A note on Engel elements in the first Grigorchuk group
نویسندگان
1 Department of Civil Engineering, Persian Gulf University, Shahid Mahini St., Bushehr, P.O. Box 75169, Iran
2 Department of Civil Engineering, Persian Gulf University, Shahid Mahini St., Bushehr, P.O. Box 75169, Iran
doi
10.22108/ijgt.2018.109911.1470چکیده
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.
