The Smallest Positive Eigenvalue of Fibered Hyperbolic 3-Manifolds

Abstract

We study the smallest positive eigenvalue of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold which fibers over the circle. Using so-called Lipschitz model developed by Minsky and Brock-Canary-Minsky, we find a family of graphs which are uniformly quasi-isometric to such 3-manifolds. This implies that the smallest positive eigenvalue on such a graph and a manifold are uniformly comparable. Using this idea, we compute the eigenvalue on such graphs, and obtain essentially sharp upper bound. This is a joint-work with I. Gekhtman and U. Hamenstaedt.