Almost Engel compact groups

TL;DR

This paper introduces the concept of almost Engel groups, a generalization of Engel groups, and proves that compact almost Engel groups are essentially finite-by-locally nilpotent, with bounds depending on the sets involved.

Contribution

It establishes that compact almost Engel groups have a finite normal subgroup with a locally nilpotent quotient, extending the classical Engel group theory.

Findings

Almost Engel groups in compact setting are finite-by-locally nilpotent.

Bound on the size of the finite normal subgroup depends on the uniform bound of the sets.

Uses Wilson–Zelmanov theorem to connect Engel properties with local nilpotency.

Abstract

We say that a group $G$ is almost Engel if for every $g\in G$ there is a finite set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\dots ,g]\in {\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\mathscr E}(g)=\{ 1\}$ for all $g\in G$.) We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a uniform bound $|{\mathscr E}(g)|\leq m$ 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.