Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

Abstract

In this paper, we study exhaustions, referred to as ρ–restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as ρ–restricted limit sets, such that there is a Poincare series and hence an exponent of convergence δρ, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of ρ–restricted Patterson measure, and show that these measures are non–atomic, δρ–harmonic, δρ–subconformal on special sets and δρ–conformal on very special sets. Furthermore, we obtain the results that each ρ–restriction of our Kleinian group is of δρ–divergence type and that the Hausdorff dimension of the ρ–restricted limit set is equal to δρ.