Apollonian 圆填充:几何与群论 II. 超 Apollonian 群与整填充
摘要
Apollonian 圆填充通过反复以进一步的切圆填充四个两两相切圆之间的空隙而产生。此类填充可由其所包含的 Descartes 构型来描述。已观察到存在无穷多种整 Apollonian 填充,其中所有圆均具有整数曲率,且该整结构关联到 Apollonian 群的整性。此处我们考虑一个更大的离散群的作用,即超 Apollonian 群,它同样具有整结构,其轨道描述一种我们称为超填充的几何对象的 Descartes 四元组。超填充中的圆互不相交,但可任意深度地嵌套。某些 Apollonian 填充与超填充是强整的,意指所有圆的曲率均为整数且所有圆的曲率乘圆心均为整数。我们证明(在相差一个尺度下)恰有 8 种不同的(几何)强整超填充,且每一个都包含每个整 Apollonian 圆填充的一个副本(同样在相差一个尺度下)。我们证明超 Apollonian 群在 Descartes 构型参数空间的所有自同构群中具有有限体积,而该自同构群同构于 Lorentz 群 。
引用
@article{arxiv.math/0010302,
title = {Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings},
author = {R. L. Graham and J. C. Lagarias and C. L. Mallows and A. R. Wilks and C. H. Yan},
journal= {arXiv preprint arXiv:math/0010302},
year = {2007}
}
备注
37 Pages, 11 figures. The second in a series on Apollonian circle packings beginning with math.MG/0010298. Extensively revised in June, 2004. More integral properties are discussed. More revision in July, 2004: interchange sections 7 and 8, revised sections 1 and 2 to match, and added matrix formulations for super-Apollonian group and its Lorentz version. Slight revision in March 10, 2005