The Apollonian structure of Bianchi groups

TL;DR

This paper explores the geometric and arithmetic structure of Bianchi groups through a novel circle packing called Schmidt arrangements, revealing connections to Apollonian packings and introducing new thin groups with conjectured properties.

Contribution

It provides a geometric characterization of K-Apollonian packings within Bianchi groups and introduces K-Apollonian groups, extending the understanding of circle packings in arithmetic settings.

Findings

Schmidt arrangements form complex circle packings related to imaginary quadratic fields.

K-Apollonian packings generalize classical Apollonian packings to imaginary quadratic fields.

A conjecture on the curvatures of K-Apollonian packings extends the local-global conjecture.

Abstract

We study the orbit of $\widehat{\mathbb{R}}$ under the M\"obius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit $\mathcal{S}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of $\mathcal{S}_K$ generalizing Apollonian circle packings, and show that $\mathcal{S}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.