There is a well known fact:



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> If $G$ is a finitely generated group. Then $|G’| < \infty$ iff $[G:Z(G)]<\infty$.
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Suppose $\mathfrak{U}$ is a group variety. Let’s denote the corresponding verbal subgroup as a $V\_{\mathfrak{U}}(G)$ and the corresponding [marginal subgroup](https://groupprops.subwiki.org/wiki/Marginal_subgroup) as $M\_{\mathfrak{U}}(G)$. Note, that for the variety of all abelian groups $\mathfrak{A}$ (defined for the word $[x, y]$) we have $V\_{\mathfrak{A}}(G) = G’$ and $M\_{\mathfrak{A}}(G) = Z(G)$.


My question is:


Can the aforementioned statement be generalized to the following one:



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> Suppose $G$ is a finitely generated group and $\mathfrak{U}$ is a finitely based variety. Then $|V\_{\mathfrak{U}}(G)| < \infty$ iff $[G:M\_{\mathfrak{U}}(G)]<\infty$.
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?


[This question on MSE with several useful comments](https://math.stackexchange.com/q/3444714/407165)



Is finite verbal subgroup equivalent to finite index of marginal subgroup?
It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^\*(G)$ of infinite index (that answered a question of Ph. Hall). 


