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.

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