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; 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 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.
Cite
@article{arxiv.1607.00054,
title = {Group orderings, dynamics, and rigidity},
author = {Kathryn Mann and Cristobal Rivas},
journal= {arXiv preprint arXiv:1607.00054},
year = {2017}
}