Discrete subgroups of locally definable groups

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