G-linear sets and torsion points in definably compact groups

TL;DR

This paper establishes a connection between the dimension of definable sets in a definably compact group and the existence of torsion points, introducing a theory of G-linear sets within o-minimal structures.

Contribution

It develops a general theory of G-linear sets and explores the properties of definable sets containing subgroups in definably compact groups.

Findings

Definable sets with dimension less than the group contain torsion points.

Introduction of G-linear sets theory in o-minimal structures.

Characterization of definable sets containing subgroups.

Abstract

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.