Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions |
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Authors: | Ronald L Graham Jeffrey C Lagarias Colin L Mallows Allan R Wilks Catherine H Yan |
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Institution: | (1) Department of Computer Science and Engineering, University of California at San Diego, La Jolla, CA 92110, USA;(2) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA;(3) Avaya Labs, Basking Ridge, NJ 07920, USA;(4) AT&T Labs, Florham Park, NJ 07932-0971, USA;(5) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA |
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Abstract: | This 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). |
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