Words of Engel type are concise in residually finite groups. Part II

Abstract

This work is a natural follow-up of the article \[5]. Given a group-word $w$ and a group $G$, the verbal subgroup $w(G)$ is the one generated by all $w$-values in $G$. The word $w$ is called concise if $w(G)$ is finite whenever the set of $w$-values in $G$ is finite. It is an open question whether every word is concise in residually finite groups. Let $w=w(x\_1,\ldots,x\_k)$ be a multilinear commutator word, $n$ a positive integer and $q$ a prime power. In the present article we show that the word $\[w^q,\_ny]$ is concise in residually finite groups (Theorem 1.2) while the word $\[w,\_ny]$ is boundedly concise in residually finite groups (Theorem 1.1).