Self-similarity of some soluble relatively free groups

TL;DR

This paper investigates the self-similarity properties of certain algebraic groups, proving that free nilpotent groups are transitive self-similar while free metabelian groups are not.

Contribution

It establishes the transitive self-similarity of free nilpotent groups and the lack of this property in free metabelian groups of rank at least 2.

Findings

Free nilpotent groups of finite rank are transitive self-similar.

Free metabelian groups of rank ≥ 2 are not transitive self-similar.

Abstract

In this paper we prove that a free nilpotent group of finite rank is transitive self-similar. In contrast, we prove that a free metabelian group of rank $r \geq 2$ is not transitive self-similar.