Compact groups with countable Engel sinks

TL;DR

This paper proves that in compact Hausdorff groups, if every element has a countable Engel sink, then the group is essentially a finite extension of a locally nilpotent group, answering a question posed by J. S. Wilson.

Contribution

It establishes a structural result linking countable Engel sinks of elements to the group's overall composition, specifically the existence of a finite normal subgroup with a locally nilpotent quotient.

Findings

Groups with countable Engel sinks have a finite normal subgroup with a locally nilpotent quotient.

The result applies to compact Hausdorff groups.

It resolves a question posed by J. S. Wilson.

Abstract

An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. This settles a question suggested by J. S. Wilson.