Abstract
Let $Gamma$ be the first Grigorchuk group. According to a result of Bar-thol-di, 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. The natural question arises whether this is also the case 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.
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