Commutators in groups definable in o-minimal structures

TL;DR

This paper investigates the properties of commutator subgroups in groups definable within o-minimal structures, establishing their definability and finite commutator width, with implications for derived and central series.

Contribution

It demonstrates the definability and finiteness of commutator width in groups definable in o-minimal structures, including derived and lower central series.

Findings

Commutator subgroups are definable in o-minimal groups.

Finiteness of the commutator width is established.

Results apply to derived and lower central series of solvable groups.

Abstract

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.