Strongly scale-invariant virtually polycyclic groups

TL;DR

This paper proves that strongly scale-invariant virtually polycyclic groups are necessarily virtually nilpotent, extending the understanding of such groups and their endomorphisms using algebraic and geometric methods.

Contribution

It demonstrates that the existence of certain endomorphisms in virtually polycyclic groups implies they are virtually nilpotent, and characterizes these groups explicitly.

Findings

Strongly scale-invariant virtually polycyclic groups are virtually nilpotent.

Characterization of virtually nilpotent groups with such endomorphisms.

Generalization of results on Reidemeister numbers for infra-solvmanifolds.

Abstract

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