Group orderings, dynamics, and rigidity
Kathryn Mann, Cristóbal Rivas · Annales de l’institut Fourier · 2018
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 S 1 and ℝ . As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO ( F 2 n ) 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 P 3 , answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.