On hierarchical hyperbolicity of cubical groups

TL;DR

This paper establishes conditions under which a group acting on a CAT(0) cube complex is hierarchically hyperbolic, advancing understanding of the structure and boundaries of such complexes.

Contribution

It introduces three new conditions on group actions that guarantee hierarchical hyperbolicity and proves one condition is necessary, partially answering open questions.

Findings

Conditions ensure the existence of a factor system in the cube complex

Under these conditions, the group is shown to be hierarchically hyperbolic

Results affirm a conjecture on cube complex boundaries and exclude convex staircases

Abstract

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to show that $G$ is a hierarchically hyperbolic group; this partially answers questions raised by those authors. Under any of these conditions, our results also affirm a conjecture of BehrstockHagen on boundaries of cube complexes, which implies that X cannot contain a convex staircase. The conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.