This talk explores how the fundamental group of an n-dimensional closed hyperbolic manifold can be approximated through cubulations. Learn about joint research by Eduardo Reyes and Nic Brody that successfully approximates the asymptotic geometry of actions on hyperbolic space Hn using CAT(0) cube complexes, solving a conjecture by Futer and Wise. The presentation focuses on cases where n is at most 3 or when the manifold is arithmetic of simplest type, employing a codimension-1 generalization of Bonahon's space of geodesic currents as the primary analytical tool. This IAS/Princeton University Groups and Dynamics Seminar talk from Yale University mathematician Eduardo Reyes offers insights into geometric group theory and hyperbolic geometry.