Dehn filling hyperbolic 3-manifolds

Colin C. Adams

Vol. 165 (1994), No. 2, 217–238

DOI: 10.2140/pjm.1994.165.217

Abstract

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

(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 ancestor 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.

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M25, 57M50

Milestones

Received: 8 April 1992

Revised: 4 February 1993

Accepted: 12 March 1993

Published: 1 October 1994