[Paper Review] Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces

G. A. Margulis, Amir Mohammadi|arXiv (Cornell University)|Feb 19, 2019

Geometric and Algebraic Topology15 references7 citations

TL;DR

This paper proves that a closed hyperbolic 3-manifold containing infinitely many totally geodesic surfaces must be arithmetic, establishing a strong link between geometric complexity and arithmetic structure. Using superrigidity and ergodic theory, the authors show that such a manifold's fundamental group is commensurable with an arithmetic lattice in PGL₂(ℂ), resolving a long-standing question in 3-manifold topology and arithmetic geometry.

ABSTRACT

We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.

Motivation & Objective

• To establish a converse to Reid's result that arithmetic hyperbolic 3-manifolds contain either zero or infinitely many totally geodesic surfaces.
• To resolve a question posed by Reid and McMullen on whether infinite geodesic surface count implies arithmeticity.
• To prove that the presence of infinitely many totally geodesic surfaces forces the fundamental group to be arithmetic.
• To extend superrigidity techniques to the rank-one setting (SO(3,1)) in the context of geometric complexity.

Proposed method

• Apply superrigidity theorems to representations of the fundamental group into PGL₂(ℂ) and its completions at places of a number field.
• Use ergodic theory and measure-theoretic arguments to analyze the behavior of geodesic currents and their images under equivariant maps.
• Construct a measurable equivariant map Ψ from the space of geodesics to the flag variety, analyzing its image using the action of the lattice Γ.
• Leverage the weak approximation theorem and commensurability properties of arithmetic groups to deduce that the fundamental group must be arithmetic.
• Use the fact that the image of Ψ is almost surely contained in a single orbit under PGL₂(ℝ) to deduce that the lattice is arithmetic.
• Apply a refined version of the measure-concentration argument via Lemma 6.5 to show that the image of almost every geodesic under Ψ lies in a single cross or line, implying algebraic rigidity.

Experimental results

Research questions

• RQ1 Does the presence of infinitely many totally geodesic surfaces in a closed hyperbolic 3-manifold imply that the manifold is arithmetic?
• RQ2 Can superrigidity techniques be adapted to prove arithmeticity in the rank-one case (SO(3,1)) when the lattice is not known to be arithmetic a priori?
• RQ3 Is there a measurable or geometric rigidity condition that forces a lattice in PGL₂(ℂ) to be arithmetic?
• RQ4 Can the structure of the space of geodesics and their images under equivariant maps detect arithmeticity?
• RQ5 What is the role of the commensurator in characterizing arithmetic lattices in the context of infinite geodesic surface counts?

Key findings

• A closed hyperbolic 3-manifold containing infinitely many totally geodesic surfaces is necessarily arithmetic.
• The fundamental group Γ of such a manifold is commensurable with an arithmetic lattice in PGL₂(ℂ).
• The image of the geodesic current map Ψ is almost surely contained in a single orbit under the action of PGL₂(ℝ), implying algebraic rigidity.
• The proof relies on a measurable superrigidity argument, showing that the equivariant map Ψ must factor through a single algebraic subgroup.
• The result confirms that the only way for a closed hyperbolic 3-manifold to contain infinitely many totally geodesic surfaces is if its fundamental group is arithmetic.
• The index of the fundamental group in its commensurator is infinite, consistent with Theorem C on arithmeticity via commensurability.