  Let G be a group acting geometrically on a CAT(0) cube complex X. We prove
first that G is hyperbolic relative to the collection P of subgroups if and
only if the simplicial boundary of X is the disjoint union of a nonempty
discrete set, together with a pairwise-disjoint collection of subcomplexes
corresponding, in the appropriate sense, to elements of P. As a special case of
this result is a new proof, in the cubical case, of a Theorem of
Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0)
spaces. Second, we relate the existence of cut-points in asymptotic cones of a
cube complex X to boundedness of the 1-skeleton of the boundary of X. We deduce
characterizations of thickness and strong algebraic thickness of a group G
acting properly and cocompactly on the CAT(0) cube complex X in terms of the
structure of, and nature of the G-action on, the boundary of X. Finally, we
construct, for each n,k, infinitely many quasi-isometry types of group G such
that G is strongly algebraically thick of order n, has polynomial divergence of
order n+1, and acts properly and cocompactly on a k-dimensional CAT(0) cube
complex.

Group Theory;Geometric Topology