Counting visible circles on the sphere and Kleinian groups

TL;DR

This paper analyzes the asymptotic count of circles in a packing on the sphere invariant under Kleinian groups, providing insights into geometric structures and their distribution.

Contribution

It introduces a method to compute the asymptotic number of circles with bounded curvature in invariant packings under Kleinian groups.

Findings

Derived asymptotic formulas for circle counts

Applied results to specific Kleinian group packings

Enhanced understanding of geometric circle distributions

Abstract

For a circle packing P on the sphere invariant under a geometrically finite Kleinian group, we compute the asymptotic of the number of circles in P of spherical curvature at most $T$ which are contained in any given region.