On hierarchical hyperbolicity of cubical groups

Mark F. Hagen, Timothy Susse

Research output: Contribution to journalArticle (Academic Journal)peer-review

7 Citations (Scopus)
37 Downloads (Pure)


Let χ 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 χ has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that G is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that χ cannot contain a convex staircase. The necessary 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.
Original languageEnglish
JournalIsrael Journal of Mathematics
Publication statusPublished - 17 Jan 2020


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