[Groupes ordonnés, dynamique et rigidité]
Soit un groupe dénombrable. Nous montrons qu’il y a une relation topologique entre l’espace des ordres cycliques sur et l’espace des actions de 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 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 , 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.
Let be a countable group. We show there is a topological relationship between the space of circular orders on and the moduli space of actions of 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 on and .
As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces 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 , answering a question of Navas. We also give a detailed analysis of circular orders on free groups, characterizing isolated orders.
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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
CC-BY-ND 4.0
@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},
pages = {1399--1445},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {68},
number = {4},
year = {2018},
doi = {10.5802/aif.3191},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3191/}
}
TY - JOUR AU - Mann, Kathryn AU - Rivas, Cristóbal TI - Group orderings, dynamics, and rigidity JO - Annales de l'Institut Fourier PY - 2018 SP - 1399 EP - 1445 VL - 68 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3191/ DO - 10.5802/aif.3191 LA - en ID - AIF_2018__68_4_1399_0 ER -
%0 Journal Article %A Mann, Kathryn %A Rivas, Cristóbal %T Group orderings, dynamics, and rigidity %J Annales de l'Institut Fourier %D 2018 %P 1399-1445 %V 68 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3191/ %R 10.5802/aif.3191 %G en %F AIF_2018__68_4_1399_0
Mann, Kathryn; Rivas, Cristóbal. Group orderings, dynamics, and rigidity. Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1399-1445. doi : 10.5802/aif.3191. https://aif.centre-mersenne.org/articles/10.5802/aif.3191/
[1] Spaces of invariant circular orders of groups (2016) (https://arxiv.org/abs/1508.02661, to appear in Groups Geom. Dyn.)
[2] Circular groups, planar groups, and the Euler class, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7 (2004), pp. 431-491 | Zbl
[3] Laminations and groups of homeomorphisms of the circle, Invent. Math., Volume 152 (2003) no. 1, pp. 149-204 | Zbl
[4] Geodesic and horocyclic trajectories, Universitext, Springer; EDP Sciences, 2011, xii+176 pages (Translated from the 2007 French original) | Zbl
[5] Ordering braids, Mathematical Surveys and Monographs, 148, American Mathematical Society, 2008, x+323 pages | Zbl
[6] Groups, orders and dynamics (https://arxiv.org/abs/1408.5805, to appear)
[7] Groups acting on the circle, Enseign. Math., Volume 47 (2001) no. 3-4, pp. 329-407 | Zbl
[8] Representations of cyclically ordered groups, Čas. Pěstování Mat., Volume 113 (1988) no. 2, pp. 184-196 | Zbl
[9] Rigidity and flexibility of group actions on (2015) (https://arxiv.org/abs/1510.00728, to appear in Handbook of Group Actions)
[10] Spaces of surface group representations, Invent. Math., Volume 201 (2015) no. 2, pp. 669-710 | Zbl
[11] Free subgroups of the homeomorphism group of the circle, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 9, pp. 669-674 | Zbl
[12] Some remarks on foliated bundles, Invent. Math., Volume 90 (1987) no. 2, pp. 343-358 | Zbl
[13] Basic partitions and combinations of group actions on the circle: a new approach to a theorem of Kathryn Mann, Enseign. Math., Volume 62 (2016) no. 1-2, pp. 15-47 | Zbl
[14] Free lattice-ordered groups represented as - transitive -permutation groups, Trans. Am. Math. Soc., Volume 290 (1985) no. 1, pp. 69-79 | Zbl
[17] On spaces of Conradian group orderings, J. Group Theory, Volume 13 (2010) no. 3, pp. 337-353 | Zbl
[18] Left-orderings on free products of groups, J. Algebra, Volume 350 (2012), pp. 318-329 | Zbl
[19] On the space of left-orderings of virtually solvable groups, Groups Geom. Dyn., Volume 10 (2016) no. 1, pp. 65-90 | Zbl
[20] Topology on the spaces of orderings of groups, Bull. Lond. Math. Soc., Volume 36 (2004) no. 4, pp. 519-526 | Zbl
[21] 3–manifolds, foliations and circles II (unpublished preprint)
[22] Cyclically ordered groups, Sib. Mat. Zh., Volume 17 (1976) no. 5, pp. 1046-1051 | Zbl
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