**Mathematical Surveys and Monographs**

Volume: 148;
2008;
323 pp;
Hardcover

MSC: Primary 20;
Secondary 06; 57; 68

Print ISBN: 978-0-8218-4431-1

Product Code: SURV/148

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1375-0
Product Code: SURV/148.E**

List Price: $97.00

Individual Member Price: $77.60

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#### Supplemental Materials

# Ordering Braids

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*Patrick Dehornoy; Ivan Dynnikov; Dale Rolfsen; Bert Wiest*

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.

#### Table of Contents

# Table of Contents

## Ordering Braids

- Contents v7 free
- Preface ix11 free
- Introduction 113 free
- Chapter I. Braid Groups 921 free
- Chapter II. A Linear Ordering of Braid 1931
- Chapter III. Applications of the Braid Ordering 4355
- Chapter IV. Self-distributivity 5567
- Chapter V. Handle Reduction 8799
- Chapter VI. Connection with the Garside Structure 107119
- Chapter VII. Alternating Decompositions 129141
- Chapter VIII. Dual Braid Monoids 153165
- Chapter IX. Automorphisms of a Free Group 173185
- Chapter X. Curve Diagrams 185197
- Chapter XI. Relaxation Algorithms 195207
- Chapter XII. Triangulations 221233
- Chapter XIII. Hyperbolic Geometry 247259
- Chapter XIV. The Space of all Braid Orderings 265277
- Chapter XV. Bi-ordering the Pure Braid Groups 273285
- Chapter XVI. Open Questions and Extensions 291303
- Key Definitions 309321
- Bibliography 311323
- Index of Notation 319331 free
- Index 321333 free

#### Readership

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

#### Reviews

...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