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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:
 2.2e+41 Discrete subgroups of Lie groups
##### Keywords:
hyperbolic manifolds; arithmeticity; superrigidity
Full Text:
##### References:
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