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 $r^{-\delta }$ many horoballs of size approximately $r$, where $\delta$ is the Poincaré 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 $r\lesssim R^2$, 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 $s>\delta$ and use this to examine continuity properties of $s$-conformal measures at $s=\delta$.