Algebraic & Geometric Topology

A new characterization of Conrad's property for group orderings, with applications

Andrés Navas and Cristóbal Rivas

Full-text: Open access

Abstract

We provide a pure algebraic version of the first-named author’s dynamical characterization of the Conrad property for group orderings. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 4 (2009), 2079-2100.

Dates
Received: 3 March 2009
Revised: 28 August 2009
Accepted: 31 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797077

Digital Object Identifier
doi:10.2140/agt.2009.9.2079

Mathematical Reviews number (MathSciNet)
MR2551663

Zentralblatt MATH identifier
1211.06009

Subjects
Primary: 06F15: Ordered groups [See also 20F60] 20F60: Ordered groups [See mainly 06F15]
Secondary: 57S25: Groups acting on specific manifolds

Keywords
group orders Conrad's property

Citation

Navas, Andrés; Rivas, Cristóbal. A new characterization of Conrad's property for group orderings, with applications. Algebr. Geom. Topol. 9 (2009), no. 4, 2079--2100. doi:10.2140/agt.2009.9.2079. https://projecteuclid.org/euclid.agt/1513797077


Export citation

References

  • R Botto Mura, A Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, New York (1977)
  • A Clay, Isolated points in the space of left orderings of a group, preprint (2008)
  • P Conrad, Right-ordered groups, Michigan Math. J. 6 (1959) 267–275
  • J Crisp, L Paris, The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math. 145 (2001) 19–36
  • P Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994) 115–150
  • P Dehornoy, I Dynnikov, D Rolfsen, B Wiest, Why are braids orderable?, Panoramas et Synthèses 14, Société Mathématique de France, Paris (2002)
  • T V Dubrovina, N I Dubrovin, On braid groups, Mat. Sb. 192 (2001) 53–64
  • V M Kopytov, N Y Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, New York (1996)
  • P A Linnell, Noncommutative localization in group rings, from: “Non-commutative localization in algebra and topology”, London Math. Soc. Lecture Note Ser. 330, Cambridge Univ. Press, Cambridge (2006) 40–59
  • S H McCleary, An even better representation for free lattice-ordered groups, Trans. Amer. Math. Soc. 290 (1985) 81–100
  • D W Morris, Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006) 2509–2518
  • A Navas, Grupos de difeomorfismos del cí rculo, Ensaios Matemáticos 13, Sociedade Brasileira de Matemática, Rio de Janeiro (2007)
  • A Navas, On the dynamics of left-orderable groups, preprint (2007)
  • A S Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004) 519–526
  • A V Zenkov, On groups with an infinite set of right orders, Sibirsk. Mat. Zh. 38 (1997) 90–92, ii