Real Algebraic and Analytic Geometry
Submission: 2012, September 5.
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.
Mathematics Subject Classification (2010): 03C64, 03C68, 22B99.
Keywords and Phrases: Locally definable groups, covers, discrete subgroups.
Full text: http://arxiv.org/abs/1202.5649