Let Γ be the first Grigorchuk group. According to a result of Bar-thol-di, the only left Engel elements of Γ are the involutions. This implies that the set of left Engel elements of Γ 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 Γ. Motivated by this, we prove that these three subsets of Γ coincide with the identity subgroup.
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