Group orderings, dynamics, and rigidity

2016/06/30 by Mann, Kathryn, Rivas, Cristobal
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1607.00054

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; as well as 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(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 P₃, answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.

Related

• Ping-pong configurations and circular orders on free groups
• Dynamics of isolated orders
• On the dynamics of (left) orderable groups
• On the number of circular orders on a group
• Orderable groups