Varieties of groups and the problem on conciseness of words

A group-word w is concise in a class of groups 𝒳 if and only if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G ∈ 𝒳. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word w in residually finite groups is equivalent to that in the class of virtually pro-p groups. This is used to show that if q, n are positive integers and w is a multilinear commutator word, then the words w^q and [w^q,ₙ y] are concise in residually finite groups. Earlier this was known only in the case where q is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law w^q ≡ 1, or [δₖ^q,ₙ y] ≡ 1, are varieties.