Dehn filling hyperbolic 3-manifolds

Abstract

Define a complete family of parent (ancestor) manifolds to be a set of compact 3-manifolds such that every closed orient able 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:

Hyperbolic 3-manifolds with two generators

“…Each pair of parabolic generators corresponds to a pair of curves in the cusp boundary as above and a connecting geodesic between cusps that lifts to a connector of length less than ln(4). Lifting one of the cusps to a horoball centered at oo of height 1 above the boundary plane in the upper-half-space model of if 3 , each of these connecting geodesies lifts to a connector that connects the horoball centered at oo with a horoball of diameter at least 1/4. Since each distinct horoball of diameter at least 1/4 requires a certain amount of area in the fundamental domain for the cusp subgroup fixing oo and since the area of the fundamental domain of the cusp is finite, there can be at most finitely many distinct connecting geodesies.…”

“…that begins at the basepoint, travels along a path UJ passing though the complement of the interior of the cusps from the basepoint back to a point on the cusp boundary, follows a closed loop in the cusp boundary, call it 5, and then returns to the basepoint along a; -1 . Lifting the path a; to a path a/ in iJ 3 , we obtain a path that begins and ends in the horospheres covering the cusp boundary. If u/ begins and ends on the same horosphere, a!…”

“…In this case, we will homotope a small segment of 8 slightly into the complement of the cusps at each of their points of intersection, so that they are now disjoint. Now we take the preimage of a' and f3 n = 7^7" 1 in iif 3 and look at the graph K that results. First note that K is connected.…”

“…Perhaps the most interesting result is a proof that if M is a finite volume hyperbolic 3-manifold that is generate by two parabolics, then it must be the complement of a two-bridge link in S 3 . This proof does depend on Thurston's Orbifold Theorem, which, at the time of this writing, has not appeared in print.…”

“…Such a cusp lifts to a set of horoballs in i? 3 , some of which are tangent but all of which have disjoint interiors. We denote the volume of the resulting cusp by v c (M).…”