The Apollonian structure of Bianchi groups

Abstract

We study the orbit of R under the Möbius action of the Bianchi group PSL 2 (O K ) on C, where O K is the ring of integers of an imaginary quadratic field K. The orbit 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 S K generalizing Apollonian circle packings, and show that 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.

Local-global principles in circle packings

“…In Section 9, in order the demonstrate the variety of examples to which our work applies, we verify that the hypotheses of Theorem 1.6 hold for the K-Apollonian packings of the second-named author [28], and also for an explicit example of a cuboctahedral packing (which also arises in the work of Kontorovich and Nakamura; Figure 1).…”

“…In this paper, we identify the key necessary conditions for these methods to work, which, when satisfied, guarantee an asymptotic local-to-global principle for an integral circle packing or, viewed differently, an orbit of a thin subgroup of . As a consequence, we immediately have that an asymptotic local-to-global principle holds for the - Apollonian packings described by the second-named author [Sta18a] and for superintegral polyhedral packings described by Kontorovich and Nakamura [KN17]. We provide a concrete example of such a packing and give more details on the packings of Stange and of Kontorovich and Nakamura in § 9.…”

Quaternion orders and sphere packings

“…It remains to show that the sphere collectionsŜ O,j enumerated in Theorem 2.1 are superpackings of corresponding integral crystallographic packings. Our approach is to introduce the notion of (O, j)-Apollonian packings; this terminology is taken from Stange [Sta15], where she uses a similar construction for the circle packings introduced in [Sta17]. We begin with a definition.…”

Orders of quaternion algebras with involution

“…The interest in this subject comes from the problem of constructing certain arithmetic sphere packings in R 3 . In R 2 , one can construct [Sta14] [Sta15] circle packings by considering the action of a Bianchi group SL(2, O K ) on the real line, where O K is the ring of integers of some imaginary quadratic field K. This same method can be used to produce interesting sphere packings in R 3 , but requires replacing the Bianchi group SL(2, O) with an appropriate analog SL ‡ (2, O), where O is a maximal ‡-order (see [She]).…”