We examine a graph Γ encoding the intersection of hyperplane carriers
in a CAT(0) cube complex X. The main result is that Γ is
quasi-isometric to a tree. This implies that a group G acting properly and
cocompactly on X is weakly hyperbolic relative to the hyperplane
stabilizers. Using disc diagram techniques and Wright's recent result on the
aymptotic dimension of CAT(0) cube complexes, we give a generalization of a
theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs
of asymptotically finite-dimensional groups. More precisely, we prove
asymptotic finite-dimensionality for finitely-generated groups acting on
finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded
asymptotic dimension. Finally, we apply contact graph techniques to prove a
cubical version of the flat plane theorem stated in terms of complete bipartite
subgraphs of Γ.Comment: Corrections in Sections 2 and 4. Simplification in Section