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On hierarchical hyperbolicity of cubical groups

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Abstract

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.

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Author notes

  1. Mark F. Hagen

    Present address: Current address: School of Mathematics, University of Bristol, Beacon House, Queens Road, Bristol, BS8 1QU, UK

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Mark F. Hagen

  2. Mathematics Department, Bard College at Simon’s Rock, 84 Alford Rd., Great Barrington, MA, 01230, USA

    Tim Susse

Authors
  1. Mark F. Hagen
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  2. Tim Susse
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Corresponding author

Correspondence to Tim Susse.

Additional information

Hagen was supported by the Engineering and Physical Sciences Research Council grant of Henry Wilton.

Susse was partially supported by National Science Foundation grant DMS-1313559.

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Hagen, M.F., Susse, T. On hierarchical hyperbolicity of cubical groups. Isr. J. Math. 236, 45–89 (2020). https://doi.org/10.1007/s11856-020-1967-2

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  • DOI: https://doi.org/10.1007/s11856-020-1967-2

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