On hierarchical hyperbolicity of cubical groups

By Mark F. Hagen, Tim Susse. Israel Journal of Mathematics, 2016

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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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7349f9f6a4b4336150cf3eed6f367ff3aa51f6b3

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Citations in the corpus

Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, background): “2The first question was partially answered positively in [HS16] after this paper was first posted.”