A NOTE ON ENGEL ELEMENTS IN THE FIRST GRIGORCHUK GROUP

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. 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.