Group orderings, dynamics and rigidity

Abstract

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-2n ) 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. Keywords. Author Keywords:Orderable groups; actions on the circle; spaces of orders. KeyWords Plus:SPACES