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 $[x^m,z_1,\dots ,z_r]$ and $[x,y,y,z_1,\dots ,z_r]$, where $m\in \{2,3\}$ and $r⩾1$.

中文翻译：

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有限群体的简洁性

组词 $w$ 据说在课堂上非常简洁 $\mathcal{C}$ 有限组，如果每个组 $G$ 在 $\mathcal{C}$ 这样 $w$ 花费少于 $2^{{\aleph }_0}$ 值 $G$，语言亚组 $w（G）$是有限的。Detomi，Morigi和Shumyatsky确立了多线性换向器词-以及特定的词$X^2$ 和 $[X^2，ÿ]$ —具有以下性质：在限定词组中相应的言语子群是有限的 $G$ 只要单词最多包含许多值 $G$。他们推测，实际上，每个单词都应该如此。尤其是，他们的猜想包括强词和恩格尔词作为开放案例。在本文中，我们通过参数化的词采取了一种新方法，从而获得了更好的结果。首先，我们证明多线性换向器词在所有有限群的类中都非常简洁。然后我们确定，在幂等有限组的类别中，每个组词都非常简洁。例如，据此我们推断出$w$ 是组词之一 $X^2$， $X^3$， $X^6$， $[X^3，ÿ]$ 要么 $[X，ÿ，ÿ]$， 然后 $w$在所有有限组的类别中非常简洁。确实，对于无限家族的所有单词都可以得出相同的结论$[X^{米}，{ž}_{\mathrm{1个}}，\dots ，{ž}_{\mathrm{[R}}]$ 和 $[X，ÿ，ÿ，{ž}_{\mathrm{1个}}，\dots ，{ž}_{\mathrm{[R}}]$，在哪里 $米\in \{2，3\}$ 和 $\mathrm{[R}⩾\mathrm{1个}$。