We use the projection complex machinery of Bestvina-Bromberg-Fujiwara to study hierarchically hyperbolic groups.In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex.We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.
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Hagen et al. (2022) studied this question.
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