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.

中文翻译：

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算术，超刚性和完全测地子流形| 数学年鉴

假设$ \ Gamma $是$ \ mathrm {SO} _0（n，1）$中的晶格。我们证明，如果关联的局部对称空间包含无限大的至少为$ 2 $的最大全测地子空间，则$ \ Gamma $是算术运算。这回答了里德关于双曲$ n $流形的问题，而麦克默伦则独立地回答了双曲$ 3 $流形的问题。我们通过证明此类格的某些表示形式的超刚性定理来证明这些结果。我们的超刚性定理的证明使用的是齐次动力学的均等分布结果，我们的主要结果也接受了该语言的提法。