Ordering braids.

*(English)*Zbl 1163.20024
Mathematical Surveys and Monographs 148. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4431-1/hbk). ix, 323 p. (2008).

This fascinating book discusses a variety of aspects of the ordering of the braid groups \(B_n\), \(B_\infty\). Besides a half dozen or so algorithms for establishing the order of a pair of elements (all leading to the same ordering on the group), other topics arising in the book include:

The set of all orderings on the groups, including the ordinal type and topology of various orderings,

The efficiency (complexity) of various algorithms for ordering,

Free groups, their automorphisms and orderable groups in general,

Ordering the subgroup of pure braids,

Self-distributive systems (sets with a binary operation \(*\) satisfying \(x*(y*z)=(x*y)*(x*z)\)),

Counting words in the braid group,

Mapping class groups of surfaces and the hyperbolic plane,

Biorderings.

Let \(B_n^+\) and \(B_n^{*+}\) denote the monoids of products of the Artin generators and the Birman-Ko-Lee generators, respectively. These monoids contain only positive elements. The central theme of the book is, as the authors state; proving and explaining the following theorem: The Artin braid group \(B_n\) is left-orderable, by an ordering which is a well-ordering when restricted to the braid monoids \(B_n^+\) and \(B_n^{*+}\).

A recurring task set for themselves by the authors, after each algorithm to determine positivity is defined, is the repeated examination of what they call “Properties A, B and C”. Property A – Positive braids are non-trivial. Property B – Every braid word is positive or negative. Property C – The Artin generators and their conjugates are positive.

Appropriate references are too numerous to list, definitions of orderings deriving from work of the authors as well as Garside, Mosher, Bressaud, E. Artin, Burckel, Magnus, Nielson-Thurston, and Birman-Ko-Lee and others.

There are many examples and instructive diagrams to illustrate principles involved. There are proofs of some results, sketches of proofs of others, and references for the rest. Each chapter is nearly self-contained, given some basic notation. The list below of chapter titles should be helpful in further indicating the book’s content.

Chapter I Braid Groups, Chapter II A Linear Ordering of Braids, Chapter III Applications of the Braid Ordering, Chapter IV Self-distributivity, Chapter V Handle Reduction, Chapter VI Connection with the Garside Structure, Chapter VII Alternating Decompositions, Chapter VIII Dual Braid Monoids, Chapter IX Automorphisms of a Free Group, Chapter X Curve Diagrams, Chapter XI Relaxation Algorithms, Chapter XII Triangulations, Chapter XIII Hyperbolic Geometry, Chapter XIV The Space of all Braid Orderings, Chapter XV Bi-ordering the Pure Braid Groups, Chapter XVI Open Questions and Extensions.

The set of all orderings on the groups, including the ordinal type and topology of various orderings,

The efficiency (complexity) of various algorithms for ordering,

Free groups, their automorphisms and orderable groups in general,

Ordering the subgroup of pure braids,

Self-distributive systems (sets with a binary operation \(*\) satisfying \(x*(y*z)=(x*y)*(x*z)\)),

Counting words in the braid group,

Mapping class groups of surfaces and the hyperbolic plane,

Biorderings.

Let \(B_n^+\) and \(B_n^{*+}\) denote the monoids of products of the Artin generators and the Birman-Ko-Lee generators, respectively. These monoids contain only positive elements. The central theme of the book is, as the authors state; proving and explaining the following theorem: The Artin braid group \(B_n\) is left-orderable, by an ordering which is a well-ordering when restricted to the braid monoids \(B_n^+\) and \(B_n^{*+}\).

A recurring task set for themselves by the authors, after each algorithm to determine positivity is defined, is the repeated examination of what they call “Properties A, B and C”. Property A – Positive braids are non-trivial. Property B – Every braid word is positive or negative. Property C – The Artin generators and their conjugates are positive.

Appropriate references are too numerous to list, definitions of orderings deriving from work of the authors as well as Garside, Mosher, Bressaud, E. Artin, Burckel, Magnus, Nielson-Thurston, and Birman-Ko-Lee and others.

There are many examples and instructive diagrams to illustrate principles involved. There are proofs of some results, sketches of proofs of others, and references for the rest. Each chapter is nearly self-contained, given some basic notation. The list below of chapter titles should be helpful in further indicating the book’s content.

Chapter I Braid Groups, Chapter II A Linear Ordering of Braids, Chapter III Applications of the Braid Ordering, Chapter IV Self-distributivity, Chapter V Handle Reduction, Chapter VI Connection with the Garside Structure, Chapter VII Alternating Decompositions, Chapter VIII Dual Braid Monoids, Chapter IX Automorphisms of a Free Group, Chapter X Curve Diagrams, Chapter XI Relaxation Algorithms, Chapter XII Triangulations, Chapter XIII Hyperbolic Geometry, Chapter XIV The Space of all Braid Orderings, Chapter XV Bi-ordering the Pure Braid Groups, Chapter XVI Open Questions and Extensions.

Reviewer: Lee P. Neuwirth (Princeton)

##### MSC:

20F36 | Braid groups; Artin groups |

20F60 | Ordered groups (group-theoretic aspects) |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

03E10 | Ordinal and cardinal numbers |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20F38 | Other groups related to topology or analysis |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

51M09 | Elementary problems in hyperbolic and elliptic geometries |

68Q25 | Analysis of algorithms and problem complexity |

68Q70 | Algebraic theory of languages and automata |

68W30 | Symbolic computation and algebraic computation |

20F05 | Generators, relations, and presentations of groups |

20N02 | Sets with a single binary operation (groupoids) |

06F15 | Ordered groups |