Local-global principles in circle packings

TL;DR

This paper extends local-global principles for circle packings to a broad class of Kleinian groups, demonstrating that curvatures in these packings follow an almost local-to-global pattern, with implications for number theory and geometric group theory.

Contribution

It introduces a new framework linking Kleinian groups with integral circle packings and proves a spectral gap property for a wide class of these groups.

Findings

Established an almost local-to-global principle for circle packing curvatures.

Proved spectral gap property for certain Kleinian groups.

Linked geometric group properties with number-theoretic circle packings.

Abstract

We generalize work of Bourgain-Kontorovich and Zhang, proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $\mathcal A\leq\textrm{PSL}_2(K)$ satisfying certain conditions, where $K$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof of this is that $\mathcal A$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $\textrm{PSL}_2(\mathcal{O}_K)$ containing a Zariski dense subgroup of $\textrm{PSL}_2(\mathbb{Z})$.