On the dynamics of (left) orderable groups

Navas, Andrés

Navas, Andrés

Annales de l'Institut Fourier, Volume 60 (2010) no. 5, p. 1685-1740

Abstract
Résumé

We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid groups. In particular, we show that the positive cone of the Dehornoy ordering is not finitely generated as a semigroup. To do this, we define the Conradian soul of an ordering as the maximal convex subgroup restricted to which the ordering is Conradian, and we elaborate on this notion.

Nous développons des méthodes dynamiques pour étudier les groupes ordonnables ainsi que leurs espaces d’ordres associés. Nous donnons des preuves nouvelles et élémentaires de théorèmes dus à Linnell (si un groupe ordonnable possède une infinité d’ordres, alors il en possède une infinité non dénombrable) et McCleary (l’espace des ordres du groupe libre est un ensemble de Cantor). Nous montrons que ce dernier résultat est valable aussi pour les groupes nilpotents dénombrables et sans torsion qui ne sont pas abéliens de rang un. Finalement, nous appliquons nos méthodes au cas des groupes de tresses. En particulier, nous démontrons que le cone positif de l’ordre de Dehornoy n’est pas de type fini en tant que semi-groupe. Pour ce faire, nous définissons le noyau conradien d’un ordre comme étant le plus grand sous-groupe convexe sur lequel la relation est conradienne, et nous travaillons avec cette notion.

MR 2766228 | Zbl 1316.06018 | Zbl pre05822118 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.2570
Classification: 06F15, 20F36, 20F60, 22F50
Keywords: Orderable groups, Conradian ordering, actions on the line

@article{AIF_2010__60_5_1685_0,
     author = {Navas, Andr\'es},
     title = {On the dynamics of (left) orderable groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {5},
     year = {2010},
     pages = {1685-1740},
     doi = {10.5802/aif.2570},
     zbl = {1316.06018},
     mrnumber = {2766228},
     zbl = {pre05822118},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_5_1685_0}
}

Navas, Andrés. On the dynamics of (left) orderable groups. Annales de l'Institut Fourier, Volume 60 (2010) no. 5, pp. 1685-1740. doi : 10.5802/aif.2570. http://www.numdam.org/item/AIF_2010__60_5_1685_0/

References

[1] Beklaryan, L. A. Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants, Uspekhi Mat. Nauk, Tome 59 (2004), pp. 4-66 (English translation: Russian Math. Surveys, 59 (2004), 599-660) | MR 2106645 | Zbl 1073.54018

[2] Bergman, George M. Right orderable groups that are not locally indicable, Pacific J. Math., Tome 147 (1991) no. 2, pp. 243-248 | MR 1084707 | Zbl 0677.06007

[3] Botto Mura, Roberta; Rhemtulla, Akbar Orderable groups, Marcel Dekker Inc., New York (1977) (Lecture Notes in Pure and Applied Mathematics, Vol. 27) | MR 491396 | Zbl 0358.06038

[4] Boyer, Steven; Rolfsen, Dale; Wiest, Bert Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble), Tome 55 (2005) no. 1, pp. 243-288 | Article | Numdam | MR 2141698 | Zbl 1068.57001

[5] Brin, Matthew G. The chameleon groups of Richard J. Thompson: automorphisms and dynamics, Inst. Hautes Études Sci. Publ. Math. (1996) no. 84, pp. 5-33 | Article | Numdam | MR 1441005 | Zbl 0891.57037

[6] Brodskiĭ, S. D. Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh., Tome 25 (1984) no. 2, pp. 84-103 (English translation: Siberian Math. Journal, 25 (1984), 235-251) | MR 741011 | Zbl 0579.20020

[7] Buttsworth, R. N. A family of groups with a countable infinity of full orders, Bull. Austral. Math. Soc., Tome 4 (1971), pp. 97-104 | Article | MR 279013 | Zbl 0223.06008

