Conciseness on normal subgroups and new concise words from outer commutator words (Q6595723)

Let \(G\) be a group and let \(w=w(x_{1},\ldots, x_{r}) \in F_{\infty}(x_{1}, \ldots, x_{r}, \ldots)\) be a word in a free group \(F_{\infty}\) of countably infinite rank. Let \(w\{G\}=\{w(g_{1}, \ldots , g_{r} \mid g_{i} \in G \} \subseteq G\) and \(w(G)=\langle w\{G\}\rangle\). The word \(w\) is concise if whenever the set \(w\{G\}\) is finite in a group \(G\), then also the subgroup \(w(G)\) is finite. Outer commutator words \(w\) are defined recursively, starting from single variables, in such a way that if \(\alpha\) and \(\beta\) are outer commutators in different variables, then \(w=[\alpha, \beta]\) is also an outer commutator and imposing the condition \(w \in [F_{\infty},F_{\infty}]\).\N\NThe goal of the paper under review is to extend the main results of the authors [J. Iran. Math. Soc. 4, No. 2, 189--206 (2023; Zbl 1535.20158)] from the case of lower central and derived words to the more general setting of outer commutator words. More precisely, the authors prove the following results:\N\NTheorem A: Let \(w=w(x_{1}, \ldots , x_{n})\) be an outer commutator word. If \(u_{1}, \ldots, u_{r}\) are non-commutator words in disjoint sets of variables, then the word \(w(u_{1}, \ldots, u_{r})\) is concise. In particular, the word \(w(x_{1}^{n_{1}}, \ldots , x_{r}^{n_{r}})\) is concise whenever \(n_{1}, \ldots , n_{r} \in \mathbb{Z} \setminus \{0\}\).\N\NTheorem B. Let \(w=w(x_{1},\ldots, x_{r}\)) be an outer commutator word. Assume that \(\mathbf{N}=(N_{1}, \ldots, N_{r})\) is a \(r\)-tuple of normal subgroups of a group \(G\) such that the set \(w\{\mathbf{N}\}=\{w(h_{1}, \ldots, h_{i}) \mid h_{i} \in N_{i} \}\) is finite. Then the subgroup \(w(\mathbf{N})=\langle w\{\mathbf{N}\} \rangle\) is also finite.