On bounded conciseness of Engel-like words in residually finite groups

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 said to be boundedly concise if for each positive integer m there exists a number depending only on m and w bounding the order of w(G) whenever the set of w-values in a group G has size at most m. In the present article we show that various generalizations of the Engel word are boundedly concise in residually finite groups.