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

Abstract

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 $x,y\in X$ 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 vertex and edge groups are all 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.