Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

Abstract

B have a lowest common right multiple. Moreover, B has left and right cancellation properties, namely, ab ac implies b c, and ba ca implies b c. Ore's criterion says: if a monoid M has left and right cancellation properties, and if any two elements of M have a common right multiple, then M embeds in its group of (right) fractions (see [10, Theorem 1.23]). This group is M Ã M À1 =, where M À 1 is the dual monoid of M, and is the congruence relation generated by the pairs xx À1 ; 1 and x À1 x; 1, with x in M. By the previous considerations, B satis®es Ore's conditions, and, therefore, embeds in its group of fractions. This is the braid group on n 1 strings.The fundamental element of B , usually denoted by D, is the lowest common right multiple of x 1 ; . . . ; x n . It is also the lowest common left multiple of x 1 ; . . . ; x n , and D 2 generates the centre of the braid group. Furthermore, the set of left divisors of D is equal to the set of right divisors of D.This situation was simultaneously generalised by Brieskorn and Saito [5], and by Deligne [15], to a family of monoids and groups called ®nite Coxeter type Artin monoids and groups. Like the braid groups, these groups have nice normal forms (see [5] and [15]), have fast word problem solutions (see [28]), and are biautomatic (see [8] and [9]), all these properties being proved through a deep study of the Artin monoids.In this paper, we shall extend the previous results to a larger class of monoids and groups, which we naturally propose to term Garside. These groups are 1991 Mathematics Subject Classi®cation: primary 20F05, 20F36; secondary 20B40, 20M05.Proc. London Math. Soc. (3) 79 (1999) 569±604. G R S; f respectively. In [14], the ®rst author shows that, under certain conditions described in § 3, the monoid M R S; f is left Gaussian.In § 4 we prove that the converse is true, namely, if M is a right Gaussian 570 patrick dehornoy and luis paris monoid, then it has the form M R S; f for some S and f that satisfy the conditions mentioned above. Then we describe necessary, suf®cient, and effective conditions for M R S; f to be a Garside monoid. This is applied in § 5 to exhibit in®nite families of Garside groups that include torus knot groups, fundamental groups of complements of complex lines through the origin, and some`braid groups' associated with complex re¯ection groups. Our main tool in this paper is an algorithmic process, called the word reversing process. It is described in § 3. It was ®rst introduced in [11] and [12] in order to study a special group related to the self-distributive identity, and was developed in [14]. It is shown in [14] that this word reversing process gives rise to very simple algorithms which solve the word problem in a group of the form G R S; f , whenever S; f satis®es the conditions mentioned before. In particular, they apply to the braid groups and have, in this case, a quadratic complexity. Tatsuoka uses in [28] a similar algorithmic process for showing that ®nite Coxeter type Artin groups have quadratic isoperimetric inequ...

Finite complex reflection arrangements are K(pi,1)

“…Garside's approach [36] to the word and conjugacy problem in the classical braid group B n was a key ingredient in Deligne's paper [31]. It was later axiomatized as a generic combinatorial group theory notion [28], rephrased with a geometric group theory viewpoint [11,24] and generalized to groupoids [39].…”

The cyclic sliding operation in Garside groups

“…Garside groups were defined by Dehornoy and Paris [12]. For a detailed introduction to these groups, see [13]; a shorter introduction, containing all the details needed for this paper can be found in [5] ( §1.1 and the beginning of §1.2).…”

A class of Garside groupoid structures on the pure braid group

“…Good overviews on Garside groups are [DehPar99], [Deh00], [Deh02]. Our summary is mainly based on [Deh00], but our terminology is mainly taken from [Deh02].…”