Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds

Uri Bader; David Fisher; Nick Miller; Matthew Stover

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arXiv · 1903.08467

Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds

Uri Bader·David Fisher·Nick Miller·Matthew Stover

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Abstract

Let $Γ$ 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 $Γ$ 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.

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BibTeX RIS

Uri Bader, David Fisher, Nick Miller, Matthew Stover. 2020-04-26. Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds. https://arxiv.org/abs/1903.08467

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