  We construct "pushing maps" on the cube complexes that model right-angled
Artin groups (RAAGs) in order to study filling problems in certain subsets of
these cube complexes. We use radial pushing to obtain upper bounds on higher
divergence functions, finding that the k-dimensional divergence of a RAAG is
bounded by r^{2k+2}. These divergence functions, previously defined for
Hadamard manifolds to measure isoperimetric properties "at infinity," are
defined here as a family of quasi-isometry invariants of groups; thus, these
results give new information about the QI classification of RAAGs. By pushing
along the height gradient, we also show that the k-th order Dehn function of a
Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs
called "orthoplex groups" which show that each of these upper bounds is sharp.
