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

TL;DR

This paper proves that definable groups in o-minimal structures are finitely generated, generalizing properties of algebraic groups and providing insights into their structure, including bounds on definable rank and solutions to Wiegold's problem.

Contribution

It establishes that all definable groups in o-minimal structures are finitely generated and introduces bounds on their definable rank, also solving Wiegold's problem in this context.

Findings

Definable groups are finitely generated within o-minimal structures.

The dimension bounds the definable rank, often strictly.

Every perfect definable group is normally monogenic.

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.