Gaussian Groups Are Torsion Free

Assume that G is a group of fractions of a cancellative monoid where lower common multiples exist and divisibility has no infinite descending chain. Then G is torsion free. The result applies in particular to all finite Coxeter type Artin groups. Finding an elementary proof for the fact that Artin’s braid groups are torsion free has been reported to be a longstanding open question [9]. The existence of a linear ordering of the braids that is left compatible with product [4] has provided such a proof—see also [10]. The argument applies to Artin groups of type Bn as well, but it remains rather specific, and there seems to be little hope to extend it to a much larger family of groups. On the other hand, we have observed in [5] and [6] that Garside’s analysis of the braids [8] applies to a large family of groups, namely all groups of fractions associated with certain monoids where divisibility has a lattice structure or, equivalently, all groups that admit a presentation of a certain syntactic form. Such groups have been called Gaussian in [6]. It is shown in the latter paper that all finite Coxeter type Artin groups, as well as a number of other groups like torus knot groups or some complex reflection groups, are Gaussian. In the present paper, we give an extremely simple argument proving that all Gaussian groups are torsion-free. However, the argument applies to an even larger family of groups of fractions. Assume that M is a monoid. For a, b in M , we say that b is a proper right divisor of a—or that a is a proper left multiple of b—if there exist c = 1 such that a is cb. We say that M is right Noetherian if the relation of being a proper right divisor has no infinite descending chain. By standard arguments, this is equivalent to the existence of a mapping ρ of M to the ordinals such that ρ(cb) > ρ(b) holds whenever c is not 1. We say that the monoid M is right Gaussian if it is right Noetherian, left cancellative, and every pair of elements (a, b) in M admits a right lower common multiple, i.e., there exists an element c that is a right multiple both of a and b and every common right multiple of a and b is a right multiple of c. The present notion of a right Gaussian monoid is slightly more general than the one considered in [6], which essentially corresponds to the special case where the rank function ρ mentioned above has integer values. Left Gaussian monoids are defined symmetrically. A Gaussian monoid is a monoid that is both left and