Comprehensive introduction to the geometric and topological structure of Kleinian groups. Starts with Möbius transformations acting on the (extended) complex plane. The next chapter concerns discrete groups of hyperbolic isometries in nspace. This part ends with a discussion of elementary Kleinian groups, and the Fuchsian groups. The second part of the book treats more advanced topics such as geometrically finite groups.
Author(s): Bernard Maskit
Series: Grundlehren der mathematischen Wissenschaften v. 287
Publisher: Springer
Year: 1987
Language: English
Pages: 344
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Introduction......Page 8
Table of Contents......Page 12
I.A. Basic Concepts......Page 16
I. B. Classification of Fractional Linear Transformations......Page 19
I.C. Isometric Circles......Page 23
I. D. Commutators......Page 26
I. E. Fractional Reflections......Page 27
I.F. Exercises......Page 28
II.A. Discontinuous Groups......Page 30
II.B. Area, Diameter, and Convergence......Page 31
II.C. Inequalities for Discrete Groups......Page 33
II.D. The Limit Set......Page 36
II.E. The Partition of C......Page 38
II.F. Riemann Surfaces......Page 40
II.G. Fundamental Domains......Page 44
II.H. The Ford Region......Page 47
II.I. Precisely Invariant Sets......Page 50
II.J. Isomorphisms......Page 51
II. K. Exercises......Page 52
II.L. Notes......Page 54
III.A. Coverings......Page 56
III.B. Regular Coverings......Page 57
III.C. Lifting Loops and Regions......Page 60
III.D. Lifting Mappings......Page 61
III.E. Pairs of Regular Coverings......Page 63
III.F. Branched Regular Coverings......Page 64
III.G. Exercises......Page 66
IV.A. The Basic Spaces and their Groups......Page 68
IV.B. Hyperbolic Geometry......Page 74
IV.C. Classification of Elements of I_"......Page 77
IV.D. Convex Sets......Page 80
IV.E. Discrete Groups of Isometrics......Page 81
IV.F. Fundamental Polyhedrons......Page 83
IV.G. The Dirichlet and Ford Regions......Page 85
IV.H. Poincare's Polyhedron Theorem......Page 88
IV.I. Special Cases......Page 93
IV.J. Exercises......Page 95
IV.K. Notes......Page 98
V.A. Basic Signatures......Page 99
V.B. Half-Turns......Page 100
V.C. The Finite Groups......Page 102
V.D. The Euclidean Groups......Page 106
V.E. Applications to Non-Elementary Groups......Page 110
V.F. Groups with Two Limit Points......Page 114
V.G. Fuchsian Groups......Page 118
V.H. Isomorphisms......Page 124
V.I. Exercises......Page 126
V.J. Notes......Page 129
VI.A. The Boundary at Infinity of a Fundamental Polyhedron......Page 130
VI.B. Points of Approximation......Page 137
VI.C. Action near the Limit Set......Page 139
VI.D. Essentially Compact 3-Manifolds......Page 143
VI.E. Applications......Page 146
VI.F. Exercises......Page 147
VI.G. Notes......Page 149
VII.A. Combinatorial Group Theory - I......Page 150
VII.B. Blocks and Spanning Discs......Page 154
VII.C. The First Combination Theorem......Page 164
VII.D. Combinatorial Group Theory - II......Page 171
VII.E. The Second Combination Theorem......Page 175
VII.F. Exercises......Page 183
VII.G. Notes......Page 185
VIII.A. The Circle Packing Trick......Page 186
VIII.B. Simultaneous Uniformization......Page 190
VIII.C. Elliptic Cyclic Constructions......Page 192
VIII.D. Fuchsian Groups of the Second Kind......Page 200
VIII.E. Loxodromic Cyclic Constructions......Page 203
VIII.F. Strings of Beads......Page 215
VIII.G. Miscellaneous Examples......Page 220
VIII.H. Exercises......Page 225
VIII.I. Notes......Page 227
IX.A. An Inequality......Page 229
IX.B. Similarities......Page 231
IX.C. Rigidity of Triangle Groups......Page 232
IX. D. B-Group Basics......Page 235
IX.E. An Isomorphism Theorem......Page 241
IX.F. Quasifuchsian Groups......Page 247
IX.G. Degenerate Groups......Page 251
IX.H. Groups with Accidental Parabolic Transformations......Page 258
IX.I. Exercises......Page 261
IX.J. Notes......Page 263
X.A. The Planarity Theorem......Page 264
X.B. Panels Defined by Simple Loops......Page 270
X.C. Structure Subgroups......Page 273
X.D. Signatures......Page 286
X.E. Decomposition......Page 297
X.F. Existence......Page 306
X.G. Similarities and Deformations......Page 314
X.H. Schottky Groups......Page 326
X.I. Fuchsian Groups Revisited......Page 329
X.J. Exercises......Page 331
X.K. Notes......Page 333
Bibliography......Page 334
Special Symbols......Page 338
Index......Page 339
