Patrick Dehornoy

The present volume follows a book, "Why Are Braids Orderable?", written by the same authors and published in 2002 by the Société Mathématique de France. 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.

In the decade since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different methods have been applied to understand this phenomenon. This book is an account of those techniques, including self-distributive algebra, finite trees, combinatorial group theory, mapping class groups, laminations and hyperbolic geometry.

The aim of this text is to give a first synthesis of recent works that connect Artin's braid groups and left self-distributive algebra, defined as the study of those algebraic systems that involve a binary operation satisfying the left self-distributivity identity x(yz)=(xy)(xz). The emphasis is put on the geometric features, as illustrated in the slogan: "The geometry of braids is a projection of the geometry of left self-distributivity". The text is an introduction to four objects, which had never been considered twelve years ago, but which have such simple definitions and such rich properties that they seem to deserve some attention. These objects are:

• the free left self-distributive system of rank 1, with its canonical ordering, its normal forms, and its distinguished realizations, one as exponentiation of braids, and one as the iterations of an elementary embedding in set theory;
• the canonical linear ordering of braids, with its several equivalent characterizations, the comparison algorithms and the well-ordering of positive braids;
• the finite self-distributive Laver tables A_n, with the mysterious asymptotic behaviour of their periods that is known only assuming an unprovable large cardinal axiom;
• the group that describes the geometry of self-distributivity, with its connection with braids that explains both the existence of braid exponentiation and of the linear ordering of braids.