On hierarchical hyperbolicity of cubical groups

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.

Minor changes in response to referee report. Streamlined the proof of Lemma 5.2, and added an examples of non-rotational actions

References in corpus (13)

• Rank rigidity for CAT(0) cube complexes
• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Groups acting on CAT(0) cube complexes
• Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
• Cubulating hyperbolic free-by-cyclic groups: the general case
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Weak hyperbolicity of cube complexes and quasi-arboreal groups
• The simplicial boundary of a CAT(0) cube complex
• Top dimensional quasiflats in $CAT(0)$ cube complexes
• Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups
• Acylindrical hyperbolicity from actions on CAT(0) cube complexes: a few criteria

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