Engel elements in some fractal groups

TL;DR

This paper investigates the structure of Engel elements in certain fractal groups acting on p-adic trees, establishing conditions under which these elements are trivial and providing examples illustrating the necessity of these conditions.

Contribution

It proves that in fractal groups with infinite derived subgroup over the stabilizer, the set of left Engel elements is trivial, extending understanding of Engel elements in complex group actions.

Findings

Engel elements are trivial in specified fractal groups.

The conditions on the derived subgroup and fractality are necessary.

Examples show dropping these conditions changes the outcome.

Abstract

Let $p$ be a prime and let $G$ be a subgroup of a Sylow pro-$p$ subgroup of the group of automorphisms of the $p$-adic tree. We prove that if $G$ is fractal and $|G':\mathrm{st}_G(1)'|=\infty$, then the set $L(G)$ of left Engel elements of $G$ is trivial. This result applies to fractal nonabelian groups with torsion-free abelianization, for example the Basilica group, the Brunner-Sidki-Vieira group, and also to the GGS-group with constant defining vector. We further provide two examples showing that neither of the requirements $|G':\mathrm{st}_G(1)'|=\infty$ and being fractal can be dropped.