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Introduction 1 A meeting of two classical subjects 1 A convergence of approaches 3 Organization of the text 5 Guidelines to the reader 7 Acknowledgements 8 Chapter I. Braid Groups 9 1. The Artin presentation 9 2. Isotopy classes of braid diagrams 10 3. Mapping class groups 12 4. Positive braids 14 Chapter II. A Linear Ordering of Braids 19 1. The a-ordering of B, 19 2. Local properties of the a-ordering 26 3. Global properties of the a-ordering 29 4. The a-ordering of positive braids 35 Chapter III. Applications of the Braid Ordering 43 1. Consequences of orderability 44 2. Applications of more specific properties 46 3. Application of well-orderability 50 Chapter IV. Self-distributivity 55 1. Colouring positive braids 56 2. Colouring arbitrary braids 66 3. The group of left self-distributivity 76 4. Normal forms in free LD-systems 81 5. Appendix: Iterations of elementary embeddings in set theory 84 Chapter V. Handle Reduction 87 1. Description of handle reduction 87 2. Convergence of handle reduction 92 3. Special cases and variants 102 Chapter VI. Connection with the Garside Structure 107 1. The degree of a positive braid 108 2. Proving Property C using a counting argument 113 3. The increasing enumeration of Div(Ad) 117 Chapter VII. Alternating Decompositions 129 1. The 4nF-splitting of a braid in B+ 130 2. The 4-normal form 135 3. Burckel's approach 143 4. Applications 148 Chapter VIII. Dual Braid Monoids 153 1. Dual braid monoids 154 2. The 0-normal form on B'* 159 3. Connection between orders 163 Chapter IX. Automorphisms of a Free Group 173 1. Artin representation of a-positive braids 173 2. From an automorphism back to a braid 178 3. Pulling back orderings of free groups 182 Chapter X. Curve Diagrams 185 1. A braid ordering using curve diagrams 185 2. Proof of Properties A, C, and S 189 Chapter XI. Relaxation Algorithms 195 1. Bressaud's regular language of relaxation braids 196 2. The transmission-relaxation normal form of braids 204 Chapter XII. Triangulations 221 1. The coordinates of a braid 222 2. Triangulations and laminations 225 3. The Mosher normal form 236 Chapter XIII. Hyperbolic Geometry 247 1. Uncountably many orderings of the braid group 248 2. The classification of orderings induced by the action on R 256 3. The subword property for all Nielsen-Thurston type orderings 263 Chapter XIV. The Space of all Braid Orderings 265 1. The spaces of orderings on a group 265 2. The space of left-orderings of the braid groups 268 Chapter XV. Bi-ordering the Pure Braid Groups 273 1. Lower central series 273 2. Artin coordinates and Magnus expansion 274 3. The Magnus ordering of PB, 279 4. The ordering of positive pure braids 283 5. Incompatibility of the orderings 286 Chapter XVI. Open Questions and Extensions 291 1. General questions 291 2. More specific questions 293 3. Generalizations and extensions 301 Key Definitions 309 Bibliography 311 Index of Notation 319

Library of Congress subject headings for this publication: Braid theory, Linear orderings