Compact groups in which commutators have finite right Engel sinks

TL;DR

This paper investigates compact groups where certain commutators have finite right Engel sinks, proving the existence of a locally nilpotent open subgroup under these conditions and a finite index subgroup with bounded properties.

Contribution

It establishes new structural results about compact groups with commutators having finite or bounded right Engel sinks, linking these conditions to the group's local nilpotency.

Findings

Existence of a locally nilpotent open subgroup under finite right Engel sink conditions.

Presence of a finite index subgroup with bounded right Engel sink properties.

Structural characterization of compact groups based on commutator Engel sink conditions.

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.