Rigidity of Infinite (circle) Packings

The nerve of a packing is a graph that encodes its combinatorics. The vertices of the nerve correspond to the packed sets, and an edge occurs between two vertices in the nerve precisely when the corresponding sets of the packing intersect. The nerve of a circle packing and other well-behaved packings, on the sphere or in the plane, is a planar graph. It was an observation of Thurston [Thl, Chapter 1; 13, Th2] that Andreev's theorem [AnI, An2] implies that given a finite planar graph, there exists a packing of (geometric) circles on the sphere whose nerve is the given graph. We refer to this fact as the circle packing theorem. The circle packing theorem also has a uniqueness part to it: if the graph is actually (the I-skelaton of) a triangulation, then the circle packing is unique up to Mobius transformations. Using the circle packing theorem, Thurston proposed a method for constructing approximate maps from a given bounded planar simply connected domain to the unit disk, and conjectured that this procedure approximates the corresponding Riemann mapping. Rodin and Sullivan proved this conjecture in [RS]. One of the crucial elements in their argument is the rigidity of the hexagonal packing, the infinite packing of circles where every circle touches six others and all have the same size. (When we say that a circle packing is rigid, we mean that any other circle packing on the sphere with the same nerve is Mobius equivalent to it. In the case of the hexagonal packing, this is the same as saying that any two planar circle packings with the combinatorics of the hexagonal packing are similar.) The techniques of [RS], and the rigidity of the hexagonal packing have since been used by others [CR, He2, Rol, R02, Schl] to obtain quasiconformal and conformal maps, and to study the quality of the convergence of Thurston's approximating scheme. The existence part of the circle packing theorem is not hard to generalize to infinite, locally finite graphs, using a geometric limit. However, the uniqueness does not hold for arbitrary (locally finite) planar triangulations. In this note we