Definable rank, o-minimal groups, and Wiegold's problem

Abstract

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to o-minimal groups, of algebraic groups (over an algebraically closed field of characteristic 0) containing a Zariski-dense finitely generated subgroup. When the group is definably connected, its dimension provides an upper bound for the definable rank. The upper bound is often strict. For instance, this is always the case for 0-groups and, more generally, for solvable groups with torsion. We further prove that every perfect definable group is normally monogenic, generalizing the finite group case. This yields a positive answer to Wiegold's problem in the o-minimal setting.