On the dynamics of (left) orderable groups
Annales de l'Institut Fourier, Volume 60 (2010) no. 5, pp. 1685-1740.

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.

DOI: 10.5802/aif.2570
Classification: 06F15, 20F36, 20F60, 22F50
Keywords: Orderable groups, Conradian ordering, actions on the line
Mot clés : groupes ordonnables, ordre conradien, actions sur la droite
Navas, Andrés 1

1 Univ. de Santiago de Chile Alameda 3363 Est. Central Santiago (Chile)
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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/articles/10.5802/aif.2570/

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

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

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

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

[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 | DOI | Numdam | MR | Zbl

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

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

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

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

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

[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., Volume 44 (2009) no. 10, pp. 1529-1532 | DOI | MR | Zbl

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

[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, Volume 16 (2007) no. 3, pp. 257-266 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

[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, Volume 4 (2001) no. 2, pp. 153-168 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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, Volume 14 (2010), pp. 573-584 | DOI | MR

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

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

[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, Volume 223 (1975) no. 5, pp. 1067-1070 English translation: Soviet Math. Dokl., 16 (1976), 1037-1041 | MR | Zbl

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

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

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

[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., Volume 1 (2001), p. 311-320 (electronic) | DOI | MR | Zbl

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

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

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

[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, Volume 54 (1993) no. 2, pp. 96-98 English translation: Math. Notes, 54 (1994), 833-834 | MR | Zbl

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

[64] Tsuboi, Takashi Γ 1 -structures avec une seule feuille, Astérisque (1984) no. 116, pp. 222-234 | Numdam | MR | Zbl

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

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

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