Group orderings, dynamics, and rigidity

Mann, Kathryn; Rivas, Cristóbal

Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1399-1445.

Soit $G$ un groupe dénombrable. Nous montrons qu’il y a une relation topologique entre l’espace $\mathrm{CO}\left(G\right)$ 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 $\mathrm{CO}\left(F_{2n}\right)$ 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.

Let $G$ be a countable group. We show there is a topological relationship between the space $\mathrm{CO}\left(G\right)$ 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 $ℝ$.

As an application of our techniques, we give an explicit construction of infinitely many nonconjugate isolated points in the spaces $\mathrm{CO}\left(F_{2n}\right)$ 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.

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

[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, Geometry and Topology Publications (2004), pp. 431-491 | Zbl

[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

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

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

[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

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

[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

[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

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

[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

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

[15] Navas, Andrés On the dynamics of (left) orderable groups, Ann. Inst. Fourier, Volume 60 (2010) no. 5, pp. 1685-1740 | Zbl

[16] Navas, Andrés Sur les rapprochements par conjugaison en dimension 1 et classe $C^1$, Compos. Math., Volume 150 (2014) no. 7, pp. 1183-1195 | Zbl

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

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

[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

[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

[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