Certain residual properties of generalized Baumslag–Solitar groups

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