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

TL;DR

This paper constructs a CAT(0) cube complex on which Grigorchuk groups act properly and faithfully under certain conditions, providing a new model for their classifying space of proper actions and contrasting with fixed-point theorems.

Contribution

It introduces a novel construction of a CAT(0) cube complex for Grigorchuk groups, enabling proper and faithful actions, and extends to groups with certain set actions.

Findings

Uncountable family of Grigorchuk groups act without bounded orbit.

Proper and faithful action for sequences without repetition.

Provides a model for the classifying space of proper actions.

Abstract

On this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups $G_\omega$ act without bounded orbit. Moreover, if the sequence $\omega$ 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 $G_\omega$ with $\omega$ 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.