We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Let $\Gamma $ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers question Reid for hyperbolic $n$-manifolds and, independently, McMullen $3$-manifolds. these results by proving superrigidity theorem certain representations s...