Approximating hyperbolic lattices by cubulations

The fundamental group of an n-dimensional closed hyperbolic
manifold admits a natural isometric action on the hyperbolic space H^n by
Deck transformations. If n is at most 3 or the manifold has infinitely many
totally geodesic codimension-1 immersed hypersurfaces, then the group also
acts isometrically on CAT(0) cube complexes, which are spaces of
combinatorial nature. I will talk about a joint work with Nic Brody in
which we approximate the asymptotic geometry of the action on H^n by the
actions on these complexes, solving a conjecture of Futer and Wise. In the
3-dimensional case, some ingredients are a recent result of Al Assal about
limits of measures on the 2-plane Grassmannian of the manifold induced by
immersed minimal surfaces, and the work of Seppi about minimal disks in H^3
with prescribed quasicircles as limit sets.