Submission date: 28 February 2012.

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. 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

Keywords and phrases:

Full text arXiv 1202.5649: pdf, ps.