On conciseness of some commutator words
Let $w =w(x_{1}, \dots, x_{n})$ be a group-word in variables $x_{1}, \dots, x_{n}$ and let $G$ be a group. The verbal subgroup $w(G)$ is the subgroup of $G$ generated by the set $G_{w}=\{w(g_{1},\dots,g_{n}) \mid g_{1},\dots,g_{n} \in G \}$. A word $w$ is \textit{concise} if, for every group $G$, whenever $G_{w}$ is finite the subgroup $w(G)$ is finite. Philip Hall conjectured that every word is concise, and he proved this for every non-commutator word (a word is non-commutator if the sum of exponents of some variables involved in it is non-zero), and for lower central words. \par The purpose of the article under review is to prove Theorem 1.1: For any non-commutator words $w$ and $w'$, the word $[w,w']$ is concise.
scientific article; zbMATH DE number 7005656