On Conditions for the Approximability of the Fundamental Groups of Graphs of Groups by Root Classes of Groups

Suppose that Gamma is a non-empty connected graph, G is the fundamental group of a graph of groups over Gamma,andCis a root class of groups (the last means that Ccontains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian powers of a certain type). It is known that G is residually a C-group if it has a homomorphism onto a group from Cacting injectively on all vertex groups. We prove that, in this assertion, the words "vertex groups" can be replaced by "edge subgroups "provided all vertex groups are residually C-groups. We also show that the converse doesn't need to hold if Cconsists of periodic groups and contains at least one infinite group