Compact groups in which commutators have finite right Engel sinks

Evgeny Khukhro; Pavel Shumyatsky

paper

Compact groups in which commutators have finite right Engel sinks

arXiv:2410.05840

Abstract

A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[... [[g, x], x], \dots, x]$ . We prove that if $G$ is a compact group in which, for some $k$ , every commutator $[... [g_{1}, g_{2}], \dots, g_{k}]$ has a finite right Engel sink, then $G$ has a locally nilpotent open subgroup. If in addition, for some positive integer $m$ , every commutator $[... [g_{1}, g_{2}], \dots, g_{k}]$ has a right Engel sink of cardinality at most $m$ , then $G$ has a locally nilpotent subgroup of finite index bounded in terms of $m$ only.