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

TL;DR

This paper provides an elementary proof confirming that certain semi-direct product groups of the form ^n 7Z are strongly scale-invariant only if they are virtually nilpotent, supporting a conjecture about the structure of such groups.

Contribution

It offers an elementary proof for the Benjamini-Nekrashevych-Pete conjecture specifically for semi-direct products ^n 7Z, expanding the classes of groups where the conjecture holds.

Findings

Confirmed the conjecture for ^n 7Z groups

Showed these groups are virtually nilpotent if strongly scale-invariant

Provided a simpler proof compared to previous methods

Abstract

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 $\cap_{n\geq 0} f^n(G)$ 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. Der\'e 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 $\mathbb{Z}^n\rtimes \mathbb{Z}$.