The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds

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 $g\ge 2$. We show the existence of a constant $C>0$ only depending on $g$ so that ${\lambda }_1(M)\in [C^{-1}/\mathrm{vol}(M{)}^2,C\log \mathrm{vol}(M)/\mathrm{vol}(M{)}^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have ${\lambda }_1(M)\in [C^{-1}/\mathrm{vol}(M{)}^2,C/\mathrm{vol}(M{)}^2]$. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements.