Bounding the order of a verbal subgroup in a residually finite group

Let w be a group-word. Given a group G, we denote by w(G) the verbal subgroup corresponding to the word w, that is, the subgroup generated by the set Gw of all w-values in G. The word w is called concise in a class of groups X if w(G) is finite whenever Gw is finite for a group G ∈χ. It is a long-standing problem whether every word is concise in the class of residually finite groups. In this paper we examine several families of group-words and show that all words in those families are concise in residually finite groups.