Elder Siblings and the Taming of Hyperbolic 3-Manifolds

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.