Certain residual properties of generalized Baumslag-Solitar groups

TL;DR

This paper investigates residual properties of generalized Baumslag-Solitar groups, providing criteria for when such groups are residually within certain classes, including torsion-free, nilpotent, and free groups, thus generalizing previous results.

Contribution

It introduces new criteria for generalized Baumslag-Solitar groups to be residually within classes of groups, especially when these classes contain non-unit or non-periodic groups, extending known theorems.

Findings

Criteria for residuality in classes of periodic groups

Conditions for residual torsion-free nilpotency

Criteria for residual freeness in GBS-groups

Abstract

Let $G$ be a generalized Baumslag-Solitar group and $\mathcal{C}$ be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for every $y \in Y$. We give a criterion for $G$ to be residually a $\mathcal{C}$-group provided $\mathcal{C}$ consists only of periodic groups. We also prove that $G$ is residually a torsion-free $\mathcal{C}$-group if $\mathcal{C}$ contains at least one non-periodic group and is closed under taking homomorphic images. These statements generalize and strengthen some known results. Using the first of them, we provide criteria for a GBS-group to be a) residually nilpotent; b) residually torsion-free nilpotent; c) residually free.