Group orderings, dynamics, and rigidity
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, p. 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.
Received : 2017-01-19
Revised : 2017-07-31
Accepted : 2017-09-14
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3191
Classification:  06F15,  20F60,  37E10,  22F50
Keywords: Orderable groups, actions on the circle, spaces of orders
@article{AIF_2018__68_4_1399_0,
     author = {Mann, Kathryn and Rivas, Crist\'obal},
     title = {Group orderings, dynamics, and rigidity},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1399-1445},
     doi = {10.5802/aif.3191},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_4_1399_0}
}
Group orderings, dynamics, and rigidity. Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1399-1445. doi : 10.5802/aif.3191. https://aif.centre-mersenne.org/item/AIF_2018__68_4_1399_0/

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