Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds

Abstract: Given an infinite-volume convex cocompact hyperbolic manifold with Fuchsian ends and infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two intersecting its convex core, we show that it (infinitely) covers an arithmetic hyperbolic manifold. This answers the general version of a question of Oh. As a byproduct of our techniques, we also give a Ratner-type classification of $\mathrm{SO}_0(m,1)$-invariant Radon measures whose support intersects the frame bundle of the interior of the convex core.