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Foundations of Garside Theory
EMS Tracts in Mathematics Vol. 22

Patrick Dehornoy (Université de Caen, France)
François Digne (Université de Picardie Jules-Verne, Amiens, France)
Eddy Godelle (Université de Caen, France)
Daan Krammer (University of Warwick, Coventry, UK)
Jean Michel (Université Denis Diderot Paris 7, France)

Foundations of Garside Theory

ISBN print 978-3-03719-139-2, ISBN online 978-3-03719-639-7
DOI 10.4171/139
June 2015, 710 pages, hardcover, 17 x 24 cm.
108.00 Euro

Winner of the 2014 EMS Monograph Award!

This text is a monograph in algebra, with connections toward geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims at providing 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 turned out to lead to interesting results in a number of cases, the central notion being what the authors call a Garside family. At the moment, the study is far from complete, and the purpose of this book is both to present the current state of the theory and to be an invitation for further research.

There are two parts: the bases of a general theory, including many easy examples, are developed in Part A, whereas various more sophisticated examples are specifically addressed in Part B.

In order to make the content accessible to a wide audience of nonspecialists, exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use the current text 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, 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.

Keywords: Group, monoid, category, greedy decomposition, normal decomposition, symmetric normal decomposition, Garside family, Garside map, Garside element, Garside monoid, Garside group, word problem, conjugacy problem, braid group, Artin–Tits group, Deligne–Luzstig variety, self-distributivity, ordered group, Yang–Baxter equation, cell decomposition

Further Information

Authors' book page

Review in MR 3362691

Review in Zentralblatt MATH