Arithmeticity, superrigidity, and totally geodesic submanifolds


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.


Uri Bader

Weizmann Institute of Science, Rehovot, Israel

David Fisher

Indiana University, Bloomington, IN, USA

Nicholas Miller

University of California, Berkeley, Berkeley, CA, USA

Matthew Stover

Temple University, Philadelphia, PA, USA