Hierarchically hyperbolic spaces and quasi-cube complexes

Mark Hagen (Bristol)

Abstract: Masur and Minsky's work on the geometry of mapping class groups, combined with more recent results about the geometry of CAT(0) cube complexes, motivated the introduction of the class of hierarchically hyperbolic spaces. A metric space X is hierarchically hyperbolic if there is a set of (uniformly) Gromov-hyperbolic spaces U, each equipped with a projection from X to U, satisfying various axioms that amount to saying that the geometry of X is recoverable, up to quasi-isometry, from this projection data. Working in this context often allows one to promote facts about hyperbolic spaces to conclusions about highly non-hyperbolic spaces: mapping class groups, Teichmuller space, "most" 3-manifold groups, etc. In particular, many CAT(0) cube complexes -- including those associated to right-angled Artin and Coxeter groups -- are hierarchically hyperbolic. The relationship between CAT(0) cube complexes and hierarchically hyperbolic spaces is intriguing. Just as, in a hyperbolic space a collection of n points has quasiconvex hull quasi-isometric to a finite tree (i.e., 1-dimensional CAT(0) cube complex), in a hierarchically hyperbolic space there is a natural notion of the quasiconvex hull of a set of n points and it is quasi-isometric to a CAT(0) cube complex, by a result of Behrstock-Hagen-Sisto. The quasi-isometry constants depend on n in general. However, when each hyperbolic space U is quasi-isometric to a tree, it turns out that this dependence disappears. From this one deduces that, if X is a metric space that is hierarchically hyperbolic with respect to quasi-trees, then X is quasi-isometric to a CAT(0) cube complex. I will discuss this theorem and some of its group-theoretic consequences. This is joint work with Harry Petyt.