Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

A Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. Part I shoewed there is a natural group action on Descartes quadruples by a group Iso(Q2) which is a real Lie group conjugate to the isochronous Lorentz group O(3, 1,R), and it acts linearly on Descartes quadruples given in “curvature-center coordinates.” It contains a discrete subgroup, the Apollonian group, which consists of 4× 4 integer matrices, which leaves packings invariant and acts transitively on their (unordered) Descartes configurations. The Apollonian group is a hyperbolic Coxeter group. In this paper define a larger group inside GL(4,Z) generated by the Apollonian group A and its transpose AT , which we call the super-Apollonian group. The super-Apollonian group is shown to be a hyperbolic Coxeter group. We study an object that we call the “integer Apollonian super-gasket”, which consists of all (ordered) primitive integer Descartes configurations, i.e. four tangent circles with integer curvatures, each of whose curvature × center is a Gaussian integer, such that the integer curvatures of circles in these quadruples have no nontrivial common divisor. The group of conformal transformations that preserve the integer Apollonian super-gasket is transitive on its members. Geometrically the superApollonian group acts nearly transitively on the integral Apollonian super-gasket, having 8 orbits. We use this fact to show that the super-Apollonian group is a subgroup of index 2 in the group of integral automorphs of the Descartes quadratic form Q2(x1, x2, x3, x4) = 2(x1 +x 2 2+x 2 3+x 2 4)− (x1 +x2+x3+x4), an indefinite integral quadratic form of determinant −16.