Cartan subgroups and regular points of o-minimal groups

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.