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Free lattice-ordered groups and the space of left orderings

Abstract

For a left-orderable group G, let LO(G) denote its space of left orderings, and F(G) the free lattice-ordered group over G. This paper establishes a connection between the topology of LO(G) and the group F(G). The main result is a correspondence between the kernels of certain maps in F(G), and the closures of orbits in LO(G) under the natural G-action. The proof of this correspondence is motivated by earlier work of McCleary, which essentially shows that isolated points in LO(G) correspond to basic elements in F(G). As an application, we will study this new correspondence between kernels and the closures of orbits to show that LO(G) is either finite or uncountable. We will also show that LO(F n ) is homeomorphic to the Cantor set, where F n is the free group on n > 1 generators.

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References

  1. Arora A.K., McCleary S.H.: Centralizers in free lattice-ordered groups. Houston J. Math. 12(4), 455–482 (1986)

    MathSciNet  MATH  Google Scholar 

  2. Ault J.C.: Right-ordered locally nilpotent groups. J. Lond. Math. Soc. 4(2), 662–666 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  3. Botto Mura, R., Rhemtulla, A.: Orderable Groups. Lecture Notes in Pure and Applied Mathematics, vol. 27. Marcel Dekker Inc., New York (1977)

  4. Clay, A.: Isolated points in the space of left orderings of a group. Groups Geom. Dyn. 4(3), 517–532 (2010). doi:10.4171/GGD/93. URL http://dx.doi.org/10.4171/GGD/93

    Google Scholar 

  5. Conrad P.: Free lattice-ordered groups. J. Algebra 16, 191–203 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  6. Dehornoy P., Dynnikov I., Rolfsen D., Wiest B.: Ordering Braids, Surveys and Monographs, vol. 148. American Mathematical Society, Providence, RI (2008)

    Google Scholar 

  7. Droste M.: Representations of free lattice-ordered groups. Order 10(4), 375–381 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  8. Dubrovina T.V., Dubrovin N.I.: On braid groups. Math. Sb. 192(5), 53–64 (2001)

    MathSciNet  Google Scholar 

  9. Glass A.M.W.: Partially Ordered Groups, Series in Algebra, vol. 7. World Scientific, River Edge (1999)

    Google Scholar 

  10. Hocking J.G., Young G.S.: Topology. Addison-Wesley Publishing Co., Inc., Reading (1961)

    MATH  Google Scholar 

  11. Kopytov, V.M.: Free lattice-ordered groups. Algebr. Logika 18(4): 426–441, 508 (1979)

    Google Scholar 

  12. Kopytov, V.M.: Free lattice-ordered groups. Sibirsk. Math. Zh. 24(1), 120–124, 192 (1983)

    Google Scholar 

  13. Kopytov V.M., Medvedev N.Y.: Right-Ordered Groups. Siberian School of Algebra and Logic, Consultants Bureau, New York (1996)

    Google Scholar 

  14. Linnell, P.A.: The space of left orders of a group is either finite or uncountable. Bull. Lond. Math. Soc. doi:10.1112/blms/bdq099

  15. McCleary S.H.: Free lattice-ordered groups represented as o-2 transitive l-permutation groups. Trans. Am. Math. Soc. 290(1), 69–79 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Morris D.W.: Amenable groups that act on the line. Algebr. Geom. Topol. 6, 2509–2518 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  17. Navas A.: On the dynamics of (left) orderable groups. Annales de l’institut Fourier 60(5), 1685–1740 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  18. Navas, A., Rivas, C.: A new characterization of Conrad’s property for group orderings, with applications. Algebr. Geom. Topol. 9(4), 2079–2100 (2009). doi:10.2140/agt.2009.9.2079. URL http://dx.doi.org/10.2140/agt.2009.9.2079. With an appendix by Adam Clay

  19. Navas, A., Wiest, B.: Nielsen-Thurston Orderings and the Space of Braid Orderings. Preprint. http://arxiv.org/abs/0906.2605

  20. Rhemtulla, A., Rolfsen, D.: Local indicability in ordered groups: braids and elementary amenable groups. Proc. Am. Math. Soc. 130(9), 2569–2577 (electronic) (2002)

    Google Scholar 

  21. Sikora A.S.: Topology on the spaces of orderings of groups. Bull. Lond. Math. Soc. 36(4), 519–526 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  22. de Vries J.: Elements of Topological Dynamics, Mathematics and its Applications, vol. 257. Kluwer, Dordrecht (1993)

    Google Scholar 

  23. Zenkov, A.V.: On groups with an infinite set of right orders. Sibirsk. Math. Zh. 38(1), 90–92, ii (1997)

    Google Scholar 

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Correspondence to Adam Clay.

Additional information

Communicated by John S. Wilson.

Research partially supported by an NSERC graduate fellowship.

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Clay, A. Free lattice-ordered groups and the space of left orderings. Monatsh Math 167, 417–430 (2012). https://doi.org/10.1007/s00605-011-0305-5

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  • DOI: https://doi.org/10.1007/s00605-011-0305-5

Keywords

  • Left-ordered groups
  • Free lattice-ordered groups
  • Space of left orderings

Mathematics Subject Classification (2000)

  • 06F15
  • 20E05