Noneuclidean Tesselations and Their Groups (eBook)

Front Cover 1
Noneuclidean Tesselations and Their Groups 4
Copyright Page 5
Contents 8
Preface 10
Abbreviations and Symbols 14
CHAPTER I. ELEMENTARY CONCEPTS AND FORMULAS 18
I.1 The Group G* of Homographic Substitutions 18
I.2 Action of G* on the Closed Complex Plane C 19
I.3 Action of G* on Hyperbolic Three-Space 27
I.4 Circle Groups as Groups of Motions of Hyperbolic Two-Space 36
I.5 Notes on Elliptic and Spherical Geometry 54
I.6 Illustrations. References and Historical Remarks 58
I.7 Appendix: Hilbert’s Axioms of Geometry 60
CHAPTER II. DISCONTINUOUS GROUPS AND TRIANGLE TESSELATIONS 69
II.1 Introductory Remarks 69
II.2 Discontinuous Groups and Fundamental Regions 73
II.3 Triangle Groups, Local and Global Relations 82
II.4 Euclidean, Spherical, and Elliptic Triangle Groups 85
II.5 Hyperbolic Triangle Groups 98
II.6 Some Subgroups of Hyperbolic Triangle Groups 107
II.7 General Theorems. A Survey and References 112
CHAPTER III. NUMBER THEORETICAL METHODS 124
III.1 The Modular Group 124
III.2 Subgroups and Quotient Groups of the Modular Group 129
III.3 Groups of Units of Ternary Quadratic and Binary Hermitian Forms 140
CHAPTER IV. MISCELLANY 151
IV.1 Examples of Discontinuous Nonfuchsian Groups 151
IV.2 Fricke Characters 165
CHAPTER V. GROUPS THAT ARE DISCONTINUOUS IN HYPERBOLIC THREE-SPACE 168
V.l Linear Groups over Imaginary Quadratic Number Fields 168
V.2 Some Geometric Contructions 170
Figures 174
References 174
Index 222