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Why are braids orderable? (English) Zbl 1048.20021
Panoramas et Synthèses 14. Paris: Société Mathématique de France (ISBN 2-85629-135-X/pbk). xiii, 190 p. (2002).
It is over ten years since the discovery that Artin’s braid groups admit a left-invariant total order, extending the partial order defined in terms of positive powers of the Artin generators.
What appeared initially to be an interesting curiosity has come to have a wide influence on work involving combinatorial group theory, mapping class groups and 3-manifolds with the discovery of a number of very different ways, both algebraic and geometric, of describing this and similar orderings. It has had some unexpected consequences, such as the exclusion of zero divisors from the group ring $$\mathbb{Z} B_n$$ of the braid groups.
This monograph is an account of a wide range of the techniques and consequences, put in the more general context of group orderability, with chapters from a purely algebraic viewpoint alongside results from Nielsen theory and the general angle of mapping class groups. It is a substantial and well-written account, which maintains a clear style in the face of several authors with widely differing backgrounds.
After an initial notational chapter there are four chapters on combinatorial approaches to orderability, followed by three on topological approaches, and a final chapter on the pure braid groups, which behave more like free groups in this setting, and admit an ordering which is invariant on both sides.
The text is prefaced by a helpful introduction giving a useful overview of the ramifications of the whole topic.

##### MSC:
 20F36 Braid groups; Artin groups 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20F60 Ordered groups (group-theoretic aspects) 06F15 Ordered groups 20M05 Free semigroups, generators and relations, word problems 57M60 Group actions on manifolds and cell complexes in low dimensions