Discrete subgroups of locally definable groups

• Alessandro Berarducci ^[1] ; Mário Edmundo ^[2] ; Marcello Mamino ^[2]
  1. [1] University of Pisa

     University of Pisa

     Pisa, Italia

     
  2. [2] Universidade de Lisboa

     Universidade de Lisboa

     Socorro, Portugal

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 19, Nº. 3, 2013, págs. 719-736
• Idioma: inglés
• DOI: 10.1007/s00029-013-0123-9
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• Resumen

  • 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 GG 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 GG . Given a locally definable connected group GG (not necessarily definably generated), we prove that the nn -torsion subgroup of GG is finite and that every zero-dimensional compatible subgroup of GG has finite rank. Under a convexity hypothesis, we show that every zero-dimensional compatible subgroup of GG is finitely generated.