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.

中文翻译：

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关于立方群的层次双曲性

设 χ 是一个适当的 CAT(0) 立方体复形，允许群 G 进行适当的协同作用。我们给出了动作的三个条件，其中任何一个条件都确保 χ 具有 [BHS17] 意义上的因子系统。我们也证明这些条件之一是必要的。这与 [BHS17] 结合表明 G 是一个层次双曲群；这部分回答了 [BHS17, BHS19] 中提出的问题。在任何这些条件下，我们的结果也证实了 Behrstock-Hagen 对立方复合体边界的猜想，这意味着 χ 不能包含凸阶梯。作用的必要条件都严格弱于虚共专性，我们不知道一个共紧立方群至少不满足其中一个条件。