Discontinous Groups of Isometries

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Overview

Fuchsian groups play a central role in various important fields of mathematics. 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, Werner Fenchel (1905-1988) joined later and overtook the much of the preparation of the manuscript. Professor Asmus Schmidt (University of Copenhagen) is the editor of this first publication in book form of the Fenchel-Nielsen notes. It is on his initiative that the long and difficult way of getting the original notes into the proper shape ready for publication succeeded.
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Editorial Reviews

From the Publisher
"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
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Product Details

Meet the Author

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

Preface
Life and work of the Authors
I Mobius transformations and non-euclidean geometry 1
1 Pencils of circles - inversive geometry 1
2 Cross-ratio 4
3 Mobius transformations, direct and reversed 6
4 Invariant points and classification of Mobius transformations 8
5 Complex distance of two pairs of points 14
6 Non-euclidean metric 18
7 Isometric transformations 23
8 Non-euclidean trigonometry 27
9 Products and commutators of motions 43
II Discontinuous groups of motions and reversions 58
10 The concept of discontinuity 58
11 Groups with invariant points or lines 70
12 A discontinuity theorem 78
13 [actual symbol not reproducible]-groups. Fundamental set and limit set 82
14 The convex domain of an [actual symbol not reproducible]-group. Characteristic and isometric neighbourhood 95
15 Quasi-compactness modulo [actual symbol not reproducible] and finite generation of [actual symbol not reproducible] 115
III Surfaces associated with discontinuous groups 127
16 The surfaces [actual symbol not reproducible] module [actual symbol not reproducible] and K ([actual symbol not reproducible]) modulo [actual symbol not reproducible] 127
17 Area and type numbers 135
IV Decompositions groups 153
18 Composition of groups 153
19 Decomposition of groups 174
20 Decompositions of [actual symbol not reproducible]-groups containing reflections 196
21 Elementary groups and elementary surfaces 213
22 Complete decomposition and normal form in the case of quasi-compactness 242
23 Exhaustion in the case of non-quasi-compactness 270
V Isomorphism and homeomorphism 283
24 Topological and geometrical isomorphism 283
25 Topological and geometrical homeomorphism 308
26 Construction of g-mappings. Metric parameters. Congruent groups 318
Symbols and definitions 349
Alphabets 353
Bibliography 355
Index 361
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