  We consider two manifestations of non-positive curvature: acylindrical
actions on hyperbolic spaces and quasigeodesic stability. We study these
properties for the class of hierarchically hyperbolic groups, which is a
general framework for studying many important families of groups, including
mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold
groups, and many others. A group that admits an acylindrical action on a
hyperbolic space may admit many such actions on different hyperbolic spaces, so
it is natural to search for a "best" one. The set of all cobounded acylindrical
actions on hyperbolic spaces admits a natural poset structure; in this paper we
prove that all hierarchically hyperbolic groups admit a unique action which is
the largest in this poset. The action we construct is also universal in the
sense that every element which acts loxodromically in some acylindrical action
on a hyperbolic space does so in this one. Special cases of this result are
themselves new and interesting. For instance, this is the first proof that
right-angled Coxeter groups admit universal acylindrical actions. The notion of
quasigeodesic stability of subgroups provides a natural analogue of
quasiconvexity outside the context of hyperbolic groups. We provide a complete
classification of stable subgroups of hierarchically hyperbolic groups,
generalizing and extending results that are known for mapping class groups and
right-angled Artin groups. We also provide a characterization of contracting
quasigeodesics; interestingly, in this generality the proof is much simpler
than in the special cases where it was already known. In the appendix, it is
verified that any space satisfying the a priori weaker property of being an
"almost hierarchically hyperbolic space" is actually a hierarchically
hyperbolic space. The results of the appendix are used to streamline the proofs
in the main text.
