Words and pronilpotent subgroups in profinite groups

TL;DR

This paper proves that in certain profinite groups, if all pronilpotent subgroups generated by specific commutator words are periodic, then the subgroup generated by these words is locally finite.

Contribution

It establishes a new link between periodic pronilpotent subgroups and local finiteness of verbal subgroups in profinite groups involving multilinear commutator words.

Findings

Pronilpotent subgroups generated by $w$-values are periodic.

The verbal subgroup $w(G)$ is locally finite under given conditions.

Provides insight into structure of profinite groups with multilinear commutator words.

Abstract

Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.