Local-global principles in circle packings

Abstract

We generalize work of Bourgain-Kontorovich [6] and Zhang [32], 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 A ≤ 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 A possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL 2 (O K ) containing a Zariski dense subgroup of PSL 2 (Z).

Spinors, lattices, and classification of integral Apollonian disk packings

“…The tangency spinors are spinors for the 3-dimensional space with Minkowski quadratic form of signature (1,2). It has been a somewhat metaphorical use of the term native to mathematical physics of (1+3)-dimensional spacetime for objects in 2-dimensional geometry.…”

“…And now some basic notions organized in a form that should be easy to consult. For more on the subject see [1,2,3,5,12,15,16,17,20].…”

A geometric study of circle packings and ideal class groups

“…The Bourgain-Kontorovich result has recently been generalized by Fuchs, Stange, and Zhang. Remark that [8] restricts attention to M, N ∈ PSL 2 (K), not PGL 2 (K) as stated above. Communication with the second-and third-named authors confirmed that their proof still applies.…”

“…Consequences of Theorem 1.1 include a restriction on possible intersection angles within certain subarrangements called bugs from [12] (Corollary 4.3), a criterion for determining superintegrality (Definition 4.4, Proposition 4.6), and a geometric connection between S D and the local-global principle for curvatures in an integral circle packing (Theorem 4.11 and Corollary 4.13). Regarding the last result, the relative position of a circle packing in S D can give a sufficient condition for applying a theorem of Fuchs, Stange, and Zhang [8], which states that curvatures in certain packings have asymptotic density 1 among integers that pass a set of local obstructions. A circle packing that satisfies our sufficient condition is displayed in Figure 2.…”

Quaternion orders and sphere packings

“…In all these cases, there is a corresponding local-global conjecture, just as for the Apollonian circle packing. This conjecture is open for all such circle packings, but there are known density one results like the theorem of Bourgain and Kontorovich-see, for example, [FSZ17]. For n = 2, there is the generalization of the Apollonian circle packing to spheres due to Soddy [Sod36]; additionally, Dias [Dia14] and Nakamura [Nak14] independently constructed the orthoplicial sphere packing.…”