no. 1
Promoting circular-orderability to left-orderability
[Obtention d’un ordre à gauche sur un groupe à partir d’un ordre circulaire]
Bell, Jason1 ; Clay, Adam2 ; Ghaswala, Tyrone2
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)
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 175-201.

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×ℤ/nℤ 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.

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×ℤ/nℤ 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.

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Révisé le :
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DOI : 10.5802/aif.3394
Classification : 20F60, 37E10, 57M27
Keywords: Ordered groups, actions on the circle, 3-manifolds
Mot clés : Groupes ordonnés, actions sur le cercle, 3-variétés
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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Bell, Jason; Clay, Adam; Ghaswala, Tyrone. Promoting circular-orderability to left-orderability. Annales de l'Institut Fourier, Tome 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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