On the dynamics of (left) orderable groups

Andrés Navas[1]

  • [1] Univ. de Santiago de Chile Alameda 3363 Est. Central Santiago (Chile)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 5, page 1685-1740
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Navas, Andrés. "On the dynamics of (left) orderable groups." Annales de l’institut Fourier 60.5 (2010): 1685-1740. <http://eudml.org/doc/116319>.

@article{Navas2010,
abstract = {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.},
affiliation = {Univ. de Santiago de Chile Alameda 3363 Est. Central Santiago (Chile)},
author = {Navas, Andrés},
journal = {Annales de l’institut Fourier},
keywords = {Orderable groups; Conradian ordering; actions on the line; left-orderable groups; spaces of orderings; Conradian orders; braid groups; Dehornoy order; positive cones; actions on the real line},
language = {eng},
number = {5},
pages = {1685-1740},
publisher = {Association des Annales de l’institut Fourier},
title = {On the dynamics of (left) orderable groups},
url = {http://eudml.org/doc/116319},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Navas, Andrés
TI - On the dynamics of (left) orderable groups
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1685
EP - 1740
AB - 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.
LA - eng
KW - Orderable groups; Conradian ordering; actions on the line; left-orderable groups; spaces of orderings; Conradian orders; braid groups; Dehornoy order; positive cones; actions on the real line
UR - http://eudml.org/doc/116319
ER -

