Retraction: Hierarchical hyperbolicity of all cubical groups

Abstract

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. Then X has a factor system in the sense of Behrstock, Hagen and Sisto (Preprint, 2014). By results in Behrstock, Hagen and Sisto (Preprint, 2014), it follows that G is a hierarchically hyperbolic group; this answers questions raised in Behrstock, Sisto (Preprint, 2014, 2015). Our results also resolve a conjecture of Behrstock-Hagen on boundaries of cube complexes. In particular, X cannot contain a convex staircase.