Ordering Garside groups

Abstract

We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group [Formula: see text] and we show that if [Formula: see text] satisfies these two conditions, then [Formula: see text] has a Dehornoy structure. Then, we show that the Artin groups of type [Formula: see text] and of type [Formula: see text], [Formula: see text] satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type [Formula: see text] coincides with the Dehornoy order.