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

Abstract

Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism $\sigma$ of $X$ onto a group from $\mathcal{C}$ such that the elements $x\sigma$ and $y\sigma$ are still not equal (respectively, not conjugate in $X\sigma$). 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 $\mathcal{C}$ consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy $\mathcal{C}$-separable if and only if it is residually a $\mathcal{C}$-group. We also find a criterion for a solvable GBS-group to be conjugacy $\mathcal{C}$-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite.