Cartan subgroups and regular points of o-minimal groups

TL;DR

This paper investigates the structure of o-minimal groups, showing that Cartan subgroups are dense and characterizing regular points via Lie algebra properties, enhancing understanding of their algebraic and geometric structure.

Contribution

It establishes the density of Cartan subgroups in o-minimal groups and characterizes regular points through Lie algebra analysis, linking algebraic and geometric aspects.

Findings

Union of Cartan subgroups is dense in G

Every Cartan subalgebra corresponds to a definable Cartan subgroup

Regular points form a dense subset belonging to unique Cartan subgroups

Abstract

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly, that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup - a Cartan subgroup of G -, and secondly, that the set of regular points of G - a dense subset of G - is formed by points which belong to a unique Cartan subgroup of G.