On finiteness of some verbal subgroups in profinite groups

TL;DR

This paper investigates conditions under which certain verbal subgroups in profinite groups are finite, focusing on groups with limited $w$-values and finitely generated verbal subgroups, contributing to the understanding of group word conciseness.

Contribution

It establishes finiteness of verbal subgroups in profinite groups for specific words when the set of $w$-values is small and the subgroup is finitely generated, advancing the theory of group word conciseness.

Findings

Verbal subgroups are finite under certain conditions for specific words.

Groups with small sets of $w$-values and finitely generated verbal subgroups have finite $w(G)$.

Results relate to the broader concept of group word conciseness.

Abstract

Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the cardinality of $G_w$ is less than $2^{\aleph_0}$ and $w(G)$ is generated by finitely many $w$-values. For several families of words $w$ we show that under these assumptions $w(G)$ must be finite. Our results are related to the concept of conciseness of group words.