The Apollonian structure of Bianchi groups

Abstract

We study the orbit of $\hat{\mathbb{R}}$ under the Möbius action of the Bianchi group ${\mathrm{PSL}}_2({\mathcal{O}}_K)$ on $\hat{\mathbb{C}}$, 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.

44 pages, 22 figures, this version makes corrections to previous version, adds references