Engel elements in weakly branch groups

TL;DR

This paper investigates Engel elements in weakly branch groups, establishing their triviality or specific structure, and applies these findings to well-known groups like multi-GGS groups.

Contribution

It proves the triviality of bounded left Engel elements in weakly branch groups and characterizes right Engel elements under mild conditions, extending understanding of their structure.

Findings

Bounded left Engel elements are always trivial in weakly branch groups.

Non-trivial left Engel elements in branch groups are p-elements and imply the group is virtually a p-group.

The set of right Engel elements is trivial under mild conditions.

Abstract

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all $p$-elements and that the group is virtually a $p$-group (and so periodic) for some prime $p$. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these results to well-known families of weakly branch groups, like the multi-GGS groups.