Cartan subgroups of groups definable in o-minimal structures

TL;DR

This paper proves that groups definable in o-minimal structures possess Cartan subgroups with finitely many conjugacy classes, and describes how these subgroups cover the entire group in terms of dimension.

Contribution

It establishes the existence and conjugacy finiteness of Cartan subgroups in o-minimal definable groups, providing a detailed description of their coverage.

Findings

Existence of Cartan subgroups in definable groups

Finiteness of conjugacy classes of Cartan subgroups

Description of subgroup coverage based on dimension

Abstract

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.