On the dynamics of (left) orderable groups

Abstract

Abstract. 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.Résumé. Nous développons des méthodes dynamiques pourétudier les groupes ordonnables ainsi que leurs espaces d'ordres associés. Nous donnons des preuves nouvelles etélémentaires de théorèmes dûsà Linnell (si un groupe ordonnable possède une infinité d'ordres, alors il possède une infinité non dénombrable) et McCleary (l'espace des ordres du groupe libre est un 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 commeétant le plus grand sous-groupe convexe sur lequel la relation est conradienne, et nous travaillons avec cette notion.

On spaces of Conradian group orderings

“…On the one hand, in [11] he noticed that in (1) and (2) above one may actually take n ¼ 2. The topological counterpart of this is the fact that the space of C-orderings is compact when it is endowed with a natural topology (see §1.1).…”

Groups of Circle Diffeomorphisms

“…The proof above together with Exercise 2.2.65 show that it has a natural dynamical counterpart: roughly, the Conrad property is the algebraic counterpart to the condition of non-existence of crossed elements for the corresponding action on the real line [174,176]. [176], suppose that the opposite inequality holds and show that for the positive elements f and g = f h in Γ one has f g n ≺ g for every n ∈ N.…”

“…A quite elegant result by Tararin [141] completely describes all orderable groups admitting only finitely many orderings. If a group admits infinitely many orderings, then it necessarily admits uncountably many [146,174,176]. For higher-rank torsion-free Abelian groups [225], for non-Abelian torsion-free nilpotent groups [176], and for non-Abelian free groups [162,176], the spaces of orderings are known to be homeomorphic to the Cantor set.…”

“…[176], suppose that the opposite inequality holds and show that for the positive elements f and g = f h in Γ one has f g n ≺ g for every n ∈ N.…”

“…The family of finitely generated groups of homeomorphisms of the interval containing finitely generated subgroups that do not admit nontrivial homomorphisms into (R, +) is quite large. Indeed, an important result in the theory of orderable groups asserts that a group (of arbitrary cardinality) is locally indicable (i.e., all of its finitely generated subgroups admit nontrivial homomorphisms into (R, +)) if and only if it is C-orderable (see §2.2.6 for the notion of C-order, and see [176] for an elementary proof of this result). Therefore, although all finitely generated groups of interval homeomorphisms are topologically conjugate to groups of Lipschitz homeomorphisms of [0, 1] (c.f., Proposition 2.3.15), many of them do not embed into Diff 1 + ([0, 1[) because they fail to be locally indicable.…”