On the Separability of Abelian Subgroups of the Fundamental Groups of Graphs of Groups. II

Consider the fundamental group \( {\mathfrak{G}} \) 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 \( {\mathfrak{G}} \) 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.