Skip to main content
SpringerLink
Log in

Definably compact groups definable in real closed fields

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study definably compact definably connected groups definable in a sufficiently saturated real closed field R. Our main result is that for such a kind of groups G that are also abelian, there is a Zariski-connected R-algebraic group H such that the o-minimal universal covering group of G is, up to a locally definable isomorphism, an open connected locally definable subgroup \(\mathcal{W}\) of the o-minimal universal covering group of H(R)0. Thus, G is definably isomorphic to the definable quotient of \(\mathcal{W}\) by a discrete subgroup.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Baro and M. J. Edmundo, Corrigendum to: “Locally definable groups in o-minimal structures” [J. Algebra 301 (2006), no. 1, 194–223; mr2230327] by Edmundo, Journal of Algebra 320 (2008), 3079–3080.

    Article  MathSciNet  Google Scholar 

  2. E. Baro and M. Otero, Locally definable homotopy, Annals of Pure and Applied Logic 161 (2010), 488–503.

    Article  MathSciNet  Google Scholar 

  3. E. Barriga, Semialgebraic groups over real closed fields, Ph.D. Thesis, University of Haifa and Universida de los Andes, 2017. https://biblioteca.uniandes.edu.co/acepto201699.php?id=12264.pdf

    Google Scholar 

  4. A. Berarducci, Cohomology of groups in o-minimal structures: acyclicity of the infinites- imal subgroup, Journal of Symbolic Logic 74 (2009), 891–900.

    Article  MathSciNet  Google Scholar 

  5. A. Berarducci, M. Edmundo and M. Mamino, Discrete subgroups of locally definable groups, Selecta Mathematica 19 (2013), 719–736.

    Article  MathSciNet  Google Scholar 

  6. E. Bouscaren, The group configuration—after E. Hrushovski, in The Model Theory of Groups (Notre Dame, IN, 1985–1987), Notre Dame Mathematical Lectures, Vol. 11, University of Notre Dame Press, Notre Dame, IN, 1989, pp. 199–209.

    Google Scholar 

  7. L. van den Dries, Tame Topology and o-minimal Structures, London Mathematical Society Lecture Note Series, Vol. 248, Cambridge University Press, Cambridge, 1998.

  8. M. J. Edmundo, Covers of groups definable in o-minimal structures, Illinois Journal of Mathematics 49 (2005), 99–120.

    MathSciNet  MATH  Google Scholar 

  9. M. J. Edmundo, Locally definable groups in o-minimal structures, Journal of Algebra 301 (2006), 194–223.

    Article  MathSciNet  Google Scholar 

  10. M. J. Edmundo, Erratum to: “Covers of groups definable in o-minimal structures” [Illinois J. Math.49 (2005), no. 1, 99–120], Illinois Journal of Mathematics 51 (2007), 1037–1038.

    Article  MathSciNet  Google Scholar 

  11. M. J. Edmundo and P. E. Eleftheriou, The universal covering homomorphism in o- minimal expansions of groups, Mathematical Logic Quarterly 53 (2007), 571–582.

    Article  MathSciNet  Google Scholar 

  12. M. J. ‘Edmundo, P. E. Eleftheriou and L. Prelli, The universal covering map in o-minimal expansions of groups, Topology and its Applications 160 (2013), 1530–1556.

    Article  MathSciNet  Google Scholar 

  13. M. J. Edmundo and M. Otero, Definably compact abelian groups, Journal of Mathematical Logic 4 (2004), 163–180.

    Article  MathSciNet  Google Scholar 

  14. P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Mathematica 18 (2012), 885–903.

    Article  MathSciNet  Google Scholar 

  15. E. Hrushovski, Y. Peterzil and A. Pillay, Groups, measures, and the NIP, Journal of the American Mathematical Society21 (2008), 563–596.

    Article  MathSciNet  Google Scholar 

  16. E. Hrushovski and A. Pillay, Groups definable in local fields and pseudo-finite fields, Israel Journal of Mathematics 85 (1994), 203–262.

    Article  MathSciNet  Google Scholar 

  17. E. Hrushovski and A. Pillay, Affine Nash groups over real closed fields, Confluentes Mathematici 3 (2011), 577–585.

    Article  MathSciNet  Google Scholar 

  18. E. Hrushovski and A. Pillay, On NIP and invariant measures, Journal of the European Mathematical Society 13 (2011), 1005–1061.

    Article  MathSciNet  Google Scholar 

  19. Y. Peterzil, A self-guide to o-minimality, preprint, http://math.haifa.ac.il/kobi/o-minimal.pdf.

  20. Y. Peterzil and A. Pillay, Generic sets in definably compact groups, Fundamenta Mathematicae 193 (2007), 153–170.

    Article  MathSciNet  Google Scholar 

  21. Y. Peterzil and S. Starchenko, Definable homomorphisms of abelian groups in o-minimal structures, Annals of Pure and Applied Logic 101 (2000), 1–27.

    Article  MathSciNet  Google Scholar 

  22. A. Pillay, On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra 53 (1988), 239–255.

    Article  MathSciNet  Google Scholar 

  23. S. Shelah, Minimal bounded index subgroup for dependent theories, Proceedings of the American Mathematical Society 136 (2008), 1087–1091.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departmento de Matemáticas, Universidad de los Andes, Cra 1 No 18A-10, Bogotá, 111711, Colombia

    Eliana Barriga

Authors
  1. Eliana Barriga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eliana Barriga.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barriga, E. Definably compact groups definable in real closed fields. Isr. J. Math. 238, 121–166 (2020). https://doi.org/10.1007/s11856-020-2014-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-020-2014-z

Search

Navigation