On conciseness of the word in Olshanskii's example (Q6198029)

Let \(w=w(x_{1}, \ldots ,x_{r}\)) be a group word and let \(G\) be a group. The set of word values of \(w\) in \(G\) is defined as \(w\{G\}=\{ w(g_{1}, \ldots, g_{r}) \mid g_{1}, \ldots, g_{r} \in G \} \subseteq G\) and the subgroup \(w(G)=\langle w\{G\} \rangle\) is called the verbal subgroup of \(w\) in \(G\). The word \(w\) is called concise in a class of groups \(\mathcal{C}\) if \(w(G)\) is finite whenever \(w\{G\}\) is finite in a group \(G \in \mathcal{C}\). Philip Hall conjectured that all words are concise in the class of all groups but \textit{S. V. Ivanov}, in [Sov. Math. 33, No. 6, 59--70 (1990; Zbl 0697.20016)], refuted P. Hall's conjecture, showing that the word \([[x^{pn}, y^{pn}]^{n}, y^{pn}]^{n}\) is not concise for big enough \(n\) and \(p\), where \(n\) is odd and \(p\) is a prime. Another noteworthy word was introduced by \textit{A. Yu. Olshanskii} in [Mat. Sb., N. Ser. 126(168), No. 1, 59--82 (1985; Zbl 0574.20023)]. For positive integers \(d\) and \(n\), set \[v(x,y)=[[x^{d},y^{d}]^{d},[y^{d},x^{-d}]^{d}]\] and \[ w_{o}(x, y) = [x, y]v(x, y)^{n}[x, y]^{\epsilon_{1}} v(x,y)^{n+1} \ldots [x, y]^{\epsilon_{h-1}}v(x,y)^{n+h-1}, \] where \(\epsilon_{k}=1\) if \(k \equiv 1,2,3,5,6 \mod 10\) and \(\epsilon_{k}=-1\) if \(k \equiv 0,4,7, 8,9 \mod 10\) for \(k=1, \ldots, h\) and \(h > 5\cdot10^5\). Olshanskii showed that the parameters \(n\) and \(d\) can be chosen in such a way that the word \(w_{o}\) has several remarkable properties. In particular, the variety of groups where \(w_{o}\) is a law contains infinite non-abelian groups while all finite groups in the variety are abelian. Moreover, the word \(w_{o}\) is not concise in the class of all groups (see [\textit{A. Yu. Olshanskii}, Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.20019)], p. 493). A result due to \textit{G. A. Fernández-Alcober} and \textit{M. Morigi} ([J. Lond. Math. Soc., II. Ser. 82, No. 3, 581--595 (2010; Zbl 1227.20032)]) asserts that if a word \(w\) is concise in the class of all groups it is actually boundedly concise, that is, there is a function \(f_{w} : \mathbb{N} \rightarrow \mathbb{N}\) such that whenever \(|w\{G\}| = m\) in a group \(G\), then \(|w(G)| \leq f_{w}(m)\). On the other hand, it is an open problem whether the same phenomenon holds for words that are concise in the class of residually finite groups. A first result proved in the paper under review is (Theorem 3.1): The word \(w_{o}\) is boundedly concise in the class of residually finite groups. A variation of the notion of conciseness for profinite groups was introduced by \textit{E. Detomi}, \textit{B. Klopsch} and the second author in [J. Lond. Math. Soc., II. Ser. 102, No. 3, 977--993 (2020; Zbl 1528.20034)]: a word \(w\) is strongly concise in a class of profinite groups \(\mathcal{P}\) if the verbal subgroup \(w(G)\) is finite in any group \(G \in \mathcal{P}\) in which \(|w\{G\}| < 2^{\aleph_{0}}\). Theorem 3.4 of this paper states that the word \(w_{o}\) is strongly concise in the class \(\mathcal{P}\). The authors point out that the question whether the word \([[x^{pn}, y^{pn}]^{n}, y^{pn}]^{n}\) considered by Ivanov is concise in residually finite groups (or even strongly concise in profinite groups) remains open.