Varieties of groups and the problem on conciseness of words

TL;DR

This paper investigates the conciseness of group words in residually finite groups, establishing new results for multilinear commutator words and connecting conciseness to properties of virtually pro-p groups.

Contribution

It shows the equivalence of conciseness in residually finite groups and virtually pro-p groups, extending known results to broader classes of words.

Findings

Conciseness of certain words is equivalent in residually finite and virtually pro-p groups.

Multilinear commutator words raised to a power are concise in residually finite groups.

Certain classes satisfying specific laws form varieties.

Abstract

A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word $w$ in residually finite groups is equivalent to that in the class of virtually pro-$p$ groups. This is used to show that if $q,n$ are positive integers and $w$ is a multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are concise in residually finite groups. Earlier this was known only in the case where $q$ is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law $w^q\equiv1$, or $[\delta_k^q,{}_n\, y]\equiv1$, are varieties.