## Institute for Mathematical Sciences

## Preprint ims95-10

** R. Canary, Y. Minsky, and E. Taylor**
* Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds*

Abstract: Let $M$ be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let $\Lambda(M)$ be the
supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic
3-manifolds homeomorphic to the interior of $N$. 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 $\Lambda(M)=D(M)(2-D(M))$
if $M$ is not handlebody or a thickened torus. We characterize
exactly when $\Lambda(M)=1$ and $D(M)=1$ in terms of the
characteristic submanifold of the incompressible core of $M$.

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