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Ordering Braids

Patrick Dehornoy Université de Caen, Caen, France and Institut Universitaire de France, Paris, France
Ivan Dynnikov Moscow State University, Moscow, Russia
Dale Rolfsen University of British Columbia, Vancouver, BC, Canada
Bert Wiest Université de Rennes, Rennes, France
Available Formats:
Hardcover ISBN: 978-0-8218-4431-1
Product Code: SURV/148
List Price: $103.00 MAA Member Price:$92.70
AMS Member Price: $82.40 Electronic ISBN: 978-1-4704-1375-0 Product Code: SURV/148.E List Price:$97.00
MAA Member Price: $87.30 AMS Member Price:$77.60
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List Price: $154.50 MAA Member Price:$139.05
AMS Member Price: $123.60 Click above image for expanded view Ordering Braids Patrick Dehornoy Université de Caen, Caen, France and Institut Universitaire de France, Paris, France Ivan Dynnikov Moscow State University, Moscow, Russia Dale Rolfsen University of British Columbia, Vancouver, BC, Canada Bert Wiest Université de Rennes, Rennes, France Available Formats:  Hardcover ISBN: 978-0-8218-4431-1 Product Code: SURV/148  List Price:$103.00 MAA Member Price: $92.70 AMS Member Price:$82.40
 Electronic ISBN: 978-1-4704-1375-0 Product Code: SURV/148.E
 List Price: $97.00 MAA Member Price:$87.30 AMS Member Price: $77.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$154.50 MAA Member Price: $139.05 AMS Member Price:$123.60
• Book Details

Mathematical Surveys and Monographs
Volume: 1482008; 323 pp
MSC: Primary 20; Secondary 06; 57; 68;

In the fifteen years since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different approaches have been used to understand this phenomenon. This book is an account of those approaches, which involve such varied objects and domains as combinatorial group theory, self-distributive algebra, finite combinatorics, automata, low-dimensional topology, mapping class groups, and hyperbolic geometry. The remarkable point is that all these approaches lead to the same ordering, making the latter rather canonical.

We have attempted to make the ideas in this volume accessible and interesting to students and seasoned professionals alike. Although the text touches upon many different areas, we only assume that the reader has some basic background in group theory and topology, and we include detailed introductions wherever they may be needed, so as to make the book as self-contained as possible.

The present volume follows the book, Why are braids orderable?, written by the same authors and published in 2002 by the Société Mathématique de France. The current text contains a considerable amount of new material, including ideas that were unknown in 2002. In addition, much of the original text has been completely rewritten, with a view to making it more readable and up-to-date.

Readership

Graduate students and research mathematicians interested in braid, group theory, low-dimensional topology.

• Table of Contents

• Chapters
• Introduction
• 1. Braid groups
• 2. A linear ordering of braids
• 3. Applications of the braid ordering
• 4. Self-distributivity
• 5. Handle reduction
• 6. Connection with the Garside structure
• 7. Alternating decompositions
• 8. Dual braid monoids
• 9. Automorphisms of a free group
• 10. Curve diagrams
• 11. Relaxation algorithms
• 12. Triangulations
• 13. Hyperbolic geometry
• 14. The space of all braid orderings
• 15. Bi-ordering the pure braid groups
• 16. Open questions and extensions
• Additional Material

• Reviews

• From a review of the previous edition:

...this is a timely and very carefully written book describing important, interesting and beautiful results in this new area of research concerning braid groups. It will no doubt create much interest and inspire many more insights into these order structures.

Stephen P. Humphries for Mathematical Reviews
• Request Review Copy
• Get Permissions
Volume: 1482008; 323 pp
MSC: Primary 20; Secondary 06; 57; 68;

In the fifteen years since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different approaches have been used to understand this phenomenon. This book is an account of those approaches, which involve such varied objects and domains as combinatorial group theory, self-distributive algebra, finite combinatorics, automata, low-dimensional topology, mapping class groups, and hyperbolic geometry. The remarkable point is that all these approaches lead to the same ordering, making the latter rather canonical.

We have attempted to make the ideas in this volume accessible and interesting to students and seasoned professionals alike. Although the text touches upon many different areas, we only assume that the reader has some basic background in group theory and topology, and we include detailed introductions wherever they may be needed, so as to make the book as self-contained as possible.

The present volume follows the book, Why are braids orderable?, written by the same authors and published in 2002 by the Société Mathématique de France. The current text contains a considerable amount of new material, including ideas that were unknown in 2002. In addition, much of the original text has been completely rewritten, with a view to making it more readable and up-to-date.

Readership

Graduate students and research mathematicians interested in braid, group theory, low-dimensional topology.

• Chapters
• Introduction
• 1. Braid groups
• 2. A linear ordering of braids
• 3. Applications of the braid ordering
• 4. Self-distributivity
• 5. Handle reduction
• 6. Connection with the Garside structure
• 7. Alternating decompositions
• 8. Dual braid monoids
• 9. Automorphisms of a free group
• 10. Curve diagrams
• 11. Relaxation algorithms
• 12. Triangulations
• 13. Hyperbolic geometry
• 14. The space of all braid orderings
• 15. Bi-ordering the pure braid groups
• 16. Open questions and extensions
• From a review of the previous edition:

...this is a timely and very carefully written book describing important, interesting and beautiful results in this new area of research concerning braid groups. It will no doubt create much interest and inspire many more insights into these order structures.

Stephen P. Humphries for Mathematical Reviews
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