Compact groups all elements of which are almost right Engel

TL;DR

This paper investigates compact groups where every element is almost right Engel, proving such groups are essentially finite-by-locally nilpotent, with bounds depending on the uniformity of the associated finite sets.

Contribution

It extends the understanding of Engel-like conditions in compact groups by characterizing their structure when elements are almost right Engel, generalizing previous results on Engel elements.

Findings

Groups are finite-by-locally nilpotent when all elements are almost right Engel.

Bounded sets lead to a finite normal subgroup with order bounds.

Uses Wilson–Zelmanov theorem on Engel profinite groups.

Abstract

We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[g,x],x],\dots ,x]\in {\mathscr R}(g)$ if $x$ is repeated at least $n(x,g)$ times. Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$. We prove that if all elements of a compact (Hausdorff) group $G$ are almost right Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a uniform bound $|{\mathscr R}(g)|\leq m$ for the orders of the corresponding sets, then the subgroup $N$ can be chosen of order bounded in terms of $m$. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.