Geometry and Arithmetic of Crystallographic Sphere Packings
Metric Geometry
2017-12-04 v1 Geometric Topology
Number Theory
Abstract
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.
Cite
@article{arxiv.1712.00147,
title = {Geometry and Arithmetic of Crystallographic Sphere Packings},
author = {Alex Kontorovich and Kei Nakamura},
journal= {arXiv preprint arXiv:1712.00147},
year = {2017}
}
Comments
Research announcement. 14 pages, 5 figures