Hyperbolic 3-manifolds with two generators
The main result in this paper is that if a non-free Kleinian group is generated by two parabolic elements, then there is a bound for the ``length'' of each of these elements. The length here is measured using the fact that parabolic elements in the fundamental group can be represented by loops in canonical cusp boundaries and by taking shortest loops with respect to the hyperbolic metric. There are famous bounds of this sort for pairs of hyperbolic elements (Jorgensen's inequalities). The author produces explicit universal upper bounds on the ``distance'' between two such parabolics, by measuring distances between two horoballs for the corresponding canonical cusps. The author gives several applications, including the following: A two-generator finite volume Kleinian group has at most finitely many distinct pairs of parabolic generators, up to conjugacy of the pair. Let \(M\) be an orientable finite volume hyperbolic 3-manifold. Then the fundamental group of \(M\) is generated by two parabolics if and only if \(M\) is a 2-bridge link in \(S^3\) that is not a 2-braid. The two generators correspond to meridians. The paper includes also some new constraints on pairs of generators of non-elementary Kleinian groups which are both of hyperbolic type.
scientific article; zbMATH DE number 929682