On conciseness of the word in Olshanskii's example

TL;DR

This paper investigates the conciseness properties of a specific group-word in different classes of groups, proving it is concise in residually finite groups and strongly concise in profinite groups.

Contribution

It demonstrates that Olshanskii's non-concise word is actually concise in residually finite groups and strongly concise in profinite groups, clarifying its behavior in these classes.

Findings

The word is concise in residually finite groups.

The word is strongly concise in profinite groups.

Conciseness depends on the class of groups considered.

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.