To the Question of the Root-Class Residuality of Free Constructions of Groups

Let $$\mathcal{C}$$ be a root class of groups and $$\mathcal{\pi}_{1}(\mathcal{G})$$ be the fundamental group of a graph $$\mathcal{G}$$ of groups. We prove that if $$\mathcal{G}$$ has a finite number of edges and there exists a homomorphism of $$\mathcal{\pi}_{1}(\mathcal{G})$$ onto a group of $$\mathcal{C}$$ acting injectively on all the edge subgroups, then $$\mathcal{\pi}_{1}(\mathcal{G})$$ is residually a $$\mathcal{C}$$ -group. The main result of the paper is that the inverse statement is not true for many root classes of groups. The proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a $$\mathcal{C}$$ -group, which is of independent interest.