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On the dynamics of (left) orderable groups. (English) Zbl 1316.06018

Summary: 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.

MSC:

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
20F36 Braid groups; Artin groups
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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