References

top
  1. L. A. Beklaryan, Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants, Uspekhi Mat. Nauk 59 (2004), 4-66 Zbl1073.54018MR2106645
  2. George M. Bergman, Right orderable groups that are not locally indicable, Pacific J. Math. 147 (1991), 243-248 Zbl0677.06007MR1084707
  3. Roberta Botto Mura, Akbar Rhemtulla, Orderable groups, (1977), Marcel Dekker Inc., New York Zbl0358.06038MR491396
  4. Steven Boyer, Dale Rolfsen, Bert Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 243-288 Zbl1068.57001MR2141698
  5. Matthew G. Brin, The chameleon groups of Richard J. Thompson: automorphisms and dynamics, Inst. Hautes Études Sci. Publ. Math. (1996), 5-33 Zbl0891.57037MR1441005
  6. S. D. Brodskiĭ, Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh. 25 (1984), 84-103 Zbl0579.20020MR741011
  7. R. N. Buttsworth, A family of groups with a countable infinity of full orders, Bull. Austral. Math. Soc. 4 (1971), 97-104 Zbl0223.06008MR279013
  8. Danny Calegari, Nonsmoothable, locally indicable group actions on the interval, Algebr. Geom. Topol. 8 (2008), 609-613 Zbl1154.37015MR2443241
  9. Danny Calegari, Nathan M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003), 149-204 Zbl1025.57018MR1965363
  10. Pierre-Alain Cherix, Florian Martin, Alain Valette, Spaces with measured walls, the Haagerup property and property (T), Ergodic Theory Dynam. Systems 24 (2004), 1895-1908 Zbl1068.43007MR2106770
  11. A. Clay, Free lattice ordered groups and the topology on the space of left orderings of a group, (2009) 
  12. Adam Clay, Lawrence H. Smith, Corrigendum to: “On ordering free groups” [J. Symbolic Comput. 40 (2005) 1285–1290], J. Symbolic Comput. 44 (2009), 1529-1532 Zbl1171.20314MR2543435
  13. Paul Conrad, Right-ordered groups, Michigan Math. J. 6 (1959), 267-275 Zbl0099.01703MR106954
  14. M. A. Dabkowska, M. K. Dabkowski, V. S. Harizanov, J. H. Przytycki, M. A. Veve, Compactness of the space of left orders, J. Knot Theory Ramifications 16 (2007), 257-266 Zbl1129.57024MR2320157
  15. Mieczysław K. Dąbkowski, Józef H. Przytycki, Amir A. Togha, Non-left-orderable 3-manifold groups, Canad. Math. Bull. 48 (2005), 32-40 Zbl1065.57001MR2118761
  16. Michael R. Darnel, Theory of lattice-ordered groups, 187 (1995), Marcel Dekker Inc., New York Zbl0810.06016MR1304052
  17. Patrick Dehornoy, Braids and self-distributivity, 192 (2000), Birkhäuser Verlag, Basel Zbl0958.20033MR1778150
  18. Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, Bert Wiest, Why are braids orderable?, 14 (2002), Société Mathématique de France, Paris Zbl1048.20021MR1988550
  19. Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, Bert Wiest, Ordering braids, 148 (2008), American Mathematical Society, Providence, RI Zbl1163.20024MR2463428
  20. Bertrand Deroin, Victor Kleptsyn, Andrés Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math. 199 (2007), 199-262 Zbl1139.37025MR2358052
  21. T. V. Dubrovina, N. I. Dubrovin, On braid groups, Mat. Sb. 192 (2001), 53-64 Zbl1037.20036MR1859702
  22. Alex Furman, Random walks on groups and random transformations, Handbook of dynamical systems, Vol. 1A (2002), 931-1014, North-Holland, Amsterdam Zbl1053.60045MR1928529
  23. Étienne Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), 329-407 Zbl1044.37033MR1876932
  24. A. M. W. Glass, Partially ordered groups, 7 (1999), World Scientific Publishing Co. Inc., River Edge, NJ Zbl0933.06010MR1791008
  25. Misha Gromov, Spaces and questions, Geom. Funct. Anal. (2000), 118-161 Zbl1006.53035MR1826251
  26. P. De la Harpe, Topics in geometric group theory, (2000) Zbl0965.20025
  27. John G. Hocking, Gail S. Young, Topology, (1961), Addison-Wesley Publishing Co., Inc., Reading, Mass.-London Zbl0135.22701MR125557
  28. M. Horak, M. Stein, Partially ordered groups which act on oriented trees, (2005) Zbl1271.20032
  29. L. Jiménez, Grupos ordenables: estructura algebraica y dinámica, (2008) 
  30. Vadim A. Kaimanovich, The Poisson boundary of polycyclic groups, Probability measures on groups and related structures, XI (Oberwolfach, 1994) (1995), 182-195, World Sci. Publ., River Edge, NJ Zbl0912.60011MR1414934
  31. Christian Kassel, L’ordre de Dehornoy sur les tresses, Astérisque (2002), 7-28 Zbl1060.20033MR1886754
  32. V. M. Kopytov, N. Ya. Medvedev, The theory of lattice-ordered groups, 307 (1994), Kluwer Academic Publishers Group, Dordrecht Zbl0834.06015MR1369091
