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Hierarchically hyperbolic spaces and quasi-cube complexes

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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.