The universal covering homomorphism in o-minimal expansions of groups

Suppose G is a definably connected, definable group in an o-minimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p:U->G is a locally definable covering homomorphism and π_1(G) is isomorphic to the o-minimal fundamental group π (G) of G defined using locally definable covering homomorphisms.FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU); NSF grant DMS-02-45167 (Cholak

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 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.Comment: Final version. 17 pages. To appear in Selecta Mathematic