Refined horoball counting and conformal measure for Kleinian group actions

TL;DR

This paper refines horoball counting techniques for Kleinian groups, linking horoball distribution to conformal measures and fractal dimensions, with implications for understanding geometric and measure-theoretic properties of limit sets.

Contribution

It extends horoball counting results to localized settings within limit sets, revealing dependence on position and scale, and applies these to conformal measure analysis.

Findings

Horoball counts depend on location and scale, especially for larger sizes.

Established estimates for fractal dimensions of conformal measures.

Analyzed measure continuity at the Poincaré exponent.

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 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 $r \lesssim R^2$, 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 $s=\delta$.