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We prove that the hierarchical hull of any finite set of interior points,hierarchy rays, and boundary points in a hierarchically hyperbolic space (HHS)is quasi-median quasi-isometric to a CAT(0) cube complex of bounded dimension.Our construction extends and refines a theorem of Behrstock-Hagen-Sisto aboutmodeling hulls of interior points and our previous work with Zalloum onmodeling finite sets of rays via limits of these finite models. We furtherprove that the quasi-median quasi-isometry between the hull of a finite set ofrays or boundary points and its cubical model extends to an isomorphism betweentheir respective hierarchical and simplicial boundaries. In this sense, weprove that the hierarchical boundary of any proper HHS is locally modeled bythe simplicial boundaries of CAT(0) cube complexes. This is a purely geometricstatement, allowing one to important various topologies from the cubicalsetting. Our proof of the cubical model theorem is new, even for the interiorpoints case. In particular, we provide a concrete description of the cubicalmodel as a cubical subcomplex of a product of simplicial trees into which thehierarchical data is directly encoded. Moreover, the above boundary isomorphismis new for all non-cubical HHSes, including mapping class groups andTeichm\"uller spaces of finite-type surfaces. As an application of ourtechniques, we show that in most HHSes, including all hierarchically hyperbolicgroups, the distance between any pair of points in the top-level hyperbolicspace is coarsely the length of a maximal 0-separated chain of hyperplanesseparating them in an appropriate cubical model. For mapping class groups, thissays that these cubical models cubically encode distance in the curve graph ofthe surface.

Cubulating Infinity in Hierarchically Hyperbolic Spaces