# Elder Siblings and the Taming of Hyperbolic 3-Manifolds

 Title: Elder Siblings and the Taming of Hyperbolic 3-Manifolds Author: Freedman, Michael H.; McMullen, Curtis T. Note: Order does not necessarily reflect citation order of authors. Citation: 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. Full Text & Related Files: McMullen_ElderSiblingsTaming.pdf (194.6Kb; PDF) 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. Published Version: http://www.acadsci.fi/mathematica/ Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3446000 Downloads of this work: