# Spectral
theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

### Richard D. Canary

### Yair N. Minsky

### Edward C. Taylor

## Abstract

Let M be a compact 3-manifold whose interior admits a complete hyperbolic
structure. We let L(M) be the supremum of the bottom
l_{0}(N) of the spectrum
of the Laplacian where
N varies over all hyperbolic 3-manifolds homeomorphic to the
interior of M. Similarly, we let D(M) be the infimum of the Hausdorff
dimensions of limit sets of Kleinian groups whose quotients are
homeomorphic to the interior of M. We observe that L(M) = D(M)(2-D(M))
if M is not handlebody or a thickened torus.
We characterize exactly when L(M) = 1 and D(M) = 1
in terms of the characteristic submanifold of the incompressible core
of M.

File translated fromT_{E}Xby T_{T}H,version 2.56.

On 23 Nov 1999, 17:42.