Words of Engel type are concise in residually finite groups. II. (Q2032444)

Summary: 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).