Smith Conjecture (eBook)

Front Cover 1
The Smith Conjecture, Volume 112 4
Copyright Page 5
Contents 6
Contributors 10
Preface 12
Acknowledgments 14
List of Notation 16
PART A: INTRODUCTION 18
Chapter 1. The Smith Conjecture 20
1. Formulations 20
2. Generalizations 22
3. Some Consequences Relating to the Poincaré Conjecture 22
4. Additional Remarks 23
Chapter 2. History of the Smith Conjecture and Early Progress 24
1. History of the Smith Conjecture 24
2. Early Progress 25
Chapter 3. An Outline of the Proof 28
1. Preliminaries 28
2. First Reductions 30
3. The Argument in Brief 30
References for Part A 34
PART B: THE CASE OF NO INCOMPRESSIBLE SURFACE 36
Chapter 4. The Proof in the Case of No Incompressible Surface 38
Introduction 38
1. The Algebraic Approach to the Smith Conjecture 39
2. Hyperbolic Geometry and Algebraic Integers 40
3. The Existence of Hyperbolic Structures and the Torus Theorem 45
4. PSL2(C) and Incompressible Surfaces 48
5. History 52
References 53
Chapter 5. On Thurston’s Uniformization Theorem for Three-Dimensional Manifolds 54
Introduction 54
1. An Introduction to Hyperbolic Geometry 60
2. Kleinian Groups 64
3. Statement of the Main Theorem—The Case of Finite Volume 68
4. Hierarchies and Pared Manifolds 72
5. Statement of the Main Theorem—The General Case 77
6. Convex Hyperbolic Structures of Finite Volume 78
7. The Gluing Theorem—Statement and First Reduction 87
8. Combination Theorems 91
9. Deformation Theory 95
10. The Fixed Point Theorem 101
11. The First Step in the Proof of the Bounded Image Theorem 104
12. Completion of the Proof of the Bounded Image Theorem 112
13. Special Cases 123
14. Kleinian Groups with Torsion 124
15. Patterns of Circles 134
16. The Inductive Step in the Proof of Theorems A’ and B’ 136
References 141
Chapter 6. Finitely Generated Subgroups of GL2 144
1. The GL2-Subgroup Theorem 144
2. Arboreal Group Theory 147
3. The Tree of SL2 over a Local Field 149
4. Proof of the GL2-Subgroup Theorem 151
References 153
PART C: THE CASE OF AN INCOMPRESSIBLE SURFACE 154
Chapter 7. Incompressible Surfaces in Branched Coverings 156
1. Introduction 156
2. Terminology and Statement of Results 157
3. Proofs of Theorems 1 and 2 159
4. The Equivalent Loop Theorem for Involutions 164
References 168
Chapter 8. The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces 170
1. Morrey’s Solution for the Plateau Problem in a General Riemannian Manifold 171
2. The Existence Theorem for Manifolds with Boundary 176
3. Existence of Closed Minimal Surfaces 177
4. Existence of the Free Boundary Value Problem for Minimal Surfaces 179
References 180
PART D: GENERALIZATIONS 182
Chapter 9. Group Actions on R3 184
References 196
Chapter 10. Finite Group Actions on Homotopy 3-Spheres 198
1. Orbifolds 200
2. Two-Dimensional Orbifolds 204
3. Three-Dimensional Orbifolds 206
4. Seifert-Fibered Orbifolds 209
5. Seifert-Fibered Orbifolds and Linear Actions 213
6. Statement of the Main Results 223
7. A Special Case 224
8. Completion of the Proof 230
Appendix 233
References 242
Chapter 11. A Survey of Results in Higher Dimensions 244
1. The Montgomery-Samuelson Example 247
2. G-Complexes 247
3. The Brieskorn Examples 248
4. Oliver's Example 249
5. Local Properties: Groups of Homeomorphisms versus Groups of Diffeomorphisms 251
6. Work of Lowell Jones 252
7. Actions on Disks 253
8. Actions on Spheres 255
9. Actions on Euclidean Spaces 255
References 256
Index 258