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

2025/09/21 by Adilson A. Berlatto, Berlatto, Adilson, Tulio M. G. Santos +1
Materials Science · Mathematics · #Quasicrystal Structures and Properties #Geometric and Algebraic Topology #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2509.16947

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 \citeBG, resolves a question posed by S. Sidki in \citeLectureDusseldorf.

Citations

• Self-similarity of some soluble relatively free groups
• Which Nilpotent Groups are Self-Similar?
• Intransitive Self-similar Groups
• Virtual Endomorphisms of Nilpotent Groups

Related

• Geometrical equivalence of nilpotent torsion free groups
• Self-similarity of some soluble relatively free groups
• Which Nilpotent Groups are Self-Similar?
• Geometric equivalence of π-torsion free nilpotent groups
• On self-similarity of wreath products of abelian groups