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 > 2. We show the existence of a constant C>0 only depending on g so that lambda 1(M)is an element of[C-1/vol(M)2,Clogvol(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 lambda 1(M)is an element of[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.