On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups release_ih57yictive5foes3gqr3onp3i

by Michael Kapovich, Alex Kontorovich

Released as a article .

2021  

Abstract

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 not 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.
In text/plain format

Archived Files and Locations

application/pdf  8.8 MB
file_n3iqbhgy25c4jkayc6li2zspym
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2021-04-28
Version   v1
Language   en ?
arXiv  2104.13838v1
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: f9f71e75-9584-4d69-924f-9e8182bb9ead
API URL: JSON