Geometric Finiteness and Uniqueness for Kleinian Groups with Circle Packing Limit Sets

J. Reine Angew. Math 436 (1993), pp. 209– 219. Stony Brook IMS Preprint #1991/23 December 1991 Let G ⊂ PSL(2,C) be a geometrically finite Kleinian group, with region of discontinuity Ω(G). By Ahlfors’ finiteness theorem, the quotient, Ω(G)/G, is a finite union of Riemann surfaces of finite type. Thus on it, there are only finitely many mutually disjoint free homotopy classes of simple closed curves. It is shown in [9] and [13] that if γ1, . . . , γk are a set of mutually disjoint and simple closed curves in Ω(G)/G, represented by primitive non-elliptic and non-conjugate elements g1, . . . , gk of G, then there is a group G , and an isomorphism φ:G → G taking parabolic elements of G to parabolic elements of G, for which the images φ(g1), . . . , φ(gk) in G ′ are parabolic. Heuristically, this means that the curves γi have been “shrunk” or “pinched” to punctures. In this paper, we assume that G is a finitely generated torsion free non-elementary Kleinian group with Ω(G) 6= ∅. We show that the maximal number of elements of G that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to G may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called maximally parabolic. We show such groups exist. We state our main theorems concisely here. Full statements appear in sections 4 and 5.