JournalsjemsVol. 23, No. 11pp. 3591–3623

Finiteness of maximal geodesic submanifolds in hyperbolic hybrids

  • David Fisher

    Indiana University, Bloomington, USA

  • Jean-François Lafont

    Ohio State University, Columbus, USA

  • Nicholas Miller

    University of California at Berkeley, USA

  • Matthew Stover

    Temple University, Philadelphia, USA
Finiteness of maximal geodesic submanifolds in hyperbolic hybrids cover
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Abstract

We show that large classes of non-arithmetic hyperbolic nn-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 22 that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient nn-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

Cite this article

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. 3591–3623

DOI 10.4171/JEMS/1077