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Garside groups. (Groupes de Garside.) (French) Zbl 1017.20031

The author studies a family of monoids and groups, called Garside monoids and Garside groups. Garside monoids are natural extensions of the positive braid monoids introduced by F. A. Garside [Q. J. Math., Oxf. II. Ser. 20, 235-254 (1969; Zbl 0194.03303)]. Briefly speaking, a Garside monoid is defined to be a monoid whose left and right divisibility relations are lattice relations, plus the existence of a certain element, called Garside element, whose left and right divisors coincide, are finite, and generate the monoid. A Garside group is the group of fractions of a Garside monoid. The author explores many equivalent conditions for a monoid (resp. a group) to be a Garside monoid (resp. a Garside group), some of them being algorithmic, and proves some combinatorial properties for these groups. In particular, he shows that Garside groups are biautomatic.
Reviewer: Luis Paris (Dijon)

MSC:

20F36 Braid groups; Artin groups
20M05 Free semigroups, generators and relations, word problems
20F65 Geometric group theory
57M07 Topological methods in group theory

Citations:

Zbl 0194.03303
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