Commutators in groups definable in o-minimal structures

Abstract

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal 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.