Strong conciseness in profinite 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. Detomi, Morigi and Shumyatsky established that multilinear commutator words -- and the particular words $x^2$ and $[x^2,y]$ -- have the property that the corresponding verbal subgroup is finite in a profinite group $G$ whenever the word takes at most countably many values in $G$. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if $w$ is one of the group words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and $[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.

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