Kleinian circle packings
Let \(\Gamma\) be a finitely generated Kleinian group so that the limit set of \(\Gamma\) is the complement of a circle packing. There are some attractive graphics in this paper illustrating this sort of limit set. One can define, in a fairly straightforward fashion, a packing exponent of this specific circle packing. This is a purely geometric quantity which does not depend on the action of the Kleinian group. The purpose of this paper is to show that this exponent is equal to the Hausdorff dimension of the limit set. It was shown by Larman that for an arbitrary circle packing the Hausdorff dimension of the residual set is at least as large as the exponent. The main part of the proof is the converse inequality. This is proved by a delicate sharpening of Ahlfors' argument, which yields an upper bound for the Hausdorff dimension of the limit set in the case where \(\Gamma\) is geometrically finite.
scientific article; zbMATH DE number 794435