We show that large classes of non-arithmetic hyperbolic -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 that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient -manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.
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David Fisher, Jean-François Lafont, Nicholas Miller, Matthew Stover, Finiteness of maximal geodesic submanifolds in hyperbolic hybrids. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3591–3623DOI 10.4171/JEMS/1077