Compact groups with countable Engel sinks

Abstract

An Engel sink of an element $g$ of a group $G$ is a set $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to $\mathcal{E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose $\mathcal{E}(g)=\{1\}$.) 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.

Few minor corrections added