Cubulating Infinity in Hierarchically Hyperbolic Spaces

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 about modeling hulls of interior points and our previous work with Zalloum on modeling finite sets of rays via limits of these finite models. We further prove that the quasi-median quasi-isometry between the hull of a finite set of rays or boundary points and its cubical model extends to an isomorphism between their respective hierarchical and simplicial boundaries. In this sense, we prove that the hierarchical boundary of any proper HHS is locally modeled by the simplicial boundaries of CAT(0) cube complexes. This is a purely geometric statement, allowing one to important various topologies from the cubical setting. Our proof of the cubical model theorem is new, even for the interior points case. In particular, we provide a concrete description of the cubical model as a cubical subcomplex of a product of simplicial trees into which the hierarchical data is directly encoded. Moreover, the above boundary isomorphism is new for all non-cubical HHSes, including mapping class groups and Teichm\"uller spaces of finite-type surfaces. As an application of our techniques, we show that in most HHSes, including all hierarchically hyperbolic groups, the distance between any pair of points in the top-level hyperbolic space is coarsely the length of a maximal 0-separated chain of hyperplanes separating them in an appropriate cubical model. For mapping class groups, this says that these cubical models cubically encode distance in the curve graph of the surface.