On superintegral Kleinian sphere packings, bugs, and arithmetic groups

We develop the notion of a Kleinian Sphere Packing, a generalization of crystallographic (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 (2019), no. 2, 436-441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do superintegral such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles p/m for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

Automated summary

We introduce the concept of Kleinian Sphere Packing, a generalization of crystallographic (Apollonian-like) sphere packings. Unlike crystallographic packings, Kleinian packings exist in all dimensions, and so do superintegral ones. We extend the Arithmeticity Theorem to Kleinian packings, showing that the superintegral ones come from Q-arithmetic lattices of simplest type. Similarly, this applies to more general objects called Kleinian Bugs, where spheres may meet at finitely many dihedral angles p/m without being disjoint. We address two questions from Kontorovich and Nakamura (2019): (i) the general falseness of the Arithmeticity Theorem over number fields, and (ii) the fact that integral packings only arise from non-uniform lattices.