On the subgroup separability of the free product of groups

Abstract: Suppose that $\mathcal{C}$ is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), $G$ is the free product of residually $\mathcal{C}$-groups $A_{i}$ ($i \in \mathcal{I}$), and $H$ is a subgroup of $G$ satisfying a non-trivial identity. We prove a criterion for the $\mathcal{C}$-separability of $H$ in $G$. It follows from this criterion that, if ${\mathcal{V}<em>{j} \mid j \in \mathcal{J}}$ is a family of group varieties, each $\mathcal{V}</em>{j}$ ($j \in \mathcal{J}$) is distinct from the variety of all groups, and $$\mathcal{V} = \bigcup_{j \in \mathcal{J}} \mathcal{V}<em>{j}$$, then one can give a description of $\mathcal{C}$-separable $\mathcal{V}$-subgroups of $G$ provided such a description is known for every group $A</em>{i}$ ($i \in \mathcal{I}$).