Strongly scale-invariant virtually polycyclic groups

A finitely generated group $\Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the image $\varphi(\Gamma)$ of finite index in and subgroup $\bigcap\_{n>0}\varphi^n(\Gamma)$ finite. The only known examples such groups are virtually nilpotent, or equivalently, all have polynomial growth. question by Nekrashevych Pete asks whether these possibilities for endomorphisms, motivated positive answer due to Gromov special case expanding morphisms. In this paper, we study class polycyclic groups, i.e. solvable which every generated. Using $\mathbb{Q}$-algebraic hull, allows us extend endomorphisms certain a linear algebraic group, show that existence implies nilpotent. Moreover, fully characterize nilpotent morphism satisfying condition above, related grading on corresponding radicable group. As another application methods, generalize result Fel’shtyn Lee about maps infra-solvmanifolds can Reidemeister number iterates.