The smallest positive eigenvalue of fibered hyperbolic 3-manifolds

We study the smallest positive eigenvalue λ₁(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 ≥ 2. We show the existence of a constant C > 0 only depending on g so that λ1(M) ∈[C−1/vol(M)², C log vol(M)/vol(M)^{22g−2/(22g−2−1)}] and that this estimate is essentially sharp. We show that if M is typical or random, then we have λ1(M) ∈ [C⁻¹/vol(M)², C/vol(M)²]. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements.