Commutators in groups definable in o-minimal structures (Q2845879)

In [On the connections between groups definable in o-minimal structures and real Lie groups: the non-compact case. Siena: University of Siena (PhD Thesis) (2009)], \textit{A. Conversano} exhibits an o-minimal, definably connected group \(G\) such that \(G'\) is not definable. Furthermore, \(G\) is a central extension of a simple group by an infinite centre. In the authors' words, the paper essentially proves that central extensions of this type are the only obstructions to the definability of the commutator subgroups in the o-minimal case. More precisely, a definably connected group \(G\) definable in an o-minimal structure is said to be a \textit{strict central extension} of a definably simple group if \(Z(G)\) is infinite and \(G/Z(G)\) is infinite non-abelian and definably simple. The main theorem shows the following. Let \(G\) be a group definable in an o-minimal structure, \(A\) and \(B\) two definable subgroups normalizing each other and such that \(A^\circ B^\circ\) satisfy the following assumption: the derived subgroup \((H/K)'\) is definable for every definable section \(H/K\) of \(A^\circ B^\circ\) that is a strict central extension of a definably simple group. Then, the subgroup \([A,B]\) is definable.