HYPERBOLIC AND PARABOLIC PACKINGS

The contacts graph, or nerve, of a packing, is a combinatorial graph t hat describes the combinatorics of the packing. Let G be the 1-skeleto n of a triangulation of an open disk. G is said to be CP parabolic (re sp. CP hyperbolic) if there is a locally finite disk packing P in the plane (resp. the unit disk) with contacts graph G. Several criteria fo r deciding whether G is CP parabolic or CP hyperbolic are given, inclu ding a necessary and sufficient combinatorial criterion. A criterion i n terms of the random walk says that if the random walk on G is recurr ent, then G is CP parabolic. Conversely, if G has bounded valence and the random walk on G is transient, then G is CP hyperbolic. We also gi ve a new proof that G is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that if G is CP hyperbo lic and D is any simply connected proper subdomain of the plane, then there is a disk packing P with contacts graph G such that P is contain ed and locally finite in D.