Properness and finiteness of totally geodesic submanifolds in the convex core

TL;DR

This paper proves that in certain infinite-volume geometrically finite manifolds, maximal totally geodesic submanifolds within the convex core are finite in number and volume, revealing a rigidity phenomenon contrasting finite-volume cases.

Contribution

It establishes finiteness and properness of maximal totally geodesic submanifolds in the convex core of infinite-volume manifolds, and links their abundance to arithmeticity.

Findings

Maximal totally geodesic submanifolds in the convex core are finite in number.

Such submanifolds are properly immersed and have finite volume.

Infinite abundance of these submanifolds implies the manifold is arithmetic.

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.