The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space H^n. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on CAT(0) cube complexes. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on H^n by actions on these complexes, solving a conjecture of Futer and Wise. The main tool is a codimension-1 generalization of the space of geodesic currents introduced by Bonahon.