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Finite Subdivision Rules
 Conform. Geom. Dyn
, 2001
"... . 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 sub ..."
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Cited by 21 (8 self)
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. 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...
Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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Cited by 11 (0 self)
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
A “regular” pentagonal tiling of the plane
 Conform. Geom. Dyn
, 1997
"... Abstract. The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn il ..."
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Cited by 9 (2 self)
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Abstract. The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling selfsimilarity. In parallel with the conformal tilings, the paper develops discrete tilings based on circle packings. These faithfully reflect the key features of the theory and provide the tiling illustrations of the paper. Moreover, it is shown that under refinement the discrete tiles converge to their true conformal shapes, shapes for which no other approximation techniques are known. The paper concludes with some further examples which may contribute to the study of tilings and shinglings being carried forward by Cannon, Floyd, and Parry.
Constructing rational maps from subdivision rules, Conform
 Geom. Dyn
, 2001
"... Abstract. Suppose R is an orientationpreserving 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 2sphere. If R has mesh approaching 0 and SR is a 2sphere, it is proved in Theor ..."
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Cited by 9 (4 self)
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Abstract. Suppose R is an orientationpreserving 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 2sphere. If R has mesh approaching 0 and SR is a 2sphere, 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 orientationpreserving 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 2sphere 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 orientationpreserving 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 orientationpreserving 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 BowersStephenson 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) =
Bounded outdegree and extremal length on discrete Riemann surfaces, Conform
 Department of Mathematics, University of Georgia
, 2010
"... Abstract. Let T be a triangulation of a Riemann surface. We show that the 1skeleton of T may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from T by attaching new edges and vertices and subdivi ..."
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Cited by 3 (0 self)
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Abstract. Let T be a triangulation of a Riemann surface. We show that the 1skeleton of T may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from T by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the classical surface. We will prove a bound on the distortion of the discrete extremal lengths of path families on T under the refinement process. Our bound will depend only on the refinement and not on T. In particular, the result does not require bounded degree. 1.
Combinatorial modulus and type of graphs
 TopologyanditsApplications
"... Let a A be the 1skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A ′ , formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a ..."
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Cited by 2 (0 self)
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Let a A be the 1skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A ′ , formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a disk triangulation graph may be refined without changing its combinatorial type, provided the refinement is not too wild. We also explore the type problem in terms of disk growth, proving a parabolicity condition based on a superlinear growth rate, which we also prove optimal. We prove our results with no degree restrictions in both the EEL and VEL settings and examine type problems for more general complexes and dual graphs. 1