Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
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阿波罗圆堆积：几何和群论 III。
DOI：
10.1007/s00454-005-1197-8
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发表时间：
2000
影响因子：
0.8
通讯作者：
C. Yan
中科院分区：
文献类型：
--
作者：
R. Graham;J. Lagarias;C. Mallows;A. R. Wilks;C. Yan
本文在第一部分和第二部分中给出了阿波罗圆填充的$n元维类比。这两篇文章考虑了用笛卡尔构形描述的圆填充，它们是由四个相互接触的圆组成的集合。他们研究了具有关于圆的曲率和中心的完整性性质的填充。这里我们考虑$n$维笛卡尔构型的集合，它由$n+2$相互接触的球体组成。我们在一个坐标系统中参数化为$n$维笛卡尔构形的空间$M_D^n$中工作，即那些$(n+2)\×(n+2)$实矩阵$W$具有$W^T Q_{D，n}W=Q_{W，n}$其中$Q_{D，n}=x_1^2+\cdots+x_(n+2)^2-({1}/{n})(x_1+\cdots+
X_(n+2)^2$是$n$维笛卡尔二次型，$q_{W，n}=-8x_1x_2+2x_3^2+\cdots+2x_(n+2)^2$，$\bq_{D，n}$和$\bq_{W，n}$是它们对应的对称矩阵。关于参数空间$M_D^n$
对于增广的曲率中心矩阵，群${\it Aut}(Q_{D，n})$作用在左边，${\it Aut}(Q_{W，n})$作用在右边。这两个群都与$(n+2)$维Lorentz群$O(n+1，1)$同构，并给出两个
不同的“几何”动作。${\it Aut}(q_{W，n})$的正确操作
(本质上)对应于作用于基础
欧几里得空间$\rr^n$，而${\it Aut}(q_{D，n})$的左作用是
仅在参数空间$M_D^n$上定义。我们引入了阿波罗尼亚群、对偶阿波罗尼亚群和超阿波罗尼亚群的$n维类似。这些是${\it Aut}(q_{D，n})$中有限生成的群，其中
如下积分性质：对偶阿波罗群由所有维度的积分矩阵组成，而另外两个由有理矩阵组成，其分母具有从有限集$S$中提取的素因数取决于维度。我们证明了Apollonian群和对偶Apollonian群是有限表示的，并且是Coxeter群。我们定义阿波罗星系综是阿波罗群下的任何轨道，对另外两个群也有类似的概念。我们确定在哪些维度中存在有理阿波罗簇系综(所有曲率都是有理的)和强有理阿波罗球系综(所有增广的曲率中心坐标都是有理的)。
AbstractThis paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots +
x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$
of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two
different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$
(essentially) corresponds to Mobius transformations acting on the underlying
Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is
defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with
the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).