According to the classical uniformization theorem, every smooth Riemannian surface Z homeomorphic to the 2-sphere is conformally diffeomorphic to S 2 (the unit sphere

. 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 [respectively CP hyperbolic], if there is a locally finite disk packing P in the plane [respectively, 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 shall 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...

. We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively subdividing the given tiling. We wish to determine when this sequence of tilings is conformal in the sense of Cannon's combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be repaced by a single axiom which is implied by either of them, and that it su#ces to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed. This paper is concerned with recursive subdivisions of planar complexes. As an introductory example, we present a finite subdivision rule in Figure 1. There are two kinds of edges and three kinds of tiles. A thin edge is subdivided into five su...

. Let P be a locally finite disk pattern on the complex plane C whose combinatorics is described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function # : E # [0, #/2] on the set of edges. Let P # be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function. We show that P and P # di#er only by a euclidean similarity. In particular, when the dihedral angle function # is identically zero, this yields the rigidity theorems of B. Rodin and D. Sullivan, and of O. Schramm, whose arguments rely essentially on the pairwise disjointness of the interiors of the disks. The approach here is analytical, and uses the maximum principle, the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern. A similar rigidity property holds for locally finite disk patterns in the hyperbolic plane, wh...

3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65

Abstract. Suppose R is an orientation-preserving finite subdivision rule with an edge pairing. Then the subdivision map σR is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If R has mesh approaching 0 and SR is a 2-sphere, it is proved in Theorem 3.1 that if R is conformal then σR is realizable by a rational map. Furthermore, a general construction is given which, starting with a one tile rotationally invariant finite subdivision rule, produces a finite subdivision rule Q with an edge pairing such that σQ is realizable by a rational map. In this paper we illustrate a technique for constructing critically finite rational maps. The starting point for the construction is an orientation-preserving finite subdivision rule R with an edge pairing. For such a finite subdivision rule the CW complex SR is a surface, and the map σR: SR → SR is a branched covering. If SR is orientable, then unless σR is a homeomorphism or a covering of the torus, SR is a 2-sphere and σR is critically finite. In the latter case, SR has an orbifold structure OR and σR induces a map τR: T (OR) →T(OR) on the Teichmüller space of the orbifold. By work of Thurston, σR can be realized by a rational map exactly if τR has a fixed point. Alternatively, we prove in Theorem 3.1 that σR can be realized by a rational map if R has mesh approaching 0 and is conformal. We next give a general construction which, starting with a one tile rotationally invariant finite subdivision rule R, produces an orientation-preserving finite subdivision rule Q with an edge pairing such that Q is conformal if and only if R is conformal; we then show in Theorem 3.2 that σQ is realizable by a rational map. We next give several examples of orientation-preserving finite subdivision rules with edge pairings. For each example R for which the associated map σR can be realized by a rational map, we explicitly construct a rational map realizing it. We conclude with some questions. A motivation for this work is the Bowers-Stephenson paper [1]. In that paper they construct an expansion complex for the pentagonal subdivision rule (see Figure 4) and numerically approximate the expansion constant. In Example 4.4 we consider an associated finite subdivision rule Q with an edge pairing and construct a rational map fQ(z) =

. We study a wide class of transient planar graphs, through a geometric model given by a square tiling of a cylinder. For many graphs, the geometric boundary of the tiling is a circle, and is easy to describe in general. The simple random walk on the graph converges (with probability 1 ) to a point in the geometric boundary. We obtain information on the harmonic measure, and estimates on the rate of convergence. This allows us to extend results we previously proved for triangulations of a disc. 1991 Mathematics Subject Classification. 60J15, 60J45, 52C20. Key words and phrases. Planar Graphs, Random walks, Harmonic measure, Dirichlet problem. The first author would like to thank R. Pemantle for the friendly support via NSF grant # DMS-9353149. Typeset by A M S-T E X 1 2 BENJAMINI & SCHRAMM 1. Introduction In this paper, we continue our study of harmonic functions on planar graphs initiated in [1]. Here, we will focus on the simple random walk and the Dirichlet problem. In [...

this paper is to give a new, complete, and constructive proof of the AndreevThurston Theorem, while simultaneously extending it to incorporate packings with "polynomial -like" branch structures (Main Theorem, x7). Moreover, the proof rests on solving a general "boundary value" problem for branched circle packings for triangulations of the disc (Proposition 1, x5); this allows extension of results (such as the Discrete Schwarz and Distortion Lemmas) which have played a central role in recent studies of tangency packings. We also illustrate a branched circle packing which is not polynomial-like and pose the existence and uniqueness question for general branched packings. The authors thank Jeff van Eeuwen for valuable conversations on an earlier version of the material in this paper. The computer package CirclePack used to compute the packings illustrated in the paper is available from the second author. 1. Past and Future

. We introduce two basic notions, `transboundary extremal length' and `fat sets', and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two directions. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded by quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heart-shapes and points. Introduction The concept of extremal length (see, e.g. [21]) is a very useful tool in the study of conformal and quasiconformal mappings. Basically, one can say that its usefulness stems from two basic pro...

Abstract not found