A note on Engel elements in the first Grigorchuk group

TL;DR

This paper investigates Engel elements in the first Grigorchuk group, proving that the sets of bounded and unbounded left and right Engel elements all consist only of the identity, thus clarifying their structure.

Contribution

It establishes that the sets of bounded and unbounded left and right Engel elements in the first Grigorchuk group are all trivial, extending previous results about Engel elements.

Findings

Sets of bounded and unbounded left and right Engel elements are equal to the identity subgroup.

The set of all Engel elements in the first Grigorchuk group is not a subgroup.

The results clarify the structure of Engel elements in this specific group.

Abstract

Let $\Gamma$ be the first Grigorchuk group. According to a result of Bartholdi, the only left Engel elements of $\Gamma$ are the involutions. This implies that the set of left Engel elements of $\Gamma$ is not a subgroup. Of particular interest is to wonder whether this happens also for the sets of bounded left Engel elements, right Engel elements, and bounded right Engel elements of $\Gamma$. Motivated by this, we prove that these three subsets of $\Gamma$ coincide with the identity subgroup.