Compact groups with countable Engel sinks

Abstract

An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.