Strong conciseness of Engel words in profinite groups

TL;DR

This paper proves that for any fixed number of iterations, the n-Engel word is strongly concise in finitely generated profinite groups, advancing understanding of verbal subgroups in this class.

Contribution

It establishes the strong conciseness of n-Engel words in finitely generated profinite groups, extending previous results on multilinear commutator words.

Findings

n-Engel words are strongly concise in finitely generated profinite groups

The result applies to all fixed n-Engel words regardless of n

Supports the conjecture that all words are strongly concise in profinite groups

Abstract

A group word $w$ is said to be strongly concise in a class $\mathscr C$ of profinite groups if, for any group $G$ in $\mathscr C$, either $w$ takes at least continuum values in $G$ or the verbal subgroup $w(G)$ is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups. In the present paper we prove that for any $n$ the $n$-Engel word $[...[x,y],y],\dots y]$ (where $y$ is repeated $n$ times) is strongly concise in the class of finitely generated profinite groups.