Cubulating Infinity in Hierarchically Hyperbolic Spaces

TL;DR

This paper demonstrates that hierarchical hulls in hierarchically hyperbolic spaces can be modeled by CAT(0) cube complexes, providing new geometric insights and boundary correspondences, with applications to mapping class groups.

Contribution

It introduces a new cubical model for hierarchical hulls in HHS, extending previous results and establishing boundary isomorphisms for non-cubical HHSes like mapping class groups.

Findings

Hierarchical hulls are quasi-median quasi-isometric to bounded dimension CAT(0) cube complexes.

The boundary of hierarchical hulls corresponds to the simplicial boundaries of cube complexes.

Distance in the top-level hyperbolic space is coarsely represented by hyperplane chains in the cubical model.

Abstract

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.