New paper accepted by Proceedings of the Royal Society of Edinburgh, section A – Mathematics

New paper by Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia) “Compact groups in which all elements have countable right Engel sinks” has been accepted for publication in Proceedings of the Royal Society of Edinburgh, section A – Mathematics. The work was supported by Mathematical Center in Akademgorodok, FAPDF and CNPq-Brazil, and stems from the collaboration with University of Brasilia.

Abstract: A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[\dots [[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)={ 1}$ .) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable right Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent.