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

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 D w.x1; : : : ; xk/ be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word OEwq; ny is concise in residually finite groups (Theorem 1.2) while the word OEw; ny is boundedly concise in residually finite groups (Theorem 1.1).