Refined horoball counting and conformal measure for Kleinian group actions
Abstract
Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the `inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small r>0 there are r-δ many horoballs of size approximately r, where δ is the Poincar\'e exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately r inside a given ball B(z,R). Roughly speaking, if r R2, then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of B(z,R)). However, for larger values of r, the count depends in a subtle way on z. Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several `fractal dimensions' of certain s-conformal measures for s>δ and use this to examine continuity properties of s-conformal measures at s=δ.