Effective equidistribution of circles in the limit sets of Kleinian groups

TL;DR

This paper establishes effective equidistribution and counting results for circles in invariant packings under Kleinian groups, extending previous work and applying to limit sets with specific geometric properties.

Contribution

It provides new effective equidistribution and circle counting theorems for Kleinian group invariant packings, including non-cocompact cases with limit sets containing circles.

Findings

Effective equidistribution for small circles intersecting regular sets.

Effective estimates for circle counts in limit sets as circle size shrinks.

Extension of previous results to broader classes of Kleinian groups.

Abstract

Consider a general circle packing $\mathcal{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex-cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathcal{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah. In view of the recent result of McMullen-Mohammadi-Oh, our effective circle counting theorem applies to the circles contained in the limit set of a convex-cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover consider the circle packing $\mathcal{P}(\mathcal{T})$ of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than $t$, as $t\to 0$, effectivising the work of Oh.