Self-similarity of the generalized Baumslag-Solitar groups

TL;DR

This paper proves that certain classes of residually finite generalized Baumslag-Solitar groups and related fundamental groups are self-similar, expanding understanding of their algebraic and geometric properties.

Contribution

It establishes the self-similarity of all residually finite generalized Baumslag-Solitar groups of rank n and a broader class of fundamental groups with specific torsion-free and embedding conditions.

Findings

All residually finite generalized Baumslag-Solitar groups of rank n are self-similar.

Residually finite fundamental groups of certain graphs of groups with Heisenberg-like vertex groups are self-similar.

The results extend the class of groups known to exhibit self-similarity.

Abstract

We show that all residually finite generalized Baumslag-Solitar groups of rank $n \geq 1$, defined on a finite and connected graph, are self-similar. Furthermore we prove that all residually finite fundamental groups of (finite, connected) graph of groups where all vertex and edge groups are torsion-free and commensurable with the Heisenberg group and all edge groups properly embed in the corresponding vertex groups are self-similar.