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http://dx.doi.org/10.1090/surv/148 Ordering Braids Mathematical Surveys and Monographs Volume 148 Ordering Braids Patrick Dehornoy with Ivan Dynnikov Dale Rolfsen Bert Wiest American Mathematical Society Providence, Rhode Island Editorial Committee Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Benjamin Sudakov J. T. Stafford, Chair The original edition of this work was published under the title Why Are Braids Order- able? by Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, 2002, by the Soci´et´eMath´ematique de France, Paris, France. 2000 Mathematics Subject Classification. Primary 20F36; Secondary 06F05, 06F15, 20B30, 20F34, 20F38, 57M25, 57M50, 68Q25, 68Q70. For additional information and updates on this book, visit www.ams.org/bookpages/surv-148 Library of Congress Cataloging-in-Publication Data Dehornoy, Patrick. Ordering braids / Patrick Dehornoy ; with Ivan Dynnikov, Dale Rolfsen, Bert Wiest. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 148) Includes bibliographical references and index. ISBN 978-0-8218-4431-1 (alk. paper) 1. Braid theory. 2. Linear orderings. I. Dynnikov, Ivan, 1971– II. Rolfsen, Dale. III. Wiest, Bert. IV. Title. QA612.23.D445 2008 514.224—dc22 2008009859 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgmentofthesourceisgiven. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 131211100908 Contents Preface ix 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 σ-ordering of Bn 19 2. Local properties of the σ-ordering 26 3. Global properties of the σ-ordering 29 4. The σ-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 v vi CONTENTS d 3. The increasing enumeration of Div(∆n) 117 Chapter VII. Alternating Decompositions 129 + 1. The Φn-splitting of a braid in Bn 130 2. The Φ -normal form 135 3. Burckel’s approach 143 4. Applications 148 Chapter VIII. Dual Braid Monoids 153 1. Dual braid monoids 154 +∗ 2. The φ -normal form on Bn 159 3. Connection between orders 163 Chapter IX. Automorphisms of a Free Group 173 1. Artin representation of σ-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,andS 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 PBn 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 CONTENTS vii Key Definitions 309 Bibliography 311 Index of Notation 319 Index 321 Preface The present volume follows a book, “Why are braids orderable?”, written by the same authors and published in 2002 by the Soci´et´eMath´ematique de France in the series Panoramas et Synth`eses. We emphasize that this is not a new edition of that book. Although this book contains most of the material in the previous book, it also contains a considerable amount of new material. In addition, much of the original text has been completely rewritten, with a view to making it more readable and up-to-date. We have been able not only to include ideas that were unknown in 2002, but we have also benefitted from helpful comments by colleagues and students regarding the contents of the SMF book, and we have taken their advice to heart in writing this book. The reader is assumed to have some basic background in group theory and topology. However, we have attempted to make the ideas in this volume accessible and interesting to students and seasoned professionals alike. In fact, the question “Why are braids orderable?” has not been answered to our satisfaction, either in the book with that title or the present volume. That is, we do not understand precisely what makes the braid groups so special that they enjoy an ordering so easy to describe, so challenging to construct and with such subtle properties as are described in these pages. The best we can offer is some insight into the easier question, “How are braids orderable?” Patrick Dehornoy, Caen Ivan Dynnikov, Moscow Dale Rolfsen, Vancouver Bert Wiest, Rennes December 2007 ix Key Definitions Sigma-ordering: −1 • For β,β in B∞,therelationβ<β is true if β β is σ-positive. Φ −1 Φ • For β,β in B∞,therelationβ< β is true if β β is σ -positive. Sigma-positive braid word: • A braid word is σ-positive if the σi with lowest index occurs positively only. • Φ A braid word is σ -positive if the σi with highest index occurs positively only. Sigma-positive braid: • Abraidisσ-positive if it admits a σ-positive representative word. • AbraidisσΦ-positive if it admits a σΦ-positive representative word. Property A (Acyclicity): • A σ-positive braid is nontrivial. Property C (Comparison): • Every nontrivial braid of Bn can be represented by an n-strandbraidwordthat is σ-positive or σ-negative. Property S (Subword): • −1 Every braid of the form β σiβ is σ-positive. Complementary Definitions • −1 ±1 A braid word is σi-positive if it contains at least one σi,noσi ,noσj with j<i. • −1 ±1 ... id. ... σi-negative if ... at least one σi ,noσi,noσj with j<i. • ±1 ... id. ... σi-free if ... no σj with j i. • A braid is called σi-positive if it admits a σi-positive expression, etc. Property A (second, equivalent form): A σ1-positive braid is nontrivial. Property C (second, equivalent form): Every braid of Bn can be represented by an n-strandbraidwordthatisσ1-positive, σ1-negative, or σ1-free. 309 Bibliography 1. S.I. Adyan, Fragments of the word Delta in a braid group, Mat. Zametki Acad. Sci. SSSR 36 (1984), no. 1, 25–34, (Russian); English translation in Math. Notes of the Acad. Sci. USSR 36 (1984), no. 1, p. 505–510. 2. I. Agol, J. Hass, and W.P. Thurston, The computational complexity of knot genus and spanning area, Trans. Amer. Math. Soc. 358 (2006), no. 9, 3821–3850, (electronic). 3. I. Anshel, M. Anshel, and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), 287–291. 4. E. Artin, Theorie der Z¨opfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72. 5. , Theory of braids, Ann. of Math. 48 (1947), 101–126. 6. S. Arworn and Y. Kim, On finitely-determined total orders,preprint. 7. L. Bacardit and W. Dicks, Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, arXiv: math.GR/0705.0587. 8. V.G. Bardakov, On the theory of braid groups, Mat. Sb. 183 (1992), no. 6, 3–42, (Russian. English summary); English translation in Acad. Sci. Sb. Math. 76 (1993), no. 1, p. 123–153. 9. H. Bass and A. Lubotzky, Linear-central filtrations on groups, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math., vol. 169, American Mathematical Society, 1994, pp.
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