Selecta Mathematica

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

Discrete subgroups of locally definable groups

  • Alessandro BerarducciEmail author
  • Mário Edmundo
  • Marcello Mamino


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.


Locally definable groups Covers Discrete subgroups 

Mathematics Subject Classification (1991)

03C64 03C68 22B99 



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.


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Alessandro Berarducci
    • 1
    Email author
  • 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

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