On the separability of abelian subgroups of the fundamental groups of graphs of groups. II

Consider the fundamental group ${\frak G}$ of an arbitrary graph of groups and some root class ${\mathcal C}$ of groups, i.e., a class containing a nontrivial group and closed under subgroups, extensions, and unrestricted direct 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 each $y \in Y$. We provide some criterion for the separability by ${\mathcal C}$ of a finitely generated abelian subgroup of ${\frak G}$ valid when the group satisfies an analog of the Baumslag filtration condition. This enables us to describe the ${\mathcal C}$-separable finitely generated abelian subgroups for the fundamental groups of some graphs of groups with central edge subgroups.