Strong conciseness in profinite metabelian groups

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G \in \mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Using the notion of polynomial mappings introduced by Passi, we establish that every group word is strongly concise in the class of profinite metabelian groups. With this new approach, we also give an alternative proof for the fact that every group word is strongly concise in the class of profinite nilpotent groups.