A new characterization of Conrad's property for group orderings, with applications

TL;DR

This paper introduces an algebraic approach to Conrad's property for group orderings, enabling new proofs and applications, including demonstrating the uncountability of orderings in certain groups.

Contribution

It provides a pure algebraic characterization of Conrad's property and extends results about orderings to uncountable groups with new constructive proofs.

Findings

Algebraic characterization of Conrad's property

Uncountability of orderings in certain groups proven constructively

Example of an exotic ordering on the free group

Abstract

We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given in the Appendix.