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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3191
Classification: 06F15,  20F60,  37E10,  22F50
Keywords: Orderable groups, actions on the circle, spaces of orders
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)
@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
TI  - Group orderings, dynamics, and rigidity
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 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/
UR  - https://doi.org/10.5802/aif.3191
DO  - 10.5802/aif.3191
LA  - en
ID  - AIF_2018__68_4_1399_0
ER  - 
%0 Journal Article
%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://doi.org/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, Volume 68 (2018) no. 4, pp. 1399-1445. doi : 10.5802/aif.3191. https://aif.centre-mersenne.org/articles/10.5802/aif.3191/

[1] Baik, Hyungryul; Samperton, Eric Spaces of invariant circular orders of groups (2016) (https://arxiv.org/abs/1508.02661, to appear in Groups Geom. Dyn.)

[2] Calegari, Danny Circular groups, planar groups, and the Euler class, Proceedings of the Casson Fest (Geometry and Topology Monographs) Volume 7 (2004), pp. 431-491 | Zbl: 1181.57022

[3] Calegari, Danny; Dunfield, Nathan M. Laminations and groups of homeomorphisms of the circle, Invent. Math., Volume 152 (2003) no. 1, pp. 149-204 | Zbl: 1025.57018

[4] Dal’Bo, Françoise Geodesic and horocyclic trajectories, Universitext, Springer; EDP Sciences, 2011, xii+176 pages (Translated from the 2007 French original) | Zbl: 1205.37050

[5] Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert Ordering braids, Mathematical Surveys and Monographs, Volume 148, American Mathematical Society, 2008, x+323 pages | Zbl: 1163.20024

[6] Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal Groups, orders and dynamics (https://arxiv.org/abs/1408.5805, to appear)

[7] Ghys, Étienne Groups acting on the circle, Enseign. Math., Volume 47 (2001) no. 3-4, pp. 329-407 | Zbl: 1044.37033

[8] Jakubík, Ján; Pringerová, Gabriela Representations of cyclically ordered groups, Čas. Pěstování Mat., Volume 113 (1988) no. 2, pp. 184-196 | Zbl: 0654.06016

[9] Mann, Kathryn Rigidity and flexibility of group actions on S 1 (2015) (https://arxiv.org/abs/1510.00728, to appear in Handbook of Group Actions)

[10] Mann, Kathryn Spaces of surface group representations, Invent. Math., Volume 201 (2015) no. 2, pp. 669-710 | Zbl: 1327.57002

[11] Margulis, Gregory Free subgroups of the homeomorphism group of the circle, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 9, pp. 669-674 | Zbl: 0983.37029

[12] Matsumoto, Shigenori Some remarks on foliated S 1 bundles, Invent. Math., Volume 90 (1987) no. 2, pp. 343-358 | Zbl: 0681.58007

[13] Matsumoto, Shigenori 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: 1369.37052

[14] McCleary, Stephen H. Free lattice-ordered groups represented as o-2 transitive -permutation groups, Trans. Am. Math. Soc., Volume 290 (1985) no. 1, pp. 69-79 | Zbl: 0546.06013

[17] Rivas, Cristóbal On spaces of Conradian group orderings, J. Group Theory, Volume 13 (2010) no. 3, pp. 337-353 | Zbl: 1192.06015

[18] Rivas, Cristóbal Left-orderings on free products of groups, J. Algebra, Volume 350 (2012), pp. 318-329 | Zbl: 1261.06021

[19] Rivas, Cristóbal; Tessera, Romain On the space of left-orderings of virtually solvable groups, Groups Geom. Dyn., Volume 10 (2016) no. 1, pp. 65-90 | Zbl: 1360.06013

[20] Sikora, Adam S. Topology on the spaces of orderings of groups, Bull. Lond. Math. Soc., Volume 36 (2004) no. 4, pp. 519-526 | Zbl: 1057.06006

[21] Thurston, William Paul 3–manifolds, foliations and circles II (unpublished preprint)

[22] Zeleva, S. D. Cyclically ordered groups, Sib. Mat. Zh., Volume 17 (1976) no. 5, pp. 1046-1051 | Zbl: 0362.06022

Cited by Sources: