielsen gave a series of lectures on discontinuous groups of motions in the non-euclidean plane, and this led him - during World War II - to write the first two chapters of the book (in German). When Fenchel, who had to escape from Denmark to Sweden because of the German occupation, returned in 1945, Nielsen initiated a collaboration with him on what became known as the Fenchel-Nielsen manuscript. At that time they were both at the Technical University in Copenhagen. The first draft of the Fenchel-Nielsen manuscript (now in English) was finished in 1948 and it was planned to be published in the Princeton Mathematical Series. However, due to the rapid development of the subject, they felt that substantial changes had to be made before publication. When Nielsen moved to Copenhagen University in 1951 (where he stayed until 1955), he was much involved with the international organization UNESCO, and the further writing of the manuscript was left to Fenchel. The archives of Fenchel now deposited and catalogued at the Department of Mathematics at Copenhagen Univer- sity contain two original manuscripts: a partial manuscript (manuscript 0) in Ger- man containing Chapters I-II ( I -15), and a complete manuscript (manuscript I) in English containing Chapters I-V ( 1-27). The archives also contain part of a corre- spondence (first in German but later in Danish) between Nielsen and Fenchel, where Nielsen makes detailed comments to Fenchel's writings of Chapters III-V. Fenchel, who succeeded N. E. Nf/Jrlund at Copenhagen University in 1956 (and stayed there until 1974), was very much involved with a thorough revision of the curriculum in al- gebra and geometry, and concentrated his research in the theory of convexity, heading the International Colloquium on Convexity in Copenhagen 1965. For almost 20 years he also put much effort into his job as editor of the newly started journal Mathematica Scandinavica. Much to his dissatisfaction, this activity left him little time to finish the Fenchel-Nielsen project the way he wanted to. After his retirement from the university, Fenchel - assisted by Christian Sieben- eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - found time to finish the book Elementary Geometry in Hyperbolic Space, which was published by Walter de Gruyter in 1989 shortly after his death. Simultaneously, and with the same collaborators, he supervised a typewritten version of the manuscript (manuscript 2) on discontinuous groups, removing many of the obscure points that were in the original manuscript. Fenchel told me that he contemplated removing parts of the introductory Chapter I in the manuscript, since this would be covered by the book mentioned above; but to make the Fenchel-Nielsen book self-contained he ultimately chose not to do so. He did decide to leave out 27, entitled Thefundamental group. As editor, I started in 1990, with the consent of the legal heirs of Fenchel and Nielsen, to produce a TEX-version from the newly typewritten version (manuscript 2). I am grateful to Dita Andersen and Lise Fuldby-Olsen in my department for hav- ing done a wonderful job of typing this manuscript in AMS- TEX. I have also had much help from my colleague J0rn B0rling Olsson (himself a student of Kate Fenchel at Aarhus University) with the proof reading of the TEX-manuscript (manuscript 3) against manuscript 2 as well as with a general discussion of the adaptation to the style of TEX. In most respects we decided to follow Fenchel's intentions. However, turning the typewritten edition of the manuscript into TEX helped us to ensure that the notation, and the spelling of certain key-words, would be uniform throughout the book. Also, we have indicated the beginning and end of a proof in the usual style of TEX. With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and to my great relief and satisfaction they agreed to publish the manuscript in their series Studies in Mathematics. I am most grateful for this positive and quick reaction. One particular problem with the publication turned out to be the reproduction of the many figures which are an integral part of the presentation. Christian Siebeneicher had at first agreed to deliver these in final electronic form, but by 1997 it became clear that he would not be able to find the time to do so. However, the publisher offered a solution whereby I should deliver precise drawings of the figures (Fenchel did not leave such for Chapters IV and V), and then they would organize the production of the figures in electronic form. I am very grateful to Marcin Adamski, Warsaw, Poland, for his fine collaboration concerning the actual production of the figures. My colleague Bent Fuglede, who has personaHy known both authors, has kindly written a short biography of the two of them and their mathematical achievements, and which also places the Fenchel-Nielsen manuscript in its proper perspective. In this connection I would like to thank The Royal Danish Academy of Sciences and Letters for allowing us to include in this book reproductions of photographs of the two authors which are in the possession of the Academy. Since the manuscript uses a number of special symbols, a list of notation with short explanations and reference to the actual definition in the book has been included. Also, a comprehensive index has been added. In both cases, all references are to sections, not pages. We considered adding a complete list of references, but decided against it due to the overwhelming number of research papers in this area. Instead, a much shorter list of monographs and other comprehensive accounts relevant to the subject has been collected. My final and most sincere thanks go to Dr. Manfred Karbe from Walter de Gruyter for his dedication and perseverance in bringing this publication into existence.

فهرست مطالب

Contents
Chapter I. Möbius Transformations and Non-Euclidean Geometry.
	§1 Pencils of Circles - Inversive Geometry
	§2 Cross-ratio
	§3 Möbius Transformations, Direct and Reversed
	§4 Invariant Points and Classification of Möbius Transformations
	§5 Complex Distance of Two Pairs of Points
	§6 Non-euclidean Metric
	§7 Isometric transformations
	§8 Non-euclidean trigonometry
	§9 Products and Commutators of Motions
Chapter II. Discontinuous Groups of Motions and Reversions.
	§10 the Concept of Discontinuity
	§11 Groups with Invariant Points or Lines
	§12 A Discontinuity Theorem
	§13 ℱ-groups. Fundamental Set and Limit Set
	§14 the Convex Domain of an ℱ-group. Characteristic and Isometric Neighbourhood
	§15 Quasi-compactness Modulo ℱ and Finite Generation of ℱ
Chapter III. Surfaces Associated with Discontinuous Groups.
	§16 the Surfaces D Modulo ℭ and K(ℱ) Modulo ℱ
	§17 Area and Type Numbers
Chapter IV. Decompositions of Groups.
	§18 Composition of Groups
	§19 Decomposition of Groups
	§20 Decompositions of ℱ-groups Containing Reflections
	§21 Elementary Groups and Elementary Surfaces
	§22 Complete Decomposition and Normal Form in the Case of Quasi-compactness
	§23 Exhaustion in the Case of Non-quasi-compactness
Chapter V. Isomorphism and Homeomorphism.
	§24 Topological and Geometrical Isomorphism
	§25 Topological and Geometrical Homeomorphism
	§26 Construction of G-mappings. Metric Parameters. Congruent Groups
Symbols and Definitions
Alphabets
Bibliography
Index

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