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.
Citation
Uri Bader. David Fisher. Nicholas Miller. Matthew Stover. "Arithmeticity, superrigidity, and totally geodesic submanifolds." Ann. of Math. (2) 193 (3) 837 - 861, May 2021. https://doi.org/10.4007/annals.2021.193.3.4
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