Dehn filling hyperbolic 3-manifolds (Q1343687)

Define a complete family of parent (ancestor) manifolds to be a set of compact 3-manifolds such that every closed orientable 3-manifold can be obtained by one (or more) Dehn fillings of the manifolds in the family. In 1983, R. Myers proved that the set of 1-cusped hyperbolic 3-manifolds is a complete family of parent manifolds. We prove this result in a new way and then go on to prove: Theorem 1.1 (a) Let \(V_ 0\) be any positive real number. Then the set of 1-cusped hyperbolic 3-manifolds of volume greater than \(V_ 0\) is a complete family of parent manifolds. (b) Let \(V_ 1\) be any positive real number. Then the set of 1-cusped hyperbolic 3-manifolds of cusp volume greater than \(V_ 1\) is a complete family of parent manifolds. (c) The set of 2-cusped hyperbolic 3-manifolds containing embedded totally geodesic surfaces is a complete family of ancestor (actually grandparent) manifolds. (d) For any positive integer \(N\), the set of hyperbolic 3-manifolds, each of which shares its volume with \(N\) or more other hyperbolic 3-manifolds, is a complete family of ancector manifolds. As a corollary to Theorem 1(b), we prove that there exists a complete family of parent manifolds such that at most one Dehn filling on each manifold in the family yields a manifold of finite fundamental group.