On the LS-category of homomorphism of almost nilpotent groups

We prove the equality $\text{cat}(\phi )=\text{cd}(\phi )$ for epimorphisms $\phi :\mathrm{\Gamma }\to \mathrm{\Lambda }$ between torsion-free, finitely generated almost nilpotent groups $\mathrm{\Gamma }$ and $\mathrm{\Lambda }$. In addition, we prove the equality $\text{cat}(\phi )=\text{cd}(\phi )$ for homomorphisms $\phi :\mathrm{\Gamma }\to \mathrm{\Lambda }$ between torsion-free, finitely generated virtually nilpotent groups.