The Apollonian structure of Bianchi groups

We study the orbit of R under the Bianchi group PSL2(OK), where K is an imaginary quadratic field. The orbit SK , called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. We define certain natural subgroups whose orbits generalise Apollonian circle packings, and show that SK , considered with orientations, is a disjoint union of all of these K-Apollonian packings. These packings define a new class of thin groups called K-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture of Apollonian circle packings.