Proper actions of Grigorchuk groups on a CAT(0) cube complex

In this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups \(G_\omega \) $G_{\omega }$ act without bounded orbit. Moreover, if the sequence \(\omega \) $\omega$ does not contain repetition, we prove that the action is proper and faithful. As a consequence of this result, this cube complex is a model for the classifying space of proper actions for all the groups \(G_\omega \) $G_{\omega }$ with \(\omega \) $\omega$ without repetition. This construction works in a general way for any group acting on a set and which admits a commensurated subset. These examples of non-elliptic actions of infinite finitely generated torsion groups on a non-positively curved cube complex contrast to several established fixed-point theorems concerning actions of torsion groups.