On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups

Let C be a class of groups. A group X is said to be residually a C-group (conjugacy C-separable) if, for any elements x,y ∈ X that are not equal (not conjugate in X), there exists a homomorphism σ of X onto a group from C such that the elements xσ and yσ are still not equal (respectively, not conjugate in Xσ). A generalized Baumslag-Solitar group or GBS-group is the fundamental group of a finite connected graph of groups whose all vertex and edge groups are infinite cyclic. An ordinary Baumslag-Solitar group is the GBS-group that corresponds to a graph containing only one vertex and one loop. Suppose that the class C consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy C-separable if and only if it is residually a C-group. We also find a criterion for a solvable GBS-group to be conjugacy C-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite.