Kourovka Notebook Problem 18.35

Definitions and mathematical statement

For a group $H$, write $H'=[H,H]$ and $H''=[H',H']$. The group $H$ is metabelian if $H''=\{1\}$, equivalently if $H'$ is abelian. Let $F_2$ be the free group on two generators and let \[ G=F_2/F_2'', \] the relatively free group of rank $2$ in the variety of all metabelian groups.

For groups $A,B$, a family $\mathscr F\subseteq\operatorname{Hom}(A,B)$ is separating if \[ \forall a\in A\setminus\{1\}\;\exists f\in\mathscr F:\quad f(a)\ne1. \] It is discriminating if \[ \forall n\ge1\;\forall(a_1,\ldots,a_n)\in(A\setminus\{1\})^n\;\exists f\in\mathscr F:\quad \bigwedge_{i=1}^{n}f(a_i)\ne1. \] For $t\in T$, define its centralizer by \[ C_T(t)=\{x\in T:xt=tx\}. \]

Question. Let $T$ be any metabelian group such that $C_T(t)$ is abelian for every $t\in T\setminus\{1\}$. Suppose that $T$ admits a separating family of surjective homomorphisms $T\to G$. Must $T$ also admit a discriminating family of surjective homomorphisms $T\to G$? Equivalently, under these hypotheses, must every finite list $t_1,\ldots,t_n$ of nonidentity elements of $T$, with $n\ge1$, admit a single surjective homomorphism $\phi:T\to G$ such that $\phi(t_i)\ne1$ for every $i$? No finite generation assumption on $T$ is supplied.

The source states that a positive answer would give a metabelian analogue of a classical theorem of B. Baumslag; it does not specify that theorem or provide its citation.

Short English statement

Suppose a metabelian group $T$ has abelian centralizers of all nonidentity elements, and each nonidentity element survives under some surjection onto the free metabelian group of rank $2$. Must every finite collection of nonidentity elements survive simultaneously under one such surjection?

Sources

• Local UnsolvedMath registry, permanent ID $701464$, source problem code catalogue entry, titled Kourovka Notebook Problem 18.35. The supplied record labels the problem open; this label is source metadata, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 18.35, 18th issue (2014); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by A. Gaglione and D. Spellman in 2014. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement attributes a related classical theorem to B. Baumslag, without a theorem statement or bibliographic reference.