Abstract. We take an in-depth look at Thurston's combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S 2 → S 2 whose local degree at every critical point is 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics.

Abstract. We introduce and study finite subdivision rules. A finite subdivision rule R consists of a finite 2-dimensional CW complex S R , a subdivision R(S R ) of S R , and a continuous cellular map ϕ R : R(S R ) → S R whose restriction to each open cell is a homeomorphism. If R is a finite subdivision rule, X is a 2-dimensional CW complex, and f : X → S R is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide X to obtain an infinite sequence of tilings. 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 replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed. IntroductionThis 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 subedges (three of these are thick and two are thin) and a thick edge is subdivided into three subedges (two of these are thick and one is thin). There are three kinds of tiles: a triangle with thin edges; a quadrilateral with a pair of opposite thin edges and a pair of opposite thick edges; and a pentagon with thick edges. Each tile is subdivided into subtiles of those three kinds, and these subdivisions restrict on the boundary arcs to the subdivisions defined for the edges. Because of this, one can recursively subdivide planar complexes made up of tiles of these three kinds. For example, Figure 2 shows the second and third subdivisions of the quadrilateral tile type. This figure was produced by Kenneth Stephenson's computer program CirclePack [16].Our motivation for the above subdivision rule is illustrated in Figure 3. This figure, which was drawn from an image produced by Jeffrey Weeks's computer program SnapPea [18], shows a right-angled dodecahedron D in the Klein model of hyperbolic 3-space. The geodesic planes through the twelve faces intersect the sphere at infinity in twelve thick (red) circles. The group G generated by the reflections in these twelve geodesic planes is a discrete subgroup of Isom(H 3 ), and D is a fundamental domain for the action of G on H 3 .

2.4. The geometry of an optimal weight function: special results for quadrilaterals and rings. 2.4.1. Further definitions. 2.4.2. The correspondence. 2.4.3. Level curves and the relationship between skinny flows and fat cuts. 2.4.4. The relationship between fat flows and skinny cuts for tilings. 2.4.5. The relationship between the four moduli. 3. The finite Riemann mapping theorem. 3.1. Examples. 3.2. The Kirchhoff inequalities. 4. Algorithms which calculate the finite Riemann mapping. 4.1. The minimal path algorithm. 4.2. Proofs of the claims about the minimal path algorithm. 4.3. The efficiency of the minimal path algorithm. 4.4. Flow diagrams and a hybrid algorithm for the finite Riemann mapping problem. 4.5. An example of the effectiveness of the hybrid algorithm. 5. Optimal weight functions for 2-layer valence 3 tilings of quadrilaterals. 5.1. Basic definitions and properties. 5.2. Unique prime factorization theorem. 5.3. Prime tilings. 5.4. Preparations for factorization algorithms. 5.5. The left factor algorithm. 5.6. The joint ratio algorithm. 6. Approximating combinatorial moduli. 6.1. Skinny approximation. 6.2. The averaging trick. 6.3. Examples. 7. Variable negative curvature versus constant curvature groups.

Abstract.The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions.

Abstract. This paper is concerned with realizing Lattès maps as subdivision maps of finite subdivision rules. The main result is that the Lattès maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite subdivision rules with one tile type. An example is given of a Lattès map which is not the subdivision map of a finite subdivision rule with either i) two tile types and 1-skeleton of the subdivision complex a circle or ii) one tile type.This paper is concerned with realizing rational maps by subdivision maps of finite subdivision rules. If R is an orientation-preserving finite subdivision rule such that the subdivision complex S R is a 2-sphere, then the subdivision map σ R is a postcritically finite branched map. Furthermore, R has bounded valence if and only if σ R has no periodic critical points. In [1] and [4], Bonk-Meyer and CannonFloyd-Parry prove that if f is a postcritically finite rational map without periodic critical points, then every sufficiently large iterate of f is the subdivision map of a finite subdivision rule R with two tile types such that the 1-skeleton of S R is a circle. Since finite subdivision rules are essentially combinatorial objects, this gives good combinatorial models for these iterates. It is especially convenient to realize a postcritically finite map by the subdivision map of a finite subdivision rule with either a single tile type or with two tile types and 1-skeleton of the subdivision complex a circle.While passing to an iterate is not usually a serious obstacle, it would be preferable if one didn't need to do this. Suppose f is a postcritically finite rational map without periodic critical points. We consider the following questions.(1) Is f the subdivision map of a finite subdivision rule? (2) Is f the subdivision map of a finite subdivision rule with two tile types and 1-skeleton of the subdivision complex a circle? (3) Is f the subdivision map of a finite subdivision rule with one tile type? In this paper we consider these questions for Lattès maps. The main result of this paper is to answer question 3 in the affirmative for the maps in all but finitely many analytic conjugacy classes of Lattès maps. In addition we exhibit a Lattès map of degree 2 for which the answer to questions 2 and 3 is negative, although the answer to question 1 is positive. This paper has four sections. The first section develops the setting of Lattès maps. These results can be used to easily enumerate all Lattès maps of very small degree to test (not so easy) the above three questions for them.