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Preprint Number 427
427. Alessandro Berarducci, Mário Edmundo, Marcello Mamino
Discrete subgroups of locally definable groups
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Submission date: 28 February 2012.
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. We prove that the rank of such groups is bounded by the dimension of G. We also obtain the finiteness of the n-torsion subgroup under a divisibility assumption. Under a convexity hypothesis we show that the fundamental group of G is finitely generated.
Mathematics Subject Classification: 03C64, 03C68, 22B99
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