Conformal Tilings & Type

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2016 Conformal Tilings and Type Dane Mayhook

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COLLEGE OF ARTS AND SCIENCES

CONFORMAL TILINGS & TYPE

DANE MAYHOOK

A Dissertation submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2016

Philip L. Bowers Professor Directing Dissertation

Mark Riley University Representative

Wolfgang Heil Committee Member

Eric Klassen Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii To my family.

iii ACKNOWLEDGMENTS

The completion of this document represents the end of a long and arduous road, but it was not one that I traveled alone, and there are several people that I would like to acknowledge and offer my sincere gratitude to for their assistance in this journey. First, I would like to give my infinite thanks to my Ph.D. adviser Professor Philip L. Bowers, without whose advice, expertise, and guidance over the years this dissertation would not have been possible. I could not have hoped for a better adviser, as his mentorship—both on this piece of work, and on my career as a whole—was above and beyond anything I could have asked for. I would also like to thank Professor Ken Stephenson, whose assistance—particularly in using his software CirclePack, which produced many of the figures in this document—has been greatly appreciated. Our meetings in Knoxville over the years have been invaluable. I would like to thank Professor Wolfgang Heil, Professor Eric Klassen, and Professor Mark Riley for serving on my dissertation committee, and for their helpful suggestions. Perhaps most of all, I would like to say thank you to my mother, Deborah Parotti, father Peter Mayhook, step-father John Parotti, and sister Savanna Mayhook for their love, support, and understanding throughout the years. I could not have done it without you. Finally, I owe my gratitude to the many other family members and friends who have made this endeavor a bit easier, each in their own way. Your support has been truly appreciated.

iv TABLE OF CONTENTS

ListofFigures ...... vi Abstract...... vii

1 Introduction 1

2 Conformal Tilings 6 2.1 Planar Polygonal Complexes ...... 6 2.2 Conformal Maps ...... 7 2.3 Conformal Tilings ...... 8

3 Local Isomorphism 16 3.1 Local Isomorphism ...... 16 3.2 TheDiscreteHyperbolicPlane ...... 17 3.2.1 Symmetries of H ...... 28

4 Inﬁnite Subdivision Operators, Hierarchy, and Type 30 4.1 InﬁniteSubdivisionOperators...... 30 4.2 Hierarchies ...... 32 4.2.1 Combinatorial Hierarchies ...... 32 4.2.2 Conformal Hierarchies ...... 38 4.2.3 Fractal Hierarchies ...... 40 4.3 Type...... 50

5 Expansion Complexes 55 5.1 FiniteSubdivisionOperators ...... 55 5.2 ExpansionComplexes ...... 57 5.3 Hierarchies and Type of Expansion Complexes ...... 66 5.3.1 Addendum. The Discrete Hyperbolic Plane as an Expansion Complex . . . . 68

Bibliography ...... 71 BiographicalSketch ...... 72

v LIST OF FIGURES

2.1 The Bowers-Stephenson “regular” pentagonal tiling...... 13

3.1 The discrete hyperbolic plane complex H, embedded in C...... 19

3.2 Type 0 (left) and type 1 (right) extensions of H0,0...... 20 H 3.3 A portion of 10 embedded in C...... 22

3.4 Left: L0 embedded in L1 (dashed) with horizontal edges in red and thickened. Right: G0 embedded in G1 (dashed) with horizontal edges in red and thickened...... 27

4.1 Seeds of several simple subdivision operators. The left-hand column shows the triangular fundamental domain of the operator τ, and the right-hand column shows the resultant seed τ5 for the ﬁrst four operators, and the seed τ4 for the last four operators. 33

4.2 The cores of Z...... 34

4.3 The twisted pentagonal tiling P...... 46

4.4 The ﬁrst four stages of the fractal transition of single tile of P...... 46 →- 4.5 The fractal limit P of the twisted pentagonal tiling P...... 46

5.1 A ﬁnite subdivision operator τ that generates expansion complexes with non-simply connectedcores...... 65

5.2 The ﬁnite subdivision operator τ...... 68

The embeddings F1 ֒→ F2 ֒→ F3...... 69 5.3

5.4 The square tiling T (left) and the triangular tiling T ⋆ (right) of H...... 70

vi ABSTRACT

This paper examines a class of geometric tilings known as conformal tilings, ﬁrst introduced by Bowers and Stephenson in a 1997 paper [1], and later developed in a series of papers [2] and [3] by the same authors. These tilings carry a prescribed conformal structure in that the tiles are all conformally regular, and admit a reﬂective structure. Conformal tilings are essentially uniquely determined by their combinatorial structure, which we encode as a planar polygonal complex. It is natural to consider not just a single planar polygonal complex, but its entire local isomorphism class. We present a case study on the local isomorphism class of the discrete hyperbolic plane complex, ultimately providing a constructive description of each of its uncountably many members. Conformal tilings may tile either the plane C or the disk D, and answering the type problem motivates the remainder of the paper. Subdivision operators are used to repeatedly subdivide and amalgamate tilings, and Bowers and Stephenson prove that when a conformal tiling admits a combinatorial hierarchy manifested by an expansive, conformal subdivision operator, then that tiling is parabolic and tiles the plane C. We introduce a new, weaker, notion of hierarchy—a fractal hierarchy—and generalize their result in some cases by showing that conformal tilings which admit a combinatorial hierarchy manifested by an expansive, fractal subdivision operator are also parabolic and tile the plane C, assuming that two generic conditions for conformal tilings are true. This then answers the problem for certain expansion complexes, showing that expansion complexes for appropriate rotationally symmetric subdivision operators are necessarily parabolic.

vii CHAPTER 1

INTRODUCTION

Conformal tilings were first introduced by Bowers and Stephenson in a 1997 paper [1]. They originally arose from work on Cannon’s Conjecture—that every word-hyperbolic group with 2- sphere ideal boundary is, essentially, a cocompact Kleinian group. Since then, influenced by the classical study of aperiodic hierarchical tilings, Bowers and Stephenson have written a series of papers—[2] and [3]—developing the study of conformal tilings into a mature discipline in its own right. We will review much of the material from these first three papers, and ultimately introduce a new type of hierarchy that a conformal tiling may admit, called a fractal hierarchy. We then use this concept of a fractal hierarchy to provide a framework for generalizing a very important result in answering the type problem questions that originally motivated this study, and to provide deeper insight into conformal tilings as a whole. In the second chapter we introduce these conformal tilings—tilings of either the plane C or the disk D by conformally regular polygons with a reflective structure. Much of the necessary background material on conformal tilings developed in [2] and [3] is reviewed here. In particular, we demonstrate how to build a conformal tiling with any given (appropriate) pattern. As it turns out, this pattern encodes all of the data of the conformal tiling. That is, a tiling is essentially uniquely determined by its underlying pattern, so we quickly turn our attention away from the geometric setting and focus instead on the purely combinatorial realm. The combinatorics of our tilings will be modeled by oriented 2-dimensional regular CW-decompositions of the plane, which we refer to as the planar polygonal complex, or simply the complex, associated with the tiling. The classical type problem arises naturally here—given a planar polygonal complex K, is its associated conformal tiling TK parabolic, so that TK tiles the plane C, or is it hyperbolic, so that TK tiles the disk D? Answering the type problem for given complexes is one of the primary motivators of this paper. The study of conformal tilings has been heavily influenced by the classical study of aperiodic hierarchical tilings. One of the most fruitful concepts borrowed from that discipline to our setting

1 is that of local isomorphism. Two complexes K and L are said to be locally isomorphic if every finite connected subcomplex of K embeds isomorphically in L, and vice versa. In Chapter 3 we recall several standard results regarding local isomorphisms, perhaps the most important being that the local isomorphism class (K)—the set of planar polygonal complexes that are locally isomorphic to the complex K—is either a singleton, or uncountable. We have always found the concept of local isomorphism to be fascinating, and most of the chapter is dedicated to a detailed case study in the intricacies of local isomorphisms in the context of a single example, one that appears naturally in many areas of mathematics—the discrete hyperbolic plane H. Ultimately, we provide a constructive description of the entire local isomorphism class (H), allowing for a much deeper level of insight into H than is typically possible for general planar polygonal complexes. Chapter 4 introduces most of the theoretical machinery that has been developed thus far in answering the type problem for conformal tilings. We begin by introducing subdivision operators, which are motivated by the finite subdivision rules of Cannon, Floyd, and Parry [4]. Subdivision operators produce new planar polygonal complexes from old ones, namely by providing an algo- rithm that tells us how to subdivide each face of the complex. That is, if τ is a subdivision operator and K is a planar polygonal complex, we obtain the subdivision τK of K by subdividing each n- gon face of K according to the model n-gon seed of τ. In many ways these subdivision operators provide the natural morphisms between planar polygonal complexes, and provide the setting for discussing hierarchies of complexes. Indeed, perhaps the most important concept from this chapter is that of a hierarchy, of which we introduce several different types. A combinatorial hierarchy for a complex K that is manifested by an operator τ is a bi-infinite sequence {Kn : n ∈ Z} of planar polygonal complexes with

K = K0 and such that Kn+1 = τKn, for each n ∈ Z, and furthermore has the property that all the members of the hierarchy are locally isomorphic to one another. That is, one can move forward through the hierarchy by subdividing the faces of K according to the operator τ, and backwards by amalgamation of patches of faces in the pattern of τ, each time obtaining a new complex that is locally isomorphic to the original. A critical feature we’d like hierarchies to exhibit is that of expansivity, meaning that combinatorial patches of K can be engulfed by a core of faces—either a single face, two faces that meet along a shared edge, or the union of faces that meet at a shared

2 vertex—in the hierarchy simply by amalgamating enough times. We show that imposing a very mild combinatorial condition called shrinking on our operator τ will guarantee that hierarchies manifested by τ will be expansive. We then examine two other types of hierarchies, the ﬁrst being conformal hierarchies, which arise naturally when considering the action of dihedrally symmetric subdivision operators on tilings, and the second being fractal hierarchies, which arise instead when one considers subdivision operators that are merely rotationally symmetric. Unlike combinatorial hierarchies, these two are hierarchies of actual tilings, and not just planar polygonal complexes.

To be more precise, given a conformal tiling T whose associated complex KT admits a combinatorial hierarchy {Kn}, a conformal hierarchy for T is a sequence of conformal tilings

{Tn = TKn : n ∈ Z}, with T = T0, and such that the tilings themselves can be subdivided in situ to yield the next tiling in the hierarchy. That is, the individual tiles can be geometrically subdivided so that the resulting structure is still conformally regular and reflective. Conformal hierarchies were studied extensively by Bowers and Stephenson in [3], and it can be shown that if a conformal tiling T admits a conformal hierarchy, and the associated subdivision operator τ is dihedrally symmetric, shrinking, and simple, then T is parabolic, and so too is any tiling T ′ that is locally isomorphic to T . The third type of hierarchy arises from experimental observations, and generalizes conformal hierarchies. That is, given a conformal tiling T whose associated complex KT admits a combi- 1 natorial hierarchy, a fractal hierarchy for T is a sequence of tilings {Jn : n ∈ Z}, with three properties. The first is that T and J0 are combinatorially equivalent, and both tile the same underlying space, either C or D. The other two properties are modeled on properties exhibited by conformal hierarchies: 1) The tilings can be geometrically subdivided in situ to yield the next tiling in the hierarchy, and 2) the tilings have the property that any two combinatorially equivalent patches of tiles chosen from any two tilings in the hierarchy will always be conformally equivalent patches. These latter two properties are key in solving the type problem in the context of conformal hierarchies, and hence allow us to use similar methods of proof in this setting as well. We show that, assuming two generic conditions on conformal tilings are fulfilled, if an n-gon conformal tiling T with bounded degree admits a fractal hierarchy and the associated subdivision operator τ is

1 We omit the adjective conformal here purposesly; the tilings Jn need not be conformal tilings.

3 rotationally symmetric and shrinking, then T is parabolic, and so is any tiling T ′ that is locally isomorphic to T . Finally, we conclude with a chapter that may be viewed as an application of the concepts studied in Chapter 4. Indeed, a priori it is not at all clear that the hierarchies examined thus far exist for any but the most trivial of tilings. In this chapter we show that these types of hierarchies are in fact plentiful, and we describe a straightforward process for building such objects. Given an n-gon seed ∆, one can iteratively subdivide it using an appropriate operator τ to obtain the infinite sequence {τ n∆}, n = 0, 1, 2, ···. From this sequence we can extract a nested sequence of subcomplexes, and by taking the direct limit of this sequence we obtain a planar polygonal complex known as an expansion complex for τ. Expansion complexes have several key properties. First, by the construction it is clear that the expansion complex will contain an infinite number of copies of each τ n∆. Furthermore, we show that expansion complexes generally exhibit the features discussed in Chapter 4. That is, expansion complexes admit combinatorial hierarchies, and depending on whether or not τ is dihedrally or rotationally symmetric, this hierarchy will respectively induce either a conformal or fractal hierarchy of tilings. Finally, we show that expansion complexes for appropriate subdivision operators have all of the other combinatorial properties introduced in the previous two chapters—i.e. we show that they have bounded degree, combinatorially repetitivity, finite local complexity, and so on. The planar polygonal complexes underlying our conformal tilings in practice are typically constructed as expansion complexes for a subdivision operator. Indeed, all of the machinery developed in Chapter 4 was ultimately modeled on the observed behavior of expansion complexes. Because of this, the results of Chapter 4 can all be applied here to easily answer the type problem for conformal tilings of expansion complexes. That is, we have as an immediate corollary what is perhaps the key result of the paper—if τ is a shrinking, rotationally symmetric finite subdivision operator with one face type and bounded degree, then τ manifests a fractal hierarchy for the conformal tiling TK , and moreover, TK tiles the plane C, and conformal type is constantly parabolic across the local isomorphism class (K). This answers the original type problem for many of the conformal tilings that are naturally of interest to us. Finally, we close the chapter with a short addendum that refers back to the discrete hyperbolic plane complex H. We show that it is possible to realize H as an expansion complex for a finite

4 subdivision operator (albeit, a subdivision operator that is more general than those studied in

Chapter 4), and we show that the tiling TH is hyperbolic. In fact, we do even more—the method of proof immediately extends to show that type is constantly hyperbolic across the entire local isomorphism class (H).

5 CHAPTER 2

CONFORMAL TILINGS

Throughout this paper we will be studying tilings of two different geometries—either euclidean tilings of the plane C, or hyperbolic tilings of the disk D.A tile is a closed Jordan domain with finitely many distinguished boundary points. A tiling is a collection of tiles with disjoint interiors that meet only along full edges or along vertices, and whose union is the whole space, either C or D. A conformal tiling is a tiling with two characterizing properties—it is conformally regular, and reflective. Before defining these terms carefully, however, we first demonstrate how to build a conformal tiling from given combinatorics. We then show that the combinatorics alone contain essentially all of the data of a conformal tiling. That is, any two given conformal tilings in the same pattern are necessarily conformally equivalent. With this in mind, we begin by developing a minimal amount of combinatorial background material.

2.1 Planar Polygonal Complexes

A planar polygonal complex K, or just a complex for short, is a regular, oriented 2- dimensional CW-decomposition of the plane. That is, K is a subdivision of the plane into 2-cells with cyclic graph boundaries that meet along subgraphs of their boundaries; the intersection of two adjacent 2-cells need not be connected. The 0-skeleton K(0) is a discrete, countable set of vertices, the 1-skeleton K(1) consists of edges that span two distinct vertices, and each face of the 2-skeleton K(2) is a topological disc bounded by a finite cycle of edges, of length at least 3. If a face f has a boundary cycle of length n, we say that f is a combinatorial n-gon, or that f has polygonal type n. When we refer to either a face f or an edge e, we will be thinking of them as being closed cells. The corresponding open cells whose closures are f and e will be denoted by f o and eo respectively. The complex K of course has infinitely many faces, and may further have infinitely many differing polygonal face types. We will generally consider more tame complexes than this, so we

6 refine our focus a bit. For starters, we wish to consider only complexes K that are locally finite, meaning that each cell of K meets only finitely many other cells, which is equivalent in this setting to the requirement that each vertex have finite degree. Unless otherwise stated, all of our complexes will be assumed to be locally finite. Later in this paper we will impose even stricter conditions, such as the following: A complex K is said to have degree at most N if every face f of K has polygonal type of at most N, and every vertex v of K has degree at most N/2. The complex K has bounded degree if it has degree at most N for some N. If K is a complex with only n-gon faces, we say K is a planar n-gon complex. More generally, if K only has faces of polygonal types n1,...,nk then we say K is a planar (n1,...,nk)-gon complex.

2.2 Conformal Maps

Let Ω be an open set in C, and let f be a complex-valued function on Ω. In particular, if z = x +iy, deﬁne f by f(z) = u(x,y)+iv(x,y), where u and v are real-valued functions. The function f is said to be complex-diﬀerentiable at the point z0 ∈ Ω, with Ω open, if the quotient

f(z) − f(z0) z − z0 converges to a limit when z → z0. Here we have z ∈ Ω so that the quotient is deﬁned. If this ′ quotient converges, its limit is denoted by f (z0), called the derivative of f at z0. That is,

′ f(z) − f(z0) f (z0)= lim . z→z0 z − z0

Of course, in the above limit we have that z may approach z0 from any direction. If f is complex- diﬀerentiable at every point z ∈ Ω, then we say that f is holomorphic on Ω. If f is holomorphic on some neighborhood of a point w, then we say that f is holomorphic at w. It is well-known that f is holomorphic if and only if u and v are continuously diﬀerentiable and satisfy the famed Cauchy-Riemann equations—that is, that ux = vy and uy = −vx. It is another standard result that every holomorphic function is analytic (and vice versa), in the sense that they admit power series expansions near every point. For this reason, we will be using the term analytic as a synonym for holomorphic throughout this paper. The function f :Ω → C is said to be conformal on Ω if it is analytic on Ω, and f ′(z) 6= 0 for every z ∈ Ω. This has important geometric consequences, most notably that conformal maps are

7 angle-preserving. That is, if f is conformal on Ω, then at every point z0 ∈ Ω it will preserve the oriented angle between any two curves that intersect at z0. We will also often see anticonformal maps in this paper—these are complex conjugates of conformal maps, and hence while they also preserve the magnitude of angles, they instead reverse their orientation.

2.3 Conformal Tilings

We are now in a position to build conformal tilings from these complexes. Given a planar polygonal complex K, let K⋆ denote the star subdivision of K. Star subdivision is deﬁned by adding a new vertex—a face barycenter—to each open face of K, and for every face barycenter, adding new edges connecting the face barycenter to all of the original vertices of the face in which it is contained. The star subdivision subdivides each face of polygonal type n of K into n triangular faces in K⋆. Note that K has degree at most N if and only if every vertex of K⋆ has degree at most N. Before continuing, let us establish the following convention now: Unless otherwise stated, the terms isomorphism and isometry in this paper will always mean orientation-preserving isomorphism and orientation-preserving isometry. Next, we endow K⋆ with the star-equilateral metric. Each edge of K⋆ will be given unit length by isometrically identifying the edges with the unit interval, and this metric will be extended to the faces of K⋆ by isometrically identifying each face with a euclidean equilateral triangle with unit side lengths. Finally, this extends to a metric on all of K⋆ by taking the standard shortest path metric—that is, deﬁne the distance between two arbitrary points x,y ∈ K⋆ to be the length of the shortest path between them. We will denote the resultant metric space by |K⋆|eq. This metric space can be visualized (although it is not necessarily embeddable in R3) as the piecewise linear surface obtained by simply pasting together unit equilateral triangles in the pattern of K⋆. There is a standard way to equip a polygonal piecewise linear surface with a conformal structure.

We recall that a conformal structure is determined by an atlas A = {(Uα,φα)}, where each chart ⋆ consists of an open set Uα ⊂ |K |eq and a homeomorphism φα mapping Uα to an open subset ⋆ of the plane. Additionally, we require that the Uα cover |K |eq and that, for each α,β such that −1 Uα ∩ Uβ 6= ∅, the transition maps φβ ◦ φα : φα(Uα ∩ Uβ) → φβ(Uα ∩ Uβ) are analytic.

8 We now define such an atlas for |K⋆|eq explicitly. Our charts come in two flavors; the first will ⋆ o o o ⋆ be the edge charts. Consider an arbitrary edge e ∈ K , and let Ue = f1 ∪ e ∪ f2 ⊂|K |eq, where f1 and f2 are the two faces that meet along e. Fix two unit euclidean equilateral triangles T and T ′ in the plane so that they intersect along a common edge, and let X be the interior of T ∪ T ′. ⋆ For each edge e ∈ K , define the coordinate chart φe to be the orientation-preserving isometry

φe : Ue → X. This construction gives the edge charts (Ue,φe). Finally, observe that between edge charts the transition maps are orientation-preserving isometries. Hence, they can consist of only rotations and translations, so they are restrictions of complex maps of the form z 7→ az + c, hence clearly analytic.

Charts of the second type—the vertex charts—will resolve the cone type singularities in |K⋆|eq ⋆ ⋆ that are present at the vertices of K . Let v be a vertex, and let Uv ⊂ |K |eq be an open metric ball of radius 1/3 centered at v. If v has degree d, then there is a cycle of faces f1,f2,...,fd that meet at v. The coordinate chart φv will be deﬁned piecewise on Uv. On each |fi ∩ Uv| the map

φv is deﬁned by isometrically identifying |fi ∩ Uv| with a copy of itself in the plane, with v placed p at the origin, followed by a power map z 7→ ciz , where ci is a complex constant of modulus one.

We can ﬁx c1 = 1, and then choose each ci successively to ensure compatibility of φv across edges shared by adjacent faces fi and fi−1, for i = 2,...,d. Finally, the angle of each fi is π/3, so we need to choose p such that p · d · π/3 = 2π to ensure that φv is injective on Uv, giving p = 6/d. This deﬁnes the vertex charts (Uv,φv). Since the sets Uv are all disjoint, we need only check analyticity of transitition maps between vertex charts and overlapping edge charts. But again, in this case analyticity is clear, as the transition maps are just restrictions of the aforementioned power maps.

Together with this conformal stucture, |K⋆|eq is now realized as a simply connected, non-compact

Riemann surface, which we call SK . Our ﬁnal layer of structure will be the tiling itself. By the classical Uniformization Theorem, a Riemann surface that is non-compact and simply connected can admit only one of two conformal structures, and hence SK is conformally equivalent to G, where G is either the Poincar´edisk D, or the complex plane C. That is, there is a conformal homeomorphism Φ : SK → G, and such a map is unique up to the conformal automorphisms of G.

9 Deﬁnition 2.1. Given a planar polygonal complex K, the conformal tiling TK is the collection of tiles in G ∈ {C, D} determined by a uniformizing map Φ : SK → G, where the tiles are the images of the faces of K under Φ.

Given a complex K, if TK is a tiling of the disk D, we say that TK and K are hyperbolic. If, on the other hand, TK is a tiling of the plane C, we say that TK and K are parabolic. The classical type problem now arises—given a complex K, is it parabolic or hyperbolic? This is generally a very diﬃcult question to resolve, but answering it for certain complexes is one of the main motivations of this paper.

Let us make some observations about the structure of TK . An n-gon face f of K becomes a ⋆ piecewise linear n-gon in |K |eq, on which the dihedral group D2n of order 2n acts by isometries.

By conjugating with the uniformizing map Φ, this induces an action of D2n on the corresponding n-gon tile t in TK by conformal automorphisms. The index 2 subgroup of D2n that consists of elements that preserve the orientation of t is the subgroup that acts as conformal rotations. The remaining elements are those that reverse orientation of t; these elements act as anticonformal reflections. Each reflection has a fixed point set—the axis of the reflection—that is an analytic arc spanning across either two vertices or two edges of t if n is even, or across vertex-edge pairs if n is odd. The intersection of these axes is the conformal center of t, and is fixed under the action of D2n. Slicing t from each vertex of t to the conformal center of t along the axes divides t into n triangles, one for each edge of t. Slicing up every tile of TK in this manner gives the related ⋆ triangular tiling TK .

We will show that the tiling TK has two characterizing properties. These are conformal regularity, and reﬂectivity. We will return to the latter in a moment, but the former—regularity— means that, for each tile t of TK , if t has polygonal type n (i.e. t corresponds to a combinatorial n-gon in K), then there is a homeomorphism from t to the regular euclidean n-gon that is conformal on the interior of t, and that identiﬁes vertices.

Our observations above make it clear that TK is indeed conformally regular. Let f be an n-gon tile in TK , and ∆ be a regular euclidean n-gon. Both f and ∆ can be sliced along their axes of reflection into 2n triangles by the process detailed above. Let tf and t∆ be two of the triangles from f and ∆ respectively, and let φ : tf → t∆ be a conformal homeomorphism between these triangles that identifies their vertices, and in particular identifies the vertices corresponding to the

10 conformal centers of f and ∆.1 The map φ can be extended analytically, by Schwarz reflection, across any edge of tf to the entirety of both its neighboring triangles in f. This process can then be repeated triangle-by-triangle across all of f, giving a bijective conformal extension φ : f → ∆. We should note that the important point here was that f admitted a full dihedral group’s D2n worth of conformal symmetries, and that this process demonstrates the standard result that if D2n acts conformally on a disk Ω ⊂ C, then Ω is a conformally regular n-gon. Unfortunately, regularity alone is not a very scarce property amongst tilings of G ∈ {C, D} in the pattern of K. Indeed, there are uncountably many pairwise non-isomorphic ways to realize K as a tiling of conformally regular polygons. Fortunately, there is further underlying structure in the way in which the tiles of TK all fit together, and this additional level of structure will allow us to uniquely identify TK amongst the uncountably many conformally regular tilings in the pattern of K. This structure is our second characterizing property—reflectivity. A tiling T by conformally regular polygons is said to be reflective if, whenever two tiles t and t′ meet along a shared edge e, e is the fixed point set of an anticonformal reflection that exchanges the two triangular faces of T ⋆ that meet along e and identifies the conformal centers of t and t′.

Our earlier observations also make this property clear. A triangular face f of |K⋆|eq can be isometrically reflected across any of its edges onto any of its neighboring faces. This triangular ⋆ face f corresponds to a triangular tile t in TK , and by conjugating with the unformizing map Φ an isometric reflection of f in |K⋆|eq induces an anticonformal reflection of t onto a neighboring tile.

Finally, we note that the corners of tiles in the conformal tiling TK have interior angles depending on the vertex v to which they correspond. That is, if v has degree d, then the interior angle formed at v of all tiles meeting at v will be 2π/d—indeed, the form of the power maps at v in the conformal atlas for |K⋆|eq makes this clear. We are now in a position to deﬁne general conformal tilings.

Definition 2.2. A conformal tiling is a locally finite tiling of G ∈ {C, D} by conformally regular tiles, and that has a reflective structure.

1Such a map always exists—all 3-gons are conformally regular triangles. The interior of any 3-gon can be conformally mapped to the open disk by the Riemann mapping theorem, and this map extends continuously to the boundary by Carath´eodory. Finally, the conformal automorphisms of the disk are 3-point transitive on the boundary.

11 The next theorem shows that a conformal tiling is, essentially, uniquely determined by its combinatorics alone.

Theorem 2.1. Suppose T is a tiling of either C or D by conformally regular polygons with the combinatorics of K. If T is reﬂective, then T is conformally equivalent to the conformal tiling TK .

Proof. The proof follows similarly to the observations made above, except using Schwarz reflection across the entire tiling rather than just on a single face. Each triangular face f of K⋆ not only ⋆ ′ ⋆ coincides with a conformal triangle t in TK , but also with a conformal triangle t in T , because T has the combinatorics of K. Let φ be a conformal homeomorphism between t and t′. Because both tilings TK and T have a reflective structure, this map φ can be extended analytically, by Schwarz ⋆ reflection, across any edge of t to all of the triangles of TK that neighbor t. This process can then ⋆ be repeated triangle-by-triangle across all of TK , giving a bijective conformal map φ : TK → T , which preserves the combinatorics of TK (i.e. φ maps tiles to tiles).

Example 2.1. ...

i. Perhaps the most basic conformal tiling is the square grid tiling Z of the plane C by unit squares {[0, 1]2 +(m, n): m, n ∈ Z}. This tiling is trivially seen to be both conformally regular and reﬂective.

ii. The triangular tiling generated by the triangle group ∆(3, 3, 3) is a conformal tiling of C. It is reflective by definition, and conformally regular because, as we have seen, all triangles are conformally regular. With this in mind, the triangle groups ∆(2, 3, 6) and ∆(2, 4, 4) also generate conformal tilings of C, and there are infinitely many more hyperbolic triangle groups that generate conformal tilings of D instead. iii. Figure 2.1 shows a more interesting conformal tiling, the Bowers-Stephenson pentagonal tiling introduced in [1], and indeed the conformal tiling that ultimately led to the development of this entire topic. The combinatorics underlying this tiling are constructed by means of the expansion complexes of Chapter 5. The eye immediately picks out a great deal of structure and symmetries upon even a cursory examination of the figure, all of which will be studied in later chapters.

Examples (i) and (ii) are somewhat trivial, because they are examples of classical tilings that also coincidentally happen to be conformal. In that setting, one has a ﬁnite set A = {A1,...,Am} of prototiles, and the plane is tiled by translates of these prototiles. The rigidity of this setting

12 Figure 2.1: The Bowers-Stephenson “regular” pentagonal tiling.

13 is in stark contrast to the tiling in example (iii), where nearly every tile has a unique shape—a feature of typical conformal tilings. We close the chapter with one ﬁnal observation. The proof of Theorem 2.1 implies a similar local result on subsets of tilings. A combinatorial patch of a planar polygonal complex K is a connected subcomplex L ⊂ K such that every cell of L is contained in a closed face of L. The corresponding set of tiles in the tiling TK will be denoted TL and is called a patch of TK . Two ′ patches TL and TL′ are said to be conformally equivalent if there is a conformal homeomorphism between them that maps tiles to tiles. Obviously, a conformal equivalence between patches implies an isomorphism between their underlying combinatorial subcomplexes, and the converse is also true by the following theorem.

Theorem 2.2. If L and L′ are two combinatorially equivalent patches of K and K′, respectively, ′ ′ then TL and TL′ are conformally equivalent patches of TK and TK′ .

The proof follows analogously to to the proof of Theorem 2.1. That is, because the tilings are conformally regular and reﬂective, a conformal homeomorphism of a single tile of TL to the ′ corresponding tile in TL can be extended to all of TL by repeated applications of Schwarz reﬂection, precisely because the two patches are combinatorially equivalent. We close the chapter with an observation about conformally regular n-gons. Clearly, if a tile t is a conformally regular n-gon, then it admits a conformal rotational symmetry of order n. The following theorem shows that the converse is also true, a fact that we will use several times throughout the paper.

Theorem 2.3. Let t be an n-gon tile in G ∈ {C, D}. If t is acted on by a group G of conformal rotations of order n that stabilize the vertices of t, then t is a conformally regular n-gon.

Proof. Let t be a tile in G ∈ {C, D} with vertices v1,...,vn. Suppose t is acted on by a group G of conformal rotations of order n, and let p be the fixed point of the action. There exists a conformal homeomorphism f from the interior of the closed disk D, centered at the origin, to t that extends continuously to the boundary. By conjugating by an appropriate Möbius transformation, we can assume that p = f(0). Let g be a generator of G; we have that the map φ = f −1gf is a conformal automorphism of D that fixes 0 and does not fix any points on the boundary. It follows that φ is elliptic, and because

14 it ﬁxes the origin, it is a rigid rotation of D. In particular, φ has order n, implying that it is a rotation by 2π/n. Now let w1,...,wn be the respective preimages under f of the vertices v1,...,vn.

Since the points w1,...,wn of D are stabilized by φ, and φ is a rotation of D by 2π/n, it follows that w1,...,wn are necessarily evenly spaced along the boundary of D. Hence, D admits not just rotational symmetries that stabilize its vertices w1,...,wn, but rather the full dihedral group D2n of reﬂective symmetries as well. The result now follows: Through repeated applications of Schwarz reﬂection on triangular fundamental domains, D—and therefore t—is a conformally regular n-gon.

15 CHAPTER 3

LOCAL ISOMORPHISM

The concept of local isomorphism arises naturally in the study of classical aperiodic tilings, and we transport the concept here to the combinatorial realm of planar polygonal complexes with much beneﬁt. In [3] Bowers and Stephenson examine many of the topological properties of the local isomorphism class of (rooted) planar polygonal complexes. While we don’t delve to such lengths in this paper, we do recall some of their most important results. For our purposes, however, we consider local isomorphisms for three main reasons: First, local isomorphism turns out to be the underlying linking feature between complexes/tilings in the hierarchies studied in Section 4.2. Secondly, they act as a unifying property for the expansion complexes studied in Section 5.2. Finally, we ﬁnd the concept to be fascinating in its own right, and we explore the concept of local isomorphism as a case study in Section 3.2 in the context of a single example—the discrete hyperbolic plane.

3.1 Local Isomorphism

Given two planar polygonal complexes K and L, we say that K locally embeds in L if every ﬁnite connected subcomplex of K isomorphically embeds in L. Moreover, we say that the complexes K and L are locally isomorphic—denoted K ∼ L—if K locally embeds in L, and L locally embeds in K. The local isomorphism class of K is denoted by (K), and is the set of all planar polygonal complexes that are locally isomorphic to K. We have that (K) is always nonempty, as at the minimum it always contains K itself. Note that we are cavalierly identifying all planar polygonal complexes up to isomorphism, a practice that we will maintain for the remainder of this paper. In the case that (K) is a singleton, we say that K is singular, otherwise we say that K is plural. We have the following result on the cardinality of local isomorphism classes [3].

Theorem 3.1. If K is plural, then (K) is uncountable.

That is, if the local isomorphism class of a complex is known to have more than one element, then it must in fact consist of uncountably many members.

16 We say that a finite complex H is represented in a planar polygonal complex K if H is isomorphic to a subcomplex of K. Moreover, H is finitely represented in K if there are only finitely many subcomplexes of K that are isomorphic to H, and H is infinitely represented in K if there are infinitely many subcomplexes of K that are isomorphic to H. Lastly, we say that H is quasi-dense in K if there exists a positive integer n such that for every vertex v in K there exists an edge-path in K of length at most n that meets both v and a subcomplex isomorphic to H. Of course, quasi-denseness is a stronger condition on H than merely being infinitely represented, as quasi-denseness requires not only infinitely many copies of H, but also that they be uniformly distributed throughout K. When this condition holds for all subcomplexes of K—that is, every finite connected subcomplex H of K is quasi-dense in K—we say that K is combinatorially repetitive. In [3], the following two results are proven:

Theorem 3.2. If Aut(K) acts cocompactly on the planar polygonal complex K, then K is singular.

Theorem 3.3. If K contains a ﬁnitely represented subcomplex H, then K is singular.

Typically, when one speaks of symmetries of a complex K, they refer to global symmetries of the complex—that is, elements of the automorphism group Aut(K). Local isomorphism describes the structure seen in many of the classical hierarchical aperiodic tilings; they admit no global symmetries, and yet the eye picks out the local symmetries that abound as any one patch is immediately seen to be inﬁnitely represented. In this sense, the two above results show that plural complexes must live away from either extreme—they cannot exhibit too much global symmetry, and cannot exhibit too little local symmetry, for all of these such complexes are singular. Rather, plural complexes lie somewhere in the middle of the symmetry spectrum.

3.2 The Discrete Hyperbolic Plane

On initial consideration the concept of local isomorphism appears to be quite strong, and we have always found it fascinating that it is indeed weaker than the regular, global notion of isomorphism. That is, we find it surprising that it is possible for two complexes to share all local properties at every finite scale, and yet still be globally different.

17 Other interesting questions abound. Given a planar polygonal complex, how would one begin the process of attempting to construct another complex that is locally, but not globally, isomorphic to the original? In fact, it is tempting to think that perhaps if it is difficult to do this for a given complex then it is because it cannot be done (i.e. the complex is singular), because if it were plural, then by Theorem 3.1 the local isomorphism class is uncountable, and surely in this case finding just one other member should be a feasible problem! However, the expansion complexes of Section 5.2 show that this is not the case; it is typically straightforward to construct a handful of distinct members of the local isomorphism class of an expansion complex, and yet, despite the fact that this implies that there are uncountably many more, all except for this very small handful lie completely out of sight. As such, this section serves primarily as case study in the intricacies of local isomorphisms. We proceed in the context of a single example that appears naturally in many areas of mathematics. In the first subsection we begin by describing this single example, a planar pentagonal complex H, and we develop a method for producing other complexes that are (at least) locally isomorphic H H to . Using this procedure we construct a complex 01 that is readily seen to be nonisomorphic to H, demonstrating that H is in fact merely one member of an uncountable local isomorphism class. The following subsection further examines this constructive procedure, and concludes with a complete description of the entire local isomorphism class (H). Without further ado we present the complex H. Readers familiar with hyperbolic geometry will recognize the combinatorics of H as the discrete hyperbolic plane, the name we shall be calling it henceforth, while aperiodic tiling experts will recognize H as the hyperbolic Penrose tiling, and still others will recognize H as a single sheet of the Cayley graph for the Baumslag-Solitar group BS(1, 2). There are many ways to describe H; here we proceed in a rather pedestrian manner by embed- k ding it in C. The vertices of H are given by {vn,k : n,k ∈ Z} where vn,k is the point n2 +ik. We describe the edges using two sets. The horizontal edges are given by {[vn,k,vn+1,k]: n,k ∈ Z}, and the vertical edges are given by {[vn,k,v2n,k−1]: n,k ∈ Z}. Finally, the faces are the obvious ones— they are pentagonal with the cyclically ordered vertices vn,k,v2n,k−1,v2n+1,k−1,v2n+2,k−1,vn+1,k for the face fn,k. Because we have embedded H in C, in this chapter we will often refer to vertices simply by complex numbers. Figure 3.1 demonstrates H.

18 0

Figure 3.1: The discrete hyperbolic plane complex H, embedded in C.

We have seen that there exists a dichotomy between the cardinality of local isomorphism classes, and the ﬁrst task at hand is to prove the following:

Theorem 3.4. H is plural. Or alternatively, (H) is uncountable.

The method for proving Theorem 3.4 is straightforward—if we can ﬁnd just one other complex that is locally isomorphic but not isomorphic to H, then |(H)| > 1, and the result follows from Theorem 3.1. For arbitrary complexes this is not an easy problem, and in order to do this for H we will rely heavily on many of its special properties.

Let Tb(H) be a horizontal translation of H by b units, and consider the set {Hb,k : b ∈ R,k ∈ N} H H H H H where each b,k is given by b,k = Tb( ) ∩ Cim(z)≤k. That is, b,k is the portion of Tb( ) that is on or below the line z = x +ik. We stress the important fact here that the parameter k tells us that the uppermost level of Hb,k lies along the line z = x +ik, while the parameter b speciﬁes the locations of the vertices along this line, and that these two parameters alone uniquely determine the ∼ ∼ complex. We also want to observe that Hb,k = H0,0 for every b,k. This is clear because Hb,k = H0,k ∼ by a horizontal translation of b units, and H0,k = H0,0 by a vertical translation of k units followed by a horizontal contraction of 1/2k.

Suppose K = Hb,k for some b,k. If K is thought of as a collection of horizontal levels, then our goal here is to extend K vertically by adding another level. There are two ways to do this so that the resultant complex is isomorphic to the original. Recall that the real part of the coordinates of

19 0 0

Figure 3.2: Type 0 (left) and type 1 (right) extensions of H0,0.

the uppermost level (z = x +ik) of K are given by the set {b + n2k : n ∈ Z}. Let p be the smallest non-negative integer in this set. We deﬁne K0 to be the complex obtained by adjoining to K the vertices given by the points {p + n2k+1 +i(k + 1) : n ∈ Z}, and the adjoined edges are the obvious ones between horizontal and vertical neighbors. We call this a type 0 extension of K. On the other hand, we deﬁne K1 to be the complex obtained by adjoining the vertices given by the points {(p + 2k)+ n2k+1 +i(k + 1) : n ∈ Z}, again with the obvious additional edges. We call this a type 1 1 extension of K. Notice that K1 =∼ K0 by a horizontal translation, and K0 =∼ K by a vertical translation and horizontal dilation of 1/2. See Figure 3.2.

We continue the process. Define Kij := (Ki)j for i,j = 0, 1. For example, K01 is a type 1 extension of K0, which is a type 0 extension of K. We can then proceed inductively to define Ks where s is any finite binary string. We call Ks an s-extension of K. Of course, all of these objects constructed so far have boundary, and as such are not planar polygonal complexes. Furthermore, they are all isomorphic to one another. We now show that if we infinitely extend the finite complex H0,0 it is possible to create distinct members of (H), provided we extend cleverly enough.

Let s be an infinite binary string, and define Ks—the infinite s-extension of K—to be the direct limit of the sequence of embeddings Ks1 ֒→ Ks2 ֒→ Ks3 ··· where si is the truncation of s to its first i digits. That is, Ks = lim Ks . This extension process can be performed on any complex −→ i

Hb,k, but for the remainder of the paper we restrict our attention to the case where K = H0,0, and 2 denote by Hs the s-extension of H0,0. Notice that for every inﬁnite s-extension, Hs is noncompact,

1In a type 1 extension we are essentially defining our adjoined vertices by choosing the second smallest non-negative integer from the set {b + n2k}. An equivalent definition would be to choose the largest negative integer from that set. 2 Hs may be either an infinite or finite s-extension.

20 simply-connected, and consists of only 5-gon faces, and as such is a planar pentagonal complex that is locally isomorphic to H. The latter claim follows because any ﬁnite subcomplex of Hs is contained ∼ in some Hb,k = H0,k, which is contained in H. The opposite direction follows similarly. Also note H H that 0 = .

If s is an inﬁnite binary string let di(s) denote the ith digit of s. Then the vertices along the kth H k k i−1 level of s (i.e. along the line z = x+ik), are the points (pk +n2 )+ik where pk = P i=12 di(s).

In other words, if s is interpreted as a binary number read from left-to-right, then pk = sk, where again sk is the truncation of s to its ﬁrst k digits. For example, if s = 01110, then p1 = 0, p2 = 2, p3 = 6, and pn = 14 for n ≥ 4. The vertex vk = pk +ik has the smallest non-negative real part of all vertices along the kth level of Hs, and we call vk the minimal vertex on level k of Hs. We are now in a position to prove Theorem 3.4.

Proof of Theorem 3.4. There is a bi-infinite edge path in H along the imaginary axis consisting of only edges whose endpoints are vertices of degree 4. We will show that no such edge path exists in H the complex 10, establishing the claim. H We make a few observations. Every vertex of degree 4 in 10 (and indeed in every member of (H)) has two horizontal neighbors of degree 3, so no edge path though only vertices of degree 4 can contain a horizontal edge. However, below every vertex of degree 4 lies another vertex of degree 4, so at any degree 4 vertex v we can construct an infinite edge path from v by moving downward in the negative imaginary direction. Our claim is that this can never be extended infinitely upward in H H 10, which follows because the minimal vertices of 10 (and their negative neighbors) can be made arbitrarily far away from the imaginary axis by looking at higher and higher levels. See Figure 3.3. H H Let v be any vertex of 10 and vk be the minimal vertex on level k of 10, and let b and pk be k−1 k−2 the real parts of v and vk respectively. If k is odd then pk ≥ 2 , and if k is even then pk ≥ 2 . k−2 In general we have that pk ≥ 2 for every k. If b is positive, then we have pk >b> 0 whenever k > 2+ ⌈log2 b⌉. k k−2 k−2 Bounding on the other side, if k is odd then pk ≤ 2 − 2 = 3 · 2 , and if k is even then k−1 k−3 k−3 k−2 pk ≤ 2 − 2 = 3 · 2 . In general we have that pk ≤ 3 · 2 for every k. Since vk is minimal, ′ k its neighboring vertex vk =(pk − 2 )+ik is the closest vertex on level k to the imaginary axis with k−2 ′ negative real part. Furthermore, since pk ≤ 3 · 2 , it follows that for the real part of vk , call it ′ ′ k−2 ′ pk , we have pk ≤−2 . Hence if b is negative, then pk 2+ ⌈log2(−b)⌉.

21 0

H Figure 3.3: A portion of 10 embedded in C.

′ Combining these inequalities we have pk M where M =2+ ⌈log2 |b|⌉. Hence, H above the Mth level of 10 there are no vertices vertically above v, so any edge path constructed by moving vertically upward from v in the positive imaginary direction (and therefore through only degree 4 vertices) will be ﬁnite. H ∼ H H H Since 10 =6 , we have |( )| > 1, and it Theorem 3.1 implies that ( ) is uncountable.

′ H H H As we just saw, for the strings s = 01 and s = 0 we have that 01 and 0 = are nonisomor-

′ ∼ ′ phic. The obvious question then is, given two different infinite strings s and s , when is Hs = Hs , if ever? If s is an infinite binary string, then s can be decomposed into a finite string of length n, denoted sn, and an infinite string denoted endn(s) so that s = sn · endn(s), where sn · endn(s) is the ′ ′ juxtaposition of sn and endn(s). If s and s are two infinite strings such that endn(s) = endn(s ) for some n ∈ N, then we say that s and s′ are strongly coterminal. If s and s′ are two infinite ′ ′ strings such that endn(s) = endm(s ) for some n, m ∈ N, then we say that s and s are coterminal. We now answer the above question.

′ ∼ ′ Theorem 3.5. If s and s are coterminal, then Hs = Hs .

22 H ∼ H ′ Proof. We claim that s = endn(s). Then if s and s are coterminal, for some n, m ∈ N we have

H H ′ that endn(s) = endm(s ), and the result follows immediately from the claim because

H ∼ H H ′ ∼ H ′ s = endn(s) = endm(s ) = s

The claim can be shown by piecing together two isomorphisms, both simple enough that we H ∼ H dispatch of most of the details. First, we have endn(s) = 0···0·endn(s) by a vertical translation H ∼ H and horizontal dilation. Secondly, 0···0·endn(s) = sn·endn(s) by a horizontal translation by sn (in decimal!) units. Combining these we have the claim:

H ∼ H ∼ H H endn(s) = 0···0 ·endn(s) = sn·endn(s) = s n|{z} times

∼ ′ ′ We would hope that the converse holds. That is, if Hs = Hs then s and s are coterminal. H ∼ H Unfortunately, this is not the case. For example, it is easy to see that 0 = 1. If one uses the alternate definition of a type 1 extension—i.e. extending by choosing the vertex with largest H k negative real part—it is easy to see that the vertices of 1 are given by {(−1+n2 )+ik : n,k ∈ Z}. H ∼ H Hence 0 = 1 merely by a unit horizontal translation. This motivates the following definition. If s and s′ are two infinite strings such that there exist ′ ′ m, n ∈ N such that endm(s) = 0 and endn(s ) = 1, or endm(s) = 1 and endn(s ) = 0, then we ′ H ∼ H say that s and s are opposite. The above observation that 0 = 1, together with the proof of

′ ∼ ′ Theorem 3.5, shows that if s and s are opposite then Hs = Hs . It turns out however that this is the only case where the converse to Theorem 3.5 fails, as the following result shows.

∼ ′ Theorem 3.6. If Hs = Hs , then either:

1. s and s′ are coterminal, or

2. s and s′ are opposite.

∼ ′ ′ Lemma 3.7. If Hs = Hs by an isomorphism T and s and s are not opposite, then there is a ′ ′ minimal vertex vn of Hs′ and a minimal vertex vm of Hs with n ≥ m ≥ 0 such that T (vn )= vm.

23 Proof. Suppose Hs is isomorphic to Hs′ via an isomorphism T . A moment’s reﬂection should convince the reader that any such isomorphism T must preserve horizontal levels, and it follows that T can consist only of horizontal translations, and vertical translations together with a horizontal dilation. Hence, we can assume T maps the kth level of Hs′ to the 0th level of Hs, and furthermore we can assume that k ≥ 0.

′ H H ′ We use our usual notation of vn and vn for the minimal vertex on the nth level of s and s ′ respectively, and let ui denote the image of vk+i under T . By the assumptions of the previous ′ paragraph, ui is a vertex on the ith level of Hs. We will show that if s and s are not opposite then for some i we have ui = vi, establishing the result.

If u0 = v0 we are done, so assume Re(u0)

Let Dn be the number of edges on the nth level separating un and vn. The ﬁrst observation we ′ make is that if Dn is odd, then dk+n(s ) 6= dn(s). Secondly, note that there are three possibilities for Dn+1:

′ 1. Dn+1 = Dn/2 ⇔ dk+n(s )= dn(s), or

′ 2. Dn+1 =(Dn − 1)/2 ⇔ dk+n(s ) = 1 and dn(s) = 0, or

′ 3. Dn+1 =(Dn + 1)/2 ⇔ dk+n(s ) = 0 and dn(s) = 1

Because D0 ≥ 1 and Dn+1 ≤ (Dn + 1)/2 we can choose M large enough such that DM = 1 (and ′ ′ Re(uM )Re(v0) follows similarly, ﬁnishing the proof.

The proof of Theorem 3.6 now follows immediately.

∼ ′ ′ Proof of Theorem 3.6. If Hs = Hs via an isomorphism T and s and s are not opposite then by ′ Lemma 3.7 we have a minimal vertex vn of Hs′ and a minimal vertex vm of Hs with n ≥ m ≥ 0

24 ′ ′ ′ such that T (vn )= vm. It follows then that dn+i(s )= dm+i(s) for all i ∈ N, and hence s and s are coterminal.

A desire to combine the previous two propositions motivates the following definition. If s and s′ are infinite binary strings, define the (equivalence) relation ∼ by s ∼ s′ if either s and s′ are coterminal, or if s and s′ are opposite. With this, Theorems 3.5 and 3.6 combine to give the following statement.

∼ ′ ′ Corollary 3.8. Hs = Hs if and only if s ∼ s .

We began with the goal of describing (H), but thus far we have only considered the set {Hs : s ∈ S}. The next result rectiﬁes the situation by showing that every element of (H) can be realized as an s-extension Hs.

Theorem 3.9. If K ∈ (H) then K =∼ Hs for some inﬁnite binary string s.

Proof. Most of the work in this proof will amount to showing that if K is a planar polygonal complex that is locally isomorphic to H, then K contains within it a copy of H0,0. From there, we observe that because K is locally isomorphic to H, the pentagonal faces of K that meet the upper level (i.e. the boundary) of the subcomplex H0,0 ⊂ K must form either a type 0 or type 1 extension of H0,0. Repeating the argument ad infinitum shows that the combinatorics of K are that of Hs for some infinite binary string s. Suppose K is locally isomorphic to H. The vertices of degree 4 in H come in two types—the first has two vertical neighbors of degree 4, and the second has one vertical neighbor (above) of degree 3, and one vertical neighbor (below) of degree 4; both types have two horizontal neighbors of degree 3.

In the notation of our deﬁnition of H, let L0 be the subcomplex of H consisting of the union of the ﬁve closed faces f0,0,f0,−1,f1/2,−1,f1/4,−2,f1/2,−2. Contained in the interior of L0 is one vertex of degree 4 of the second type, and one vertex of degree 3. Call them v and w respectively. Because

K is locally isomorphic to H, the subcomplex L0 ⊂ H embeds in K via an isomorphism T0; let G0 be an isomorphic copy of L0 in K. We have usually thought of the “levels” of H as bi-inﬁnite horizontal edge paths. That is, we have called the line z = x +ik the kth level of H. In this proof when we refer to the kth level we

25 are referring to the entire bi-inﬁnite strip of faces that meet, and are below, the line z = x +ik. H The faces f0,−1 and f1/2,−1 lie along the −1th level of , and their images T (f0,−1) = g0,−1 and

T (f1/2,−1)= g1/2,−1 in G0 can be determined uniquely by examining the relationships of the faces ′ ′ of G0 relative to T (w)= w and T (v)= v . That is, respecting orientation, they are the two faces ′ ′ that meet both vertices w and v . This allows us to single out in G0 the images of each face of

L0, and moreover allows us to determine the “top” and “bottom” of each face in G0 by examining their positions relative to one another. That is, we have the concepts of horizontal, vertical, above, below, etc. in H, and we can carry these notions over to K locally in G0. ′ Since L0 is inﬁnitely represented in H (it is in fact quasi-dense), let L0 be another representation ′ of L0 in H, and let j0 be a cellular isomorphism between L0 and L0 . Let L1 denote the union of

L0 together with the faces f1,0,f−1,0,f1,−1,f−1/2,−1,f0,−2,f3/4,−2. In other words, L1 is L0 and all faces on the 0th, -1th, and -2th levels of H that meet L. The crucial observation to be made here is that j0 can always be extended to a cellular isomorphism j1 of the subcomplex L1. That is, every isomorphic copy of L0 in H shares the same neighboring faces on horizontal levels. Furthermore, if one inductively deﬁnes Lk+1 to consist of Lk together with all the faces on the 0th, -1th, and -2th ′ levels that meet Lk, and jk is a cellular isomorphism between Lk and any isomorphic copy Lk in

H, then jk can always be extended to a cellular isomorphism jk+1 on Lk+1. See Figure 3.4.

The observation of the previous paragraph implies that the isomorphism T0 : L0 → G0 can also be extended to an isomorphism T1 : L1 → G1, where in particular G0 ⊂ G1. Repeating this argument, one obtains the nested sequence G0 ֒→ G1 ֒→ G2 ֒→ ... in K, and hence the inﬁnite strip G = lim Gi =∼ L is contained in K, where L be the union of the 0th, -1th, and -2th levels of −→ H.

Our local notions of above, below, etc. in G0 also extend to each Gi, and in the limit to all of G, and so we can determine the corresponding “levels” of K. Call the uppermost level of G—i.e. the faces of G that are images of the 0th level of L—the 0th level of K, and let the faces corresponding to the -1th and -2th levels of L be the -1th and -2th levels of K respectively. Moreover, deﬁne the kth level (k ≤ −3) of K to consist of all faces of K that are adjacent to and below the (k + 1)th level. We wish to point out now in the preceding process the importance of being able to determine upper and lower edges of faces in K, and this is why we mapped over three entire levels from H,

26 g−1,0 g1,0

g0,0 f−1,0 f0,0 f1,0 g−1/2,−1 g1,−1

f−1/2,−1 f0,−1 f1/2,−1 f1,−1 g0,−1 g1/2,−1

g0,−2 g3/4,−2 f f f f 1 3 1 / / / 0 4 2 , 4 − , , , − − − 2 2 2 2 g1/4,−2 g1/2,−2

Figure 3.4: Left: L0 embedded in L1 (dashed) with horizontal edges in red and thickened. Right: G0 embedded in G1 (dashed) with horizontal edges in red and thickened. and not just one. Also, note that this labeling scheme of levels of course depends on our original choice of G0. Finally, notice that every level of H uniquely determines the entire level below it. Indeed, this is because H consists of only 5-gon faces, the vertices along any particular level alternate between degrees 3 and 4, and every vertex has one incident vertical edge below it. In K the same must be true because it is locally isomorphic to H, and since G =∼ L it follows that the combinatorics of K consisting of the kth levels, k ≤ 0 is precisely that of H0,0. That is, we have H0,0 ⊂ K. When looking at the remaining combinatorics of K, the only freedom then comes from looking above the 0th level of K. As before, deﬁne the kth level (k ≥ 0) of K to consist of all faces of K that are adjacent to and above the (k − 1)th level. Consider now the 1st level of K. Let b be the face on the 1st level that is adjacent g0,0. There are two possibilities: either b is also adjacent to g1,0, or b is also adjacent to g−1,0. Either one of these possibilities completely determines the entire 1st level, and the former will correspond to a type 0 extension, while the latter to a type 1 extension of H0,0 ⊂ K. Repeating this process for the 2nd level determines it again as either a type 0 or type 1 extension of the complex below, and continuing ad inﬁnitum we see that K can indeed be constructed as an s-extension of H0,0.

Finally, the statements of Theorems 3.5 and 3.9, and Corollary 3.8 together give the following result.

27 Theorem 3.10. (H)= {Hs : s ∈ S}/∼

′ We abuse notation a bit in the statement of Theorem 3.10, where we mean Hs ∼ Hs′ if s ∼ s . This provides a complete description of (H), and moreover provides a method for constructing every distinct member of the class, something that is typically not feasible for other plural complexes.

We also note that Theorem 3.4 follows as an immediate corollary because {Hs : s ∈ S}/∼ is clearly uncountable.

3.2.1 Symmetries of H

The complex H admits many self-symmetries; for example, one can reflect H through the imaginary axis onto itself. However, we are interested only in orientation-preserving automorphisms. With this in mind, there are still interesting automorphisms of H. One can translate H vertically up one unit, and horizontally dilate by a factor of two to obtain one such automorphism. In general, Z acts on H by automorphisms of the form x +ik 7→ x2n +i(k + n) for n ∈ Z. We observed in the proof of Lemma 3.7 that any automorphism of H must preserve its horizontal levels, so this in fact is all of them. It is H’s bi-infinite line of degree 4 vertices along the imaginary axis that make this observation so obvious. The salient question then is, do any other elements of (H) also admit symmetries, or is H unique in this sense? Using our earlier notation, S denotes the set of infinite binary strings, and if s is a binary string then di(s) denotes the ith digit of s. We say that s is rational if the infinite decimal

0.d1(s)d2(s)d3(s) ··· is rational. This is equivalent to the existence of an n such that endn(s) is repetitive, which is equivalent to the existence of n and m, n 6= m, such that endn(s) = endm(s).

We say that s is irrational otherwise. Finally, if Hs admits one or more nontrivial symmetries we say that the complex Hs is symmetric. We can now answer the above question.

Theorem 3.11. Hs is symmetric if and only if s is rational.

Proof. Suppose Hs is symmetric via an automorphism T . We have seen in the proof of Lemma 3.7 that T must preserve horizontal levels of Hs, and as such there exist a pair of minimal vertices vn and vm (with n>m>0) on the nth and mth levels of Hs respectively such that T (vn)= vm. Hence we have dn+i(s)= dm+i(s) for i ∈ N. That is, endn(s) = endm(s), and it follows that s is rational.

28 Conversely, suppose s is rational. Then there exist n>m such that endn(s) = endm(s). By H H Theorem 3.5 there exists an isomorphism R mapping s onto endn(s), and an isomorphism Q H H −1 H mapping s onto endm(s). The composition Q R is thus an automorphism of s, and is readily seen to be nontrivial because it maps the nth level of Hs to the mth level.

−1 Note that any symmetry of Hs must be of the form Q R as described in the above proof. As such, let T = Q−1R as above where n>m is as small as possible. Then the set {T k : k ∈ Z} is the set of all possible symmetries of Hs, and we have that Z acts as a group of automorphisms on all symmetric members of (H).

29 CHAPTER 4

INFINITE SUBDIVISION OPERATORS, HIERARCHY, AND TYPE

Reflecting upon the Bowers-Stephenson pentagonal tiling in Figure 4.3 for a moment, one begins to notice many local symmetries and layers of structure. For example, it is clear that if you simply blur the edges between certain patches of tiles, amalgamating several tiles into one, one can reconstruct a scaled copy of the “same” tiling. In fact, not only does this new amalgamated tiling appear to have the same (local) combinatorics as the old, but it itself appears to also be a conformally regular and reflective conformal tiling. Similar layers of symmetry appear in the classical study of tilings, known as hierarchical tilings, and we transport the concept to this setting to define several types of hierarchies of conformal tilings—combinatorial, conformal, and fractal. We then answer the type problem for these hierarchical conformal tilings. The natural way of ordering the tilings in our hierarchies is through applications of the finite subdivision rules of Cannon, Floyd, and Parry introduced in [4]. However, their finite subdivision rules allow for very exotic rules by means of their model complexes, subdivision maps, etc. Our needs are far more pedestrian, so we can do away with most of this complicated machinery and consider the similar, but much more tame, infinite subdivision operators.

4.1 Inﬁnite Subdivision Operators

An infinite subdivision operator τ is an infinite collection of seeds τn (for each n ≥ 3) such that each τn is a regular, oriented, locally-finite CW-decomposition of the combinatorial n-gon ∆n.

That is, τ is an infinite collection of seeds {τn : n = 3, 4, 5, · · ·}. Later in this paper we will consider more restricted finite subdivision operators, and distinguishing between the two concepts becomes necessary, but for now we will often drop the prefix and refer to these simply as subdivision operators, or even just as operators. There are two other conditions that are often imposed. The first is that of rotational invariance.

We say that τ is rotationally symmetric if each seed τn is rotationally symmetric. The second

30 condition is edge-compatibility. We say that τ is edge-compatible if there exists an N such that each τn subdivides every edge of ∆n into N sub-edges. In fact, we will impose the previous two conditions so often that we make the following convention now: Unless otherwise stated, the term subdivision operator will always refer to operators that are rotationally symmetric, and edge- compatible. We will also frequently enforce the condition that a subdivision operator τ is totally non-trivial (or just non-trivial) meaning that all of the seeds τn are non-trivial. That is, for every n, we have τn ≇ ∆n, where in particular τn consists