Promoting circular-orderability to left-orderability

Abstract

Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G\times \mathbb{Z}/n\mathbb{Z}$ is circularly-orderable for all $n>1$. This implies that every circularly-orderable group which is not left-orderable gives rise to a collection of positive integers that exactly encode the obstruction to left-orderability, which we call the obstruction spectrum. We precisely describe the behaviour of the obstruction spectrum with respect to torsion, and show that this same behaviour can be mirrored by torsion-free groups, whose obstruction spectra are in general more complex.

Revised version. A new section has been added to include a new result shown to us by Dave Morris. Changes have been made to improve the readability and to streamline some of the proofs. To appear in Annales de l'institut Fourier