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Arithmeticity, superrigidity, and totally geodesic submanifolds. (English) Zbl 07353243
Summary: 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.

MSC:
22E40 Discrete subgroups of Lie groups
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