Certain residual properties of generalized Baumslag-Solitar 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.

19 pages, in English

References in corpus (1)

• On the lower central series of Baumslag-Solitar groups

Cited by in corpus (2)

• Certain residual properties of HNN-extensions with central associated subgroups
• On residually nilpotence of group extensions