Finiteness of maximal geodesic submanifolds in hyperbolic hybrids

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro many their generalizations, have only finitely finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for submanifolds dimension at least $2$ are maximal, i.e., not properly contained in a proper submanifold ambient $n$-manifold. The proof is mix structure theory arithmetic groups, dynamics, geometry negative curvature.