Every hierarchically hyperbolic group that is not hyperbolic contains a Z^2 subgroup, and every virtually Z^n subgroup lies in an A-invariant uniform quasi-flat whose points are joined by hierarchy paths.
A 4-dimensional pseudo-anosov homeomorphism.Preprint 2511.10530,
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Periodic quasiflats in hierarchically hyperbolic spaces
Every hierarchically hyperbolic group that is not hyperbolic contains a Z^2 subgroup, and every virtually Z^n subgroup lies in an A-invariant uniform quasi-flat whose points are joined by hierarchy paths.