Hyperbolic 3-manifolds with two generators

Abstract

We show that if there are two parabolic elements that generate a non-elementary Kleinian group that is not free, then there is a universal upper bound of two on the "length" of each of those parabolics, length being measured in a canonical choice of cusp boundaries. Moreover, there is a universal upper bound of ln(4) on the "distance" between those parabolics, where the distance between them is the distance between a pair of horoballs corresponding to the canonical cusps. We prove a variety of results with these, the most interesting of which is: An orientable finite volume hyperbolic 3-manifold that has fundamental group generated by two parabolic isometries must be a 2-bridge knot or link complement.

The structure of 3-manifolds with two-generated fundamental group

“…We recall that a n-times punctured homotopy 3-sphere is a three-dimensional homotopy sphere minus the interior of n disjoint embedded 3-balls. and also [1]). …”

“…This is clear since in this case g 2 corresponds to a power of the fibre of Q and does therefore not correspond to the fibre of the piece M B 1 …”

“…It further follows that h n corresponds to a power of f M since M is not homeomorphic to Q. This however yields a contradiction since then h cannot be conjugated to an element of C. This implies that g, h ⊂ A, B 1 We define G x = g , G y = g 2 , h n where n is chosen minimal such that h n fixes [y, z] and G z = h . We can assume that n 2, otherwise h fixes y and we are in situation (i) of 1.…”

“…A diffeomorphism ∈ Diff + (F ) is determined, up to isotopy, by its induced action on 1 That is the necessary and sufficient condition for to be induced by a diffeomorphism ∈ Diff + (F ). Moreover, it is easy to check that 1 (F ) is generated by b, (b), 2 (b), 3 (b) .…”

“…Now the description of Heegaard genus two, compact, orientable, irreducible 3-manifolds with incompressible toral boundary follows, like in the closed case, from a careful analysis of the possible intersections of the JSJ family T and the genus two compression-bodies V 1 For example if M has one boundary component, case (1) of Theorem 2 corresponds to case 3-(b) in the proof of Theorem 2.1 in [41]. In the same way, case (2) (respectively cases (3) and (4)) of Theorem 2 corresponds to case 2-(b) (respectively 3-(a) and (1)) of the proof of Theorem 2.1 in [41].…”