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 Societe Mathematique 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.

Ordering Braids (Mathematical Surveys and Monographs)

Science & Math

Dale Rolfsen

Patrick Dehornoy

Ivan Dynnikov

and Bert Wiest

Mathematics

American Mathematical Society

This text is a monograph on algebra, with connections to geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims to provide a unified treatment for those situations in which one can find distinguished decompositions by iteratively extracting a maximal fragment lying in a prescribed family. Initiated in 1969 by F. A. Garside in the case of Artin's braid groups, this approach led to interesting results in a number of cases, the central notion being what the authors call a Garside family. The study is far from complete, and the purpose of this book is to present the current state of the theory and to invite further research. The book has two parts: In Part A, the bases of a general theory, including many easy examples, are developed. In Part B, various more sophisticated examples are specifically addressed. To make the content accessible to a wide audience of nonspecialists, the book's exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use this as a textbook both for Garside theory and for the more specialized topics investigated in Part B: Artin-Tits groups, Deligne-Lusztig varieties, groups of algebraic laws, ordered groups, and structure groups of set-theoretic solutions of the Yang-Baxter equation. The first part of the book can be used as the basis for a graduate or advanced undergraduate course. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Foundations of Garside Theory (EMS Tracts in Mathematics)

European Mathematical Society

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