Drilling cores of hyperbolic 3-manifolds to prove tameness

Abstract

We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ $\delta$-hyperbolic and with Margulis constants independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding the core, we show that there exists an exiting sequence of surfaces $\Sigma_i$. We drill out the covers of $M_i$ by a core $C$ again to make it $\delta$-hyperbolic. Then the boundary of the convex hull of $\Sigma_i$ is shown to meet the core. By the compactness argument of Souto, we show that infinitely many of $\Sigma_i$ are homotopic in $M - C^o$.