Coarse obstructions to cocompact cubulation

We provide geometric methods to both lower- and upper-bound the large-scale dimension of CAT (0) cube complexes quasiisometric to a given group G. When these bounds overlap, this provides obstructions to G being cocompactly cubulated. More strongly, it prevents G from being a coarse median space. As applications, we show that many free-by-cyclic groups cannot be cocompactly cubulated, and that any tubular group with a coarse median is virtually compact special.