Properness and finiteness of totally geodesic submanifolds in the convex core

TL;DR

The paper proves a strong rigidity phenomenon for totally geodesic submanifolds within the convex core of geometrically finite rank-one locally symmetric spaces: in the infinite-volume case, maximal totally geodesic submanifolds of dimension at least two inside the core are properly immersed with finite volume and there are only finitely many of them; infinitely many finite-volume instances force arithmeticity. The authors develop a Ratner-type dynamical framework in the renormalized frame bundle $ ext{RF} M$, combining non-divergence in infinite-volume rank-one settings, Dani–Margulis avoidance, and Mozes–Shah equidistribution to classify orbit closures as homogeneous and to deduce equidistribution when many submanifolds occur. This dynamical rigidity yields immediate geometric consequences, including finiteness of maximal finite-volume submanifolds and arithmeticity in the presence of infinitely many such submanifolds; it also links to boundary sphere packings and their counting. The work highlights a sharp contrast with the finite-volume case, where geodesic planes can be dense or exhibit fractal closures, and shows that geometric finiteness suffices to recover a robust rigidity inside the convex core.

Abstract

We study totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds. Although the infinite-volume setting can exhibit highly complicated behavior, including geodesic planes with fractal closures, we show that a strong rigidity persists inside the convex core. This rigidity has striking consequences in the infinite volume setting: every maximal totally geodesic submanifold of dimension at least two contained in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur. These results stand in sharp contrast to the behavior in the finite-volume setting. Moreover, combining this finiteness result with the work of Bader-Fisher-Miller-Stover and of Gromov-Schoen, we deduce that any geometrically finite rank-one manifold with infinitely many maximal totally geodesic submanifolds of dimension at least two and of finite volume must be arithmetic.

Figures (1)

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Theorems & Definitions (72)

• Theorem 1.1: Properness
• Corollary 1.2
• Theorem 1.3: Finiteness I
• Theorem 1.4: Finiteness II
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10: Theorem \ref{['Theoremmax0']}, Theorem \ref{['eq']}