Bounds on Eigenvalues of the Laplacian for Certain Classes of Closed Hyperbolic 3-manifolds.

Abstract

Fixing constants $epsilon$, $c$, we consider the class of all closed $epsilon$-thick hyperbolic 3-manifolds $M$ such that $pi_1(M)$ can be generated by $c$ elements. For all $k$ we prove that $lambda_k(M) sim vol^{-2}(M)$ up to a multiplicative constant depending only on $epsilon$, $c$, and $k$, where $lambda_k(M)$ is the $k$th eigenvalue of the Lapalce-Beltrami operator.