On conciseness of the word in Olshanskii’s example

Abstract

A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by \(w_o\). The problem whether every word is concise in the class of residually finite groups remains wide open. In this note, we observe that \(w_o\) is concise in residually finite groups. Moreover, we show that \(w_o\) is strongly concise in profinite groups, that is, \(w_o(G)\) is finite whenever G is a profinite group in which \(w_o\) takes less than \(2^{\aleph _0}\) values.