Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

TL;DR

This paper proves that many non-arithmetic hyperbolic manifolds, including hybrids, have only finitely many maximal totally geodesic submanifolds, using a combination of algebraic, dynamical, and geometric methods.

Contribution

It establishes finiteness results for maximal geodesic submanifolds in a broad class of non-arithmetic hyperbolic manifolds, extending previous understanding.

Findings

Finiteness of totally geodesic hypersurfaces in hybrid hyperbolic manifolds.

Finiteness of maximal geodesic submanifolds of dimension ≥ 2.

Application of structure theory, dynamics, and negative curvature geometry.

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.