Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds

Abstract

Let $\hat{I}“$ 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 $\hat{I}“$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics and our main result also admits a formulation in that language.

Corrected proof of folklore Proposition 3.1 and filled in minor omission in the proof of Lemma 3.4