7.58 (1980)

Suppose that $F$ is an absolutely free group, $R$ a normal subgroup of $F$, and let $\mathfrak{V}$ be a variety of groups. It is well-known (H. Neumann, Varieties of Groups, Springer, Berlin, 1967) that the group $F/\mathfrak{V}(R)$ is isomorphically embeddable in the $\mathfrak{V}$-verbal wreath product of a $\mathfrak{V}$-free group of the same rank as $F$ with $F/R$. Find a criterion indicating which elements of this wreath product belong to the image of this embedding. A criterion is known in the case where $\mathfrak{V}$ is the variety of all abelian groups (V. N. Remeslennikov, V. G. Sokolov, Algebra and Logic, 9, no. 5 (1970), 342–349).