Group orderings, dynamics, and rigidity

TL;DR

This paper explores the relationship between group orderings and group actions on circles and lines, providing characterizations of isolated orders and constructing examples that challenge previous conjectures.

Contribution

It establishes a topological link between order spaces and action moduli spaces, characterizes isolated orders, and constructs counterexamples to existing conjectures.

Findings

Characterization of isolated orders via action rigidity

Construction of infinitely many nonconjugate isolated circular orders on free groups

Counterexamples to conjectures on order spaces of free groups and braid groups

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 $S^1$ 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.