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...

Abstract. We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective. Mathematics Subject Classification (2000). Primary 20F36; Secondary 20F55.

This paper is a study of the subgroups of mapping class groups of Riemann surfaces, called "geometric" subgroups, corresponding to the inclusion of subsurfaces. Our analysis includes surfaces with boundary and with punctures. The centres of all the mapping class groups are calculated. We determine the kernel of inclusion-induced maps of the mapping class group of a subsurface, and give necessary and sufficient conditions for injectivity. In the injective case, we show that the commensurability class of a geometric subgroup completely determines up to isotopy the defining subsurface, and we characterize centralizers, normalizers, and commensurators of geometric subgroups.

Let A be an Artin group with standard generating set {σ s : s ∈ S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σ 2 s and σ 2 t commute whenever σ s and σ t commute, for s, t ∈ S. In this paper we prove Tits' conjecture for all Artin groups. In fact, given a number m s ≥ 2 for each s ∈ S, we show that the elements {T s = σ ms s : s ∈ S} generate a subgroup that has a finite presentation in which the only defining relations are that T s and T t commute if σ s and σ t commute.