Strong conciseness of Engel words in profinite groups

Abstract

AbstractA group wordwis said to be strongly concise in a class of profinite groups if, for any groupGin , eitherwtakes at least continuum many values inGor the verbal subgroup is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups, as well as that 2‐Engel words are strongly concise (but their approach does not seem to generalize ton‐Engel words for ). In this paper, we prove that for anyn, then‐Engel word (whereyis repeatedntimes) is strongly concise in the class of finitely generated profinite groups.