Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature) $\times$ (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system $M_ D$ consisting of those $4 \times 4$ real matrices $W$ with $W^T Q_{D} \bW = Q_{W}$ where $Q_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 - \frac{1}{2}(x_1 +x_2 +x_3 + x_4)^2$ and $Q_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. On the parameter space $M_ D$ the group $\mathop{\it Aut}(Q_D)$ acts on the left, and $\mathop{\it Aut}(Q_W)$ acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group $O(3, 1)$. The right action of $\mathop{\it Aut}(Q_W)$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^2$ while the left action of $\mathop{\it Aut}(Q_D)$ is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of $\mathop{\it Aut}(Q_D)$, which we call the Apollonian group. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in $\mathop{\it Aut}(Q_D)$, the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer $4 \times 4$ matrices. We show these groups are hyperbolic Coxeter groups.