Bi-orders do not arise from total orders
Group Theory
2021-07-01 v1 Logic
Abstract
We present some Zermelo-Fraenkel consistency results regarding bi-orderability of groups, as well as a construction of groups with Conradian orders whose every action on metric spaces has bounded orbits. A classical consequence of the ultrafilter lemma is that a group is bi-orderable if and only if it is locally bi-orderable. We show that there exists a model of ZF in which there is a group which is locally free (ergo locally bi-orderable) and not bi-orderable, and the group can be given a total order. Such a group can also exist in the presence of the principle of dependent choices. Comparable consistency results are provided for torsion-free abelian groups.
Cite
@article{arxiv.2004.13798,
title = {Bi-orders do not arise from total orders},
author = {Samuel M. Corson},
journal= {arXiv preprint arXiv:2004.13798},
year = {2021}
}