An elementary proof of the Benjamini-Nekrashevych-Pete conjecture for the semi-direct products ${\mathbb{Z}}^n⋊\mathbb{Z}$

A finitely generated group $G$ is called strongly scale-invariant if there exists an injective homomorphism $f:G\to G$ such that $f(G)$ is a finite index subgroup of $G$ and such that ${\cap }_{n\ge 0}{f}^n(G)$ is finite. Nekrashevych and Pete conjectured that all strongly scale-invariant groups are virtually nilpotent, after disproving a stronger conjecture by Benjamini. This conjecture is known to be true in some situations. Deré proved it for virtually polycyclic groups. In this paper, we provide an elementary proof for those polycyclic groups that can be written as a semi-direct product ${\mathbb{Z}}^n⋊\mathbb{Z}$.