Aimed at researchers, graduate students and undergraduates alike, this book presents a unified exposition of all the main areas and methods of the theory of Kleinian groups and the theory of uniformization of manifolds. The past 20 years have seen a rejuvenation of the field, due to the development of powerful new methods in topology, the theory of functions of several complex variables, and the theory of quasiconformal mappings. Thus this new book should provide a valuable resource, listing the basic facts regarding Kleinian groups and serving as a general guide to the primary literature, particularly the Russian literature in the field. In addition, the book includes a large number of examples, problems, and unsolved problems, many of them presented for the first time.
Author(s): S. L. Krushkal', B. N. Apanasov, N. A. Gusevski
Series: TMM 62
Publisher: American Mathematical Society
Year: 1986
Language: English
Pages: 206
Contents......Page 3
Foreword......Page 5
Editor's Preface to the English Translation......Page 7
1. Main concepts ......Page 9
2. Kleinian groups in space ......Page 15
3. Combination theorems ......Page 19
4. Quasiconformal mappings and deformations of Kleinian groups ......Page 24
5. Finitely generated Kleinian groups. Cohomology methods. Stability ......Page 29
1. What is uniformization? ......Page 35
2. The fundamental group and covering spaces. Planar coverings ......Page 38
3. Uniformization of Riemann surfaces ......Page 42
4. Uniformization by Kleinian groups ......Page 44
5. Teichmiiller spaces ......Page 49
6. Branched coverings of Riemann surfaces ......Page 58
7. Methods of three-dimensional topology ......Page 62
8. Uniformization of multidimensional manifolds ......Page 66
1. General properties of Kleinian groups (fundamental sets, sets of discontinuity) ......Page 71
2. Limit sets of Kleinian groups ......Page 85
3. Quasiconformal deformations of Kleinian groups ......Page 89
4. Kleinian groups that are not geometrically finite ......Page 98
5. Uniformization: Hyperbolic manifolds ......Page 105
6. Uniformization: Elliptic manifolds ......Page 121
7. General uniformization problems ......Page 125
1. Exercises on the geometry of Kleinian groups ......Page 137
2. Exercises on the theory of uniformization ......Page 142
3. Problems: General properties of discontinuous groups ......Page 149
4. Problems: Methods of uniformization ......Page 157
CONCLUSION. Some Unsolved Problems ......Page 179
Bibliography ......Page 183
Supplementary Bibliography ......Page 197
Subject Index ......Page 203
