The PL-methods for hyperbolic 3-manifolds to prove tameness

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Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold M with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion Mᵢ of M and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold Mᵢ negatively curved and with Margulis constant independent of i. By taking the convex hull in the cover of Mᵢ corresponding to the core, we show that there exists an exiting sequence of surfaces Σᵢ. Some of the ideas follow those of Agol. We drill out the covers of Mᵢ by a core again to make it negatively curved. Then the boundary of the convex hull of Σᵢ is shown to meet the core. By the compactness argument of Souto, we show that infinitely many of Σᵢ are homotopic in M - ᵒ. Our method should generalize to a more wider class of piecewise hyperbolic manifolds.