On the dynamics of (left) orderable groups

Navas, Andrés

Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1685-1740.

Nous développons des méthodes dynamiques pour étudier les groupes ordonnables ainsi que leurs espaces d’ordres associés. Nous donnons des preuves nouvelles et élémentaires de théorèmes dus à Linnell (si un groupe ordonnable possède une infinité d’ordres, alors il en possède une infinité non dénombrable) et McCleary (l’espace des ordres du groupe libre est un ensemble de Cantor). Nous montrons que ce dernier résultat est valable aussi pour les groupes nilpotents dénombrables et sans torsion qui ne sont pas abéliens de rang un. Finalement, nous appliquons nos méthodes au cas des groupes de tresses. En particulier, nous démontrons que le cone positif de l’ordre de Dehornoy n’est pas de type fini en tant que semi-groupe. Pour ce faire, nous définissons le noyau conradien d’un ordre comme étant le plus grand sous-groupe convexe sur lequel la relation est conradienne, et nous travaillons avec cette notion.

We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid groups. In particular, we show that the positive cone of the Dehornoy ordering is not finitely generated as a semigroup. To do this, we define the Conradian soul of an ordering as the maximal convex subgroup restricted to which the ordering is Conradian, and we elaborate on this notion.

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