  33. Valeriĭ M. Kopytov, Nikolaĭ Ya. Medvedev, Right-ordered groups, (1996), Consultants Bureau, New York Zbl0852.06005MR1393199
  34. Lucy Lifschitz, Dave Witte Morris, Isotropic nonarchimedean S -arithmetic groups are not left orderable, C. R. Math. Acad. Sci. Paris 339 (2004), 417-420 Zbl1060.20041MR2092755
  35. Lucy Lifschitz, Dave Witte Morris, Bounded generation and lattices that cannot act on the line, Pure Appl. Math. Q. 4 (2008), 99-126 Zbl1146.22014MR2405997
  36. P. Linnell, The topology on the space of left orderings of a group, (2006) 
  37. P. Linnell, The space of left orders of a group is either finite or uncountable, (2009) Zbl1215.06009
  38. Peter A. Linnell, Left ordered groups with no non-abelian free subgroups, J. Group Theory 4 (2001), 153-168 Zbl0982.06013MR1812322
  39. P. Longobardi, M. Maj, A. H. Rhemtulla, Groups with no free subsemigroups, Trans. Amer. Math. Soc. 347 (1995), 1419-1427 Zbl0833.20043MR1277124
  40. Ricardo Mañé, Introdução à teoria ergódica, 14 (1983), Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro Zbl0581.28010MR800092
  41. Stephen H. McCleary, Free lattice-ordered groups represented as o - 2 transitive l -permutation groups, Trans. Amer. Math. Soc. 290 (1985), 69-79 Zbl0546.06013MR787955
  42. Dave Witte Morris, Arithmetic groups of higher Q -rank cannot act on 1 -manifolds, Proc. Amer. Math. Soc. 122 (1994), 333-340 Zbl0818.22006MR1198459
  43. Dave Witte Morris, Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509-2518 Zbl1185.20042MR2286034
  44. Andrés Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. École Norm. Sup. (4) 35 (2002), 749-758 Zbl1028.58010MR1951442
  45. Andrés Navas, Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle, Comment. Math. Helv. 80 (2005), 355-375 Zbl1080.57002MR2142246
  46. Andrés Navas, Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal. 18 (2008), 988-1028 Zbl1201.37060MR2439001
  47. Andrés Navas, A remarkable family of left-orderable groups: Central extensions of Hecke groups, (2009) Zbl1215.06010
  48. Andrés Navas, A finitely generated, locally indicable group without faithful actions by C 1 diffeomorphisms of the interval, Geometry and Topology 14 (2010), 573-584 Zbl1197.37022MR2602845
  49. Andrés Navas, Cristóbal Rivas, A new characterization of Conrad’s property for group orderings, with applications, Algebr. Geom. Topol. 9 (2009), 2079-2100 Zbl1211.06009MR2551663
  50. Andrés Navas, Cristóbal Rivas, Describing all bi-orderings on Thompson’s group F, Groups, Geometry, and Dynamics 4 (2010), 163-177 Zbl1193.06015MR2566304
  51. Andrés Navas, B. Wiest, Nielsen-Thurston orders and the space of braid orders, (2009) Zbl1235.06017
  52. B. S. Pickelʼ, Informational futures of amenable groups, Dokl. Akad. Nauk SSSR 223 (1975), 1067-1070 Zbl0326.28027MR390176
  53. J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), 327-361 Zbl0314.57018MR391125
  54. Akbar Rhemtulla, Dale Rolfsen, Local indicability in ordered groups: Braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002), 2569-2577 (electronic) Zbl0996.20024MR1900863
  55. C. Rivas, On spaces of Conradian group orderings Zbl1192.06015
  56. C. Rivas, On left-orderable groups, (2010) 
  57. Dale Rolfsen, Bert Wiest, Free group automorphisms, invariant orderings and topological applications, Algebr. Geom. Topol. 1 (2001), 311-320 (electronic) Zbl0985.57006MR1835259
  58. Hamish Short, Bert Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), 279-312 Zbl1023.57013MR1805402
  59. Adam S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004), 519-526 Zbl1057.06006MR2069015
  60. D. M. Smirnov, Right-ordered groups, Algebra i Logika Sem. 5 (1966), 41-59 Zbl0201.36702MR206128
  61. V. Tararin, On groups having a finite number of orders, (1991) Zbl0751.20034
  62. V. M. Tararin, On the theory of right-ordered groups, Mat. Zametki 54 (1993), 96-98 Zbl0811.20042MR1244986
  63. William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347-352 Zbl0305.57025MR356087
  64. Takashi Tsuboi, Γ 1 -structures avec une seule feuille, Astérisque (1984), 222-234 Zbl0551.57014MR755173
  65. Stan Wagon, The Banach-Tarski paradox, (1993), Cambridge University Press Zbl0569.43001MR1251963
  66. A. V. Zenkov, On groups with an infinite set of right orders, Sibirsk. Mat. Zh. 38 (1997), 90-92 Zbl0880.20032MR1446675

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