Promoting circular-orderability to left-orderability
Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 175-201.

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×/n 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.

Motivé par des développements récents en topologie de basse dimension, nous fournissons un nouveau critère pour l’existence d’un ordre à gauche sur un groupe sous l’hypothèse que le groupe admet un ordre circulaire : un groupe G est ordannable à gauche si et seulement si G×/n peut être ordonné de façon circulaire pour tous les n>1. Cela implique que chaque groupe circulairement ordonné qui n’est pas ordonnable à gauche donne lieu à un ensemble d’entiers strictement positifs qui décrit exactement l’obstruction à l’existence d’un ordre à gauche, ensemble que nous appelons le spectre d’obstruction. Nous décrivons précisément le comportement du spectre d’obstruction par rapport à la torsion du groupe et montrons que ce même comportement peut être reflété par des groupes sans torsion, dont les spectres d’obstruction sont en général plus complexes.

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DOI: 10.5802/aif.3394
Classification: 20F60,  37E10,  57M27
Keywords: Ordered groups, actions on the circle, 3-manifolds
Bell, Jason 1; Clay, Adam 2; Ghaswala, Tyrone 2

1 Department of Pure Mathematics University of Waterloo Waterloo ON N2L 3G1 (Canada)
2 Department of Mathematics University of Manitoba Winnipeg MB R3T 2N2 (Canada)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Bell, Jason; Clay, Adam; Ghaswala, Tyrone. Promoting circular-orderability to left-orderability. Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 175-201. doi : 10.5802/aif.3394. https://aif.centre-mersenne.org/articles/10.5802/aif.3394/

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