Group orderings, dynamics, and rigidity
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1399-1445.

Let G be a countable group. We show there is a topological relationship between the space CO(G) of circular orders on G and the moduli space of actions of G on the circle; and an analogous relationship for spaces of left orders and actions on the line. In particular, we give a complete characterization of isolated left and circular orders in terms of strong rigidity of their induced actions of G on S 1 and .

As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces CO(F 2n ) of circular orders on free groups, disproving a conjecture from Baik–Samperton, and infinitely many nonconjugate isolated points in the space of left orders on the pure braid group P 3 , answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.

Soit G un groupe dénombrable. Nous montrons qu’il y a une relation topologique entre l’espace CO(G) des ordres cycliques sur G et l’espace des actions de G sur le cercle par homéomorphismes ; et, de manière analogue, qu’il y a une relation entre l’espace des ordres linéaires et l’espace des actions sur la droite. En particulier, nous donnons une caractérisation complète des ordres isolés par rapport à la rigidité forte de leurs actions associées.

Nous appliquons nos techniques pour construire, de manière explicite, un ensemble infini d’ordres non-conjugués et isolés dans l’espace CO(F 2n ) des ordres cycliques sur les groupes libres. Ceci donne un contre-exemple à une conjecture de Baik–Samperton. Nous donnons aussi un ensemble infini d’ordres linéaires non-conjugués et isolés sur le groupe de tresses pures P 3 , pour répondre à une question de Navas. Finalement, nous faisons une analyse détaillée des ordres cycliques sur les groupes libres qui caractérise les ordres isolés.

Published online:
DOI: 10.5802/aif.3191
Classification: 06F15, 20F60, 37E10, 22F50
Keywords: Orderable groups, actions on the circle, spaces of orders
Mot clés : Groupes ordonnables, actions sur le circle, espaces d’ordres
Mann, Kathryn 1; Rivas, Cristóbal 2

1 Dept. of Mathematics, Brown University. Box 1917, 151 Thayer St. Providence, RI 02912 (USA)
2 Depto. de Matemáticas y C.C. Universidad de Santiago de Chile Las Sophoras nº 173, Estación Central Santiago (Chile)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Mann, Kathryn; Rivas, Cristóbal. Group orderings, dynamics, and rigidity. Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1399-1445. doi : 10.5802/aif.3191.

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