Group Orderings, Dynamics, and Rigidity

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; and an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on S1 and R. As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F2n) of circular orders on free groups, disproving a conjecture from Baik–Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P3, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders. © 2018 Association des Annales de l Institut Fourier. All rights reserved.