Deficiency, commensurators and 4-dimensional infrasolvmanifolds

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Contribution

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Abstract

We show that if $\pi$ is the fundamental group of a 4-dimensional infrasolvmanifold then $-2\leq{def(\pi)}\leq0$, 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 $\mathrm{def}(G)>1$.