18.35 (2014)

A family $\mathcal{F}$ of group homomorphisms $A \to B$ is separating if for every nontrivial $a \in A$ there is $f \in \mathcal{F}$ such that $f(a) \neq 1$, and discriminating if for any finitely many nontrivial elements $a_1, \dots, a_n \in A$ there is $f \in \mathcal{F}$ such that $f(a_i) \neq 1$ for all $i = 1, \dots, n$. Let $G$ be a relatively free group of rank 2 in the variety of metabelian groups. Let $T$ be a metabelian group in which the centralizer of every nontrivial element is abelian. If $T$ admits a separating family of surjective homomorphisms $T \to G$ must it also admit a discriminating family of surjective homomorphisms $T \to G$?

A positive answer would give a metabelian analogue of a classical theorem of B. Baumslag.