On self-similarity of finitely generated torsion-free nilpotent groups

Abstract

This paper aims to investigate the self-similarity property in finitely-generated torsion-free nilpotent groups. We establish connections between geometric equivalence and self-similarity in these groups. Moreover, we show that any 2-generated torsion-free nilpotent group of class 3 has a faithful representation as a transitive self-similar group. Additionally, we provide an example of a 4-generated torsion-free nilpotent group of class 3 that does not admit such a representation. This example, sourced from the work of V. Bludov and B. Gusev in \cite{BG}, resolves a question posed by S. Sidki in \cite{LectureDusseldorf}.