Discrete subgroups of locally definable groups

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.

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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.

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Correspondence to Alessandro Berarducci.

Additional information

Partially supported by PRIN 2009WY32E8_003: O-minimalità, teoria degli insiemi, metodi e modelli non standard e applicazioni.

Partially supported by Fundação para a Ciência e a Tecnologia PEst OE/MAT/UI0209/2011.

Partially supported by Fundação para a Ciência e a Tecnologia grant SFRH/BPD/73859/2010.

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Berarducci, A., Edmundo, M. & Mamino, M. Discrete subgroups of locally definable groups. Sel. Math. New Ser. 19, 719–736 (2013). https://doi.org/10.1007/s00029-013-0123-9

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Keywords

  • Locally definable groups
  • Covers
  • Discrete subgroups

Mathematics Subject Classification (1991)

  • 03C64
  • 03C68
  • 22B99