TY - JOUR
T1 - On hierarchical hyperbolicity of cubical groups
AU - Hagen, Mark F.
AU - Susse, Timothy
PY - 2020/1/17
Y1 - 2020/1/17
N2 - 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.
AB - 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.
U2 - 10.1007/s11856-020-1967-2
DO - 10.1007/s11856-020-1967-2
M3 - Article (Academic Journal)
SN - 0021-2172
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -