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.