A note on Engel elements in the first Grigorchuk group

Document Type: Research Paper

Authors

1 Department of Mathematics (University of Salerno), Italy - Department of Mathematics and Statistic (University of the Basque Country), Spain

2 Dipartimento di Matematica, Università di Salerno - Italy

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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