Selecta Mathematica

, Volume 19, Issue 3, pp 719–736 | Cite as

Discrete subgroups of locally definable groups

  • Alessandro Berarducci
  • Mário Edmundo
  • Marcello Mamino
Article

Abstract

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group \(G\) in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of \(G\). Given a locally definable connected group \(G\) (not necessarily definably generated), we prove that the \(n\)-torsion subgroup of \(G\) is finite and that every zero-dimensional compatible subgroup of \(G\) has finite rank. Under a convexity hypothesis, we show that every zero-dimensional compatible subgroup of \(G\) is finitely generated.

Keywords

Locally definable groups Covers Discrete subgroups 

Mathematics Subject Classification (1991)

03C64 03C68 22B99 

Notes

Acknowledgments

The main results of this paper have been presented on February 2, 2012 at the Logic Seminar of the Mathematical Institute in Oxford. The first author thanks Jonathan Pila for the kind invitation. We also thank Margarita Otero, Pantelis Eleftheriou, Kobi Peterzil, and the anonymous referee for their comments.

References

  1. 1.
    Baro, E., Otero, M.: Locally definable homotopy. Ann. Pure Appl. Logic 161(4), 488–503 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Berarducci, A., Otero, M.: O-minimal fundamental group, homology and manifolds. J. Lond. Math. Soc. 2(65), 257–270 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berarducci, A., Otero, M., Peterzil, Y., Pillay, A.: A descending chain condition for groups definable in o-minimal structures. Ann. Pure Appl. Log. 134(2–3), 303–313 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dold, A.: Lectures on Algebraic Topology, Reprint of the 1980 ed, p. 377+XI. Springer, Berlin (1995)Google Scholar
  5. 5.
    Edmundo, M.J.: On Torsion Points of Locally Definable Groups in o-minimal Structures, Preprint 2003, Revised 11 Feb. 2005, pp. 1–26 (http://www.ciul.ul.pt/edmundo/)
  6. 6.
    Edmundo, M.J.: Covers of groups definable in o-minimal structures. Ill. J. Math. 49(1), 99–120 (2005)MathSciNetMATHGoogle Scholar
  7. 7.
    Edmundo, M.J.: Locally definable groups in o-minimal structures. J. Algebra 301(1), 194–223 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Edmundo, Mário J., Eleftheriou, Pantelis E.: The universal covering homomorphism in o-minimal expansions of groups. Math. Log. Q. 53(6), 571–582 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Edmundo, M.J., Otero, M.: Definably compact abelian groups. J. Math. Log. 4(2), 163–180 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Eleftheriou, P.E.: Non-standard lattices and o-minimal groups, Preprint, Jan. 24, 2012, pp. 1–16 (2012)Google Scholar
  11. 11.
    Eleftheriou, P.E., Peterzil, Y.: Definable quotients of locally definable groups. Selecta Mathematica 18, 885–903 (2012)Google Scholar
  12. 12.
    Eleftheriou, P.E., Peterzil, Y.: Lattices in locally definable subgroups of \(\langle R^{n},+\rangle \), Preprint Feb. 11 (2012), 1–12. To appear in Notre Dame Journal of Formal LogicGoogle Scholar
  13. 13.
    Hrushovski, E., Peterzil, Y., Pillay, A.: Groups, measures, and the NIP. J. Am. Math. Soc. 21(02), 563–597 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Peterzil, Ya’acov, Starchenko, Sergei: Definable homomorphisms of abelian groups in o-minimal structures. Ann. Pure Appl. Log. 101(1), 1–27 (2000)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Mimura, M., Toda, H.: Topology of Lie Groups I and II. American Mathematical Society, 1991 edition, pp. 451+iiiGoogle Scholar
  16. 16.
    Pillay, Anand: On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra 53(3), 239–255 (1988)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Shelah, Saharon: Can the fundamental (homotopy) group of a space be the rationals? Proc. Am. Math. Soc. 103(2), 627–632 (1988)MATHCrossRefGoogle Scholar
  18. 18.
    Tent, K., Ziegler, M.: A Course in Model Theory, p. x+248. Cambridge University Press, Cambridge (2012)MATHCrossRefGoogle Scholar
  19. 19.
    van den Dries, L.: Tame Topology and o-minimal Structures, London Mathematical Society Lecture Note Series, 248, p. x+180. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  20. 20.
    Whitehead, G.W.: Elements of Homotopy Theory, p. 744+xxi. Springer, Berlin (1978)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Alessandro Berarducci
    • 1
  • Mário Edmundo
    • 2
  • Marcello Mamino
    • 3
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Universidade Aberta and CMAF Universidade de LisboaLisboaPortugal
  3. 3.CMAF Universidade de LisboaLisboaPortugal

Personalised recommendations