# Arithmeticity, superrigidity, and totally geodesic submanifolds

### Abstract

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 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.

## Authors

Weizmann Institute of Science, Rehovot, Israel

David Fisher

Indiana University, Bloomington, IN, USA

Nicholas Miller

University of California, Berkeley, Berkeley, CA, USA

Matthew Stover