Discrete subgroups of locally definable groups

TL;DR

This paper investigates the structure of locally definable groups in o-minimal structures, proving a conjecture about abelian connected groups being covers of definable groups under convexity conditions, and analyzing zero-dimensional subgroups.

Contribution

It establishes necessary and sufficient convexity conditions for abelian connected groups to be covers of definable groups, and studies properties of zero-dimensional subgroups.

Findings

The n-torsion subgroup of a locally definable connected group is finite.

Every zero-dimensional compatible subgroup has finite rank.

Under convexity, zero-dimensional compatible subgroups are finitely generated.

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. 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.