  We study the acylindrical hyperbolicity of groups acting by isometries on
CAT(0) cube complexes, and obtain simple criteria formulated in terms of
stabilisers for the action. Namely, we show that a group acting essentially and
non-elementarily on a finite dimensional irreducible CAT(0) cube complex is
acylindrically hyperbolic if there exist two hyperplanes whose stabilisers
intersect along a finite subgroup. We also give further conditions on the
geometry of the complex so that the result holds if we only require the
existence of a single pair of points whose stabilisers intersect along a finite
subgroup.
