On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups [article]

Michael Kapovich, Alex Kontorovich
2021 arXiv   pre-print
We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. 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
more » ... ot be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions from [KN19]: (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.
arXiv:2104.13838v1 fatcat:ih57yictive5foes3gqr3onp3i