Elder Siblings and the Taming of Hyperbolic 3-Manifolds
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Freedman, Michael H. and Curtis T. McMullen. 1998. Elder siblings and the taming of hyperbolic 3-manifolds. Annales Academiae Scientiarum Fennicae 23: 415–428. Revised 2003.Abstract
A 3-manifold is tame if it is homeomorphic to the interior of a compact manifold with boundary. Marden’s conjecture asserts that any hyperbolic 3-manifold \(\mathbb{M} = \mathbb{H}^3/\Gamma\) with \(\pi_1(M)\) finitely-generated is tame. This paper presents a criterion for tameness. We show that wildness of \(\mathbb{M}\) is detected by large-scale knotting of orbits of \(\Gamma\). The elder sibling property prevents knotting and implies tameness by a Morse theory argument. We also show the elder sibling property holds for all convex cocompact groups and a strict form of it characterizes such groups.Other Sources
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