[8] Calegari, Danny Nonsmoothable, locally indicable group actions on the interval, Algebr. Geom. Topol., Tome 8 (2008) no. 1, pp. 609-613 | Article | MR 2443241 | Zbl 1154.37015

[9] Calegari, Danny; Dunfield, Nathan M. Laminations and groups of homeomorphisms of the circle, Invent. Math., Tome 152 (2003) no. 1, pp. 149-204 | Article | MR 1965363 | Zbl 1025.57018

[10] Cherix, Pierre-Alain; Martin, Florian; Valette, Alain Spaces with measured walls, the Haagerup property and property (T), Ergodic Theory Dynam. Systems, Tome 24 (2004) no. 6, pp. 1895-1908 | Article | MR 2106770 | Zbl 1068.43007

[11] Clay, A. Free lattice ordered groups and the topology on the space of left orderings of a group (2009) (Preprint)

[12] Clay, Adam; Smith, Lawrence H. Corrigendum to: “On ordering free groups” [J. Symbolic Comput. 40 (2005) 1285–1290], J. Symbolic Comput., Tome 44 (2009) no. 10, pp. 1529-1532 | Article | MR 2543435 | Zbl 1171.20314

[13] Conrad, Paul Right-ordered groups, Michigan Math. J., Tome 6 (1959), pp. 267-275 | Article | MR 106954 | Zbl 0099.01703

[14] Dabkowska, M. A.; Dabkowski, M. K.; Harizanov, V. S.; Przytycki, J. H.; Veve, M. A. Compactness of the space of left orders, J. Knot Theory Ramifications, Tome 16 (2007) no. 3, pp. 257-266 | Article | MR 2320157 | Zbl 1129.57024

[15] Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A. Non-left-orderable 3-manifold groups, Canad. Math. Bull., Tome 48 (2005) no. 1, pp. 32-40 | Article | MR 2118761 | Zbl 1065.57001

[16] Darnel, Michael R. Theory of lattice-ordered groups, Marcel Dekker Inc., New York, Monographs and Textbooks in Pure and Applied Mathematics, Tome 187 (1995) | MR 1304052 | Zbl 0810.06016

[17] Dehornoy, Patrick Braids and self-distributivity, Birkhäuser Verlag, Basel, Progress in Mathematics, Tome 192 (2000) | MR 1778150 | Zbl 0958.20033

[18] Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert Why are braids orderable?, Société Mathématique de France, Paris, Panoramas et Synthèses, Tome 14 (2002) | MR 1988550 | Zbl 1048.20021

[19] Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert Ordering braids, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 148 (2008) | MR 2463428 | Zbl 1163.20024

[20] Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., Tome 199 (2007) no. 2, pp. 199-262 | Article | MR 2358052 | Zbl 1139.37025

[21] Dubrovina, T. V.; Dubrovin, N. I. On braid groups, Mat. Sb., Tome 192 (2001) no. 5, pp. 53-64 | MR 1859702 | Zbl 1037.20036

[22] Furman, Alex Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam (2002), pp. 931-1014 | MR 1928529 | Zbl 1053.60045

[23] Ghys, Étienne Groups acting on the circle, Enseign. Math. (2), Tome 47 (2001) no. 3-4, pp. 329-407 | MR 1876932 | Zbl 1044.37033

[24] Glass, A. M. W. Partially ordered groups, World Scientific Publishing Co. Inc., River Edge, NJ, Series in Algebra, Tome 7 (1999) | MR 1791008 | Zbl 0933.06010

[25] Gromov, Misha Spaces and questions, Geom. Funct. Anal. (2000), pp. 118-161 | MR 1826251 | Zbl 1006.53035

[26] De La Harpe, P. Topics in geometric group theory (2000) (Univ. of Chicago Press) | Zbl 0965.20025

[27] Hocking, John G.; Young, Gail S. Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London (1961) | MR 125557 | Zbl 0135.22701

[28] Horak, M.; Stein, M. Partially ordered groups which act on oriented trees (2005) (Preprint)

[29] Jiménez, L. Grupos ordenables: estructura algebraica y dinámica (2008) (Master thesis, Univ. de Chile)

[30] Kaimanovich, Vadim A. The Poisson boundary of polycyclic groups, Probability measures on groups and related structures, XI (Oberwolfach, 1994), World Sci. Publ., River Edge, NJ (1995), pp. 182-195 | MR 1414934 | Zbl 0912.60011

[31] Kassel, Christian L’ordre de Dehornoy sur les tresses, Astérisque (2002) no. 276, pp. 7-28 (Séminaire Bourbaki, Vol. 1999/2000) | Numdam | MR 1886754 | Zbl 1060.20033

[32] Kopytov, V. M.; Medvedev, N. Ya. The theory of lattice-ordered groups, Kluwer Academic Publishers Group, Dordrecht, Mathematics and its Applications, Tome 307 (1994) | MR 1369091 | Zbl 0834.06015

[33] Kopytov, Valeriĭ M.; Medvedev, Nikolaĭ Ya. Right-ordered groups, Consultants Bureau, New York, Siberian School of Algebra and Logic (1996) | MR 1393199 | Zbl 0852.06005

[34] Lifschitz, Lucy; Morris, Dave Witte Isotropic nonarchimedean $S$ -arithmetic groups are not left orderable, C. R. Math. Acad. Sci. Paris, Tome 339 (2004) no. 6, pp. 417-420 | MR 2092755 | Zbl 1060.20041

[35] Lifschitz, Lucy; Morris, Dave Witte Bounded generation and lattices that cannot act on the line, Pure Appl. Math. Q., Tome 4 (2008) no. 1, part 2, pp. 99-126 | MR 2405997 | Zbl 1146.22014

[36] Linnell, P. The topology on the space of left orderings of a group (2006) (Preprint)

[37] Linnell, P. The space of left orders of a group is either finite or uncountable (2009) (Preprint)

[38] Linnell, Peter A. Left ordered groups with no non-abelian free subgroups, J. Group Theory, Tome 4 (2001) no. 2, pp. 153-168 | Article | MR 1812322 | Zbl 0982.06013

[39] Longobardi, P.; Maj, M.; Rhemtulla, A. H. Groups with no free subsemigroups, Trans. Amer. Math. Soc., Tome 347 (1995) no. 4, pp. 1419-1427 | Article | MR 1277124 | Zbl 0833.20043

[40] Mañé, Ricardo Introdução à teoria ergódica, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Projeto Euclides, Tome 14 (1983) | MR 800092 | Zbl 0581.28010

[41] Mccleary, Stephen H. Free lattice-ordered groups represented as $o$ - $2$ transitive $l$ -permutation groups, Trans. Amer. Math. Soc., Tome 290 (1985) no. 1, pp. 69-79 | MR 787955 | Zbl 0546.06013

[42] Morris, Dave Witte Arithmetic groups of higher $Q$ -rank cannot act on $1$ -manifolds, Proc. Amer. Math. Soc., Tome 122 (1994) no. 2, pp. 333-340 | MR 1198459 | Zbl 0818.22006

[43] Morris, Dave Witte Amenable groups that act on the line, Algebr. Geom. Topol., Tome 6 (2006), pp. 2509-2518 | Article | MR 2286034 | Zbl 1185.20042

[44] Navas, Andrés Actions de groupes de Kazhdan sur le cercle, Ann. Sci. École Norm. Sup. (4), Tome 35 (2002) no. 5, pp. 749-758 | Numdam | MR 1951442 | Zbl 1028.58010

[45] Navas, Andrés Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle, Comment. Math. Helv., Tome 80 (2005) no. 2, pp. 355-375 | Article | MR 2142246 | Zbl 1080.57002

[46] Navas, Andrés Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal., Tome 18 (2008) no. 3, pp. 988-1028 | Article | MR 2439001 | Zbl pre05565909

[47] Navas, Andrés A remarkable family of left-orderable groups: Central extensions of Hecke groups (2009) (Preprint)

[48] Navas, Andrés A finitely generated, locally indicable group without faithful actions by $C^{1}$ diffeomorphisms of the interval, Geometry and Topology, Tome 14 (2010), pp. 573-584 | Article | MR 2602845 | Zbl pre05730918

[49] Navas, Andrés; Rivas, Cristóbal A new characterization of Conrad’s property for group orderings, with applications, Algebr. Geom. Topol., Tome 9 (2009) no. 4, pp. 2079-2100 (With an appendix by Adam Clay) | Article | MR 2551663 | Zbl pre05624200

[50] Navas, Andrés; Rivas, Cristóbal Describing all bi-orderings on Thompson’s group F, Groups, Geometry, and Dynamics, Tome 4 (2010), pp. 163-177 | Article | MR 2566304 | Zbl pre05658122

[51] Navas, Andrés; Wiest, B. Nielsen-Thurston orders and the space of braid orders (2009) (Preprint)

[52] Pickelʼ, B. S. Informational futures of amenable groups, Dokl. Akad. Nauk SSSR, Tome 223 (1975) no. 5, pp. 1067-1070 (English translation: Soviet Math. Dokl., 16 (1976), 1037-1041) | MR 390176 | Zbl 0326.28027

[53] Plante, J. F. Foliations with measure preserving holonomy, Ann. of Math. (2), Tome 102 (1975) no. 2, pp. 327-361 | Article | MR 391125 | Zbl 0314.57018

[54] Rhemtulla, Akbar; Rolfsen, Dale Local indicability in ordered groups: Braids and elementary amenable groups, Proc. Amer. Math. Soc., Tome 130 (2002) no. 9, p. 2569-2577 (electronic) | Article | MR 1900863 | Zbl 0996.20024

[55] Rivas, C. On spaces of Conradian group orderings (To appear in J. Group Theory) | Zbl 1192.06015

[56] Rivas, C. On left-orderable groups, Univ. de Chile (2010) (Ph. D. Thesis)

[57] Rolfsen, Dale; Wiest, Bert Free group automorphisms, invariant orderings and topological applications, Algebr. Geom. Topol., Tome 1 (2001), p. 311-320 (electronic) | Article | MR 1835259 | Zbl 0985.57006

[58] Short, Hamish; Wiest, Bert Orderings of mapping class groups after Thurston, Enseign. Math. (2), Tome 46 (2000) no. 3-4, pp. 279-312 | MR 1805402 | Zbl 1023.57013

[59] Sikora, Adam S. Topology on the spaces of orderings of groups, Bull. London Math. Soc., Tome 36 (2004) no. 4, pp. 519-526 | Article | MR 2069015 | Zbl 1057.06006

[60] Smirnov, D. M. Right-ordered groups, Algebra i Logika Sem., Tome 5 (1966) no. 6, pp. 41-59 | MR 206128 | Zbl 0201.36702

[61] Tararin, V. On groups having a finite number of orders (1991) (Dep. Viniti (Report), Moscow)

[62] Tararin, V. M. On the theory of right-ordered groups, Mat. Zametki, Tome 54 (1993) no. 2, pp. 96-98 (English translation: Math. Notes, 54 (1994), 833-834) | MR 1244986 | Zbl 0811.20042

[63] Thurston, William P. A generalization of the Reeb stability theorem, Topology, Tome 13 (1974), pp. 347-352 | Article | MR 356087 | Zbl 0305.57025

[64] Tsuboi, Takashi $Γ_{1}$ -structures avec une seule feuille, Astérisque (1984) no. 116, pp. 222-234 | MR 755173 | Zbl 0551.57014

[65] Wagon, Stan The Banach-Tarski paradox, Cambridge University Press (1993) | MR 1251963 | Zbl 0569.43001

[66] Zenkov, A. V. On groups with an infinite set of right orders, Sibirsk. Mat. Zh., Tome 38 (1997) no. 1, pp. 90-92 (English translation: Siberian Math. Journal, 38 (1997), 76-77) | MR 1446675 | Zbl 0880.20032