Splitting definably compact groups in o-minimal structures

TL;DR

This paper extends a classical topological result about compact Lie groups to the setting of definably compact groups in o-minimal structures, highlighting both similarities and key differences.

Contribution

It proves a parallel decomposition theorem for definably compact groups in o-minimal structures and provides a counterexample to the existence of a definable semidirect complement.

Findings

Definably compact groups can be decomposed similarly to compact Lie groups.

Counterexample shows the derived subgroup may lack a definable semidirect complement.

The result bridges classical Lie group theory and o-minimal structures.

Abstract

An argument of A.Borel shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.