Hyperbolic and parabolic packings

Abstract

The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the 1-skeleton of a triangulation of an open disk. G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packing P in the plane (resp. the unit disk) with contacts graph G. Several criteria for deciding whether G is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion, A criterion in terms of the random walk says that if the random walk on G is recurrent, 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 give 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 hyperbolic 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 contained and locally finite in D.

Schwarz’s lemma for the circle packings with the general combinatorics

“…The notion resembles the extremal length of curve families in Riemann surfaces, and was introduced and studied by Cannon and others (see [2,5,12]). (A further extension to the idea of "transboundary extremal length" appeared in [13]) Below we will recall the basics.…”

Bounded Valence Excess and the Parabolicity of Tilings

“…About the same time, Cannon was using a discrete version of conformal modulus to prove his Combinatorial Riemann Mapping Theorem; the paper appeared in 1994 [1]. Cannon et al (see [2]- [5]), as well as He and Schramm [6] later employed the same concept in various ways.…”

“…Following He and Schramm in [6], we define a disk triangulation graph to be the 1-skeleton of a triangulation of an open topological disk. If G is a disk triangulation graph and v is a vertex of G, consider the tiling T dual to G. If t v is the interior of the tile corresponding to v, then let T v = T \{t v }.…”

“…We can estimate the fat flow moduli of these sub-annuli in terms of their vertex valences. By using the Layer Theorem of Cannon et al[2], we sum the estimates for these sub-annuli to prove a conjecture of He and Schramm [6]. The result is a new parabolicity criterion.…”

“…[2], we sum the estimates for these sub-annuli to prove a conjecture of He and Schramm [6]. The result is a new parabolicity criterion.…”