On superintegral Kleinian sphere packings, bugs, and arithmetic groups

Abstract

Abstract 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, 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 {{\mathbb{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 π m {\frac{\pi}{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.