Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

David Fisher; Jean-François Lafont; Nicholas Miller; Matthew Stover

doi:10.4171/jems/1077

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arXiv · 1802.04619

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

David Fisher·Jean-François Lafont·Nicholas Miller·Matthew Stover

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Abstract

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least $2$ that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient $n$-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

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BibTeX RIS

David Fisher, Jean-François Lafont, Nicholas Miller, Matthew Stover. 2018-02-13. Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids. https://doi.org/10.4171/jems%2F1077

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