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

Abstract 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 $$G_\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> </mml:math> act without bounded orbit. Moreover, if the sequence $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> 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 $$G_\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> </mml:math> with $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> 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.