Deficiency, commensurators and 4-dimensional infrasolvmanifolds

We show that if $\pi$ is the fundamental group of a 4-dimensional infrasolvmanifold then $-2\le def(\pi )\le 0$, and give examples realizing each of these values. We also determine the abstract commensurators of such groups. Finally we show that if $G$ is a finitely generated group the kernel of the natural homomorphism from $G$ to its abstract commensurator $Comm(G)$ is locally nilpotent by locally finite, and is finite if $\mathrm{d}\mathrm{e}\mathrm{f}(G)>1$.