The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds

TL;DR

This paper investigates the behavior of the smallest positive eigenvalue of the Laplace-Beltrami operator on fibered hyperbolic 3-manifolds, establishing bounds related to volume and demonstrating typical case behavior.

Contribution

It provides new bounds for the eigenvalue depending on volume and fiber genus, and analyzes typical and random cases, connecting spectral properties with geometric and dynamical features.

Findings

Eigenvalue bounds depend on volume and genus

Typical and random manifolds have eigenvalues inversely proportional to volume squared

Recurrent properties of axes of random pseudo-Anosov elements are established

Abstract

We study the smallest positive eigenvalue $\lambda_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $\lambda_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $\lambda_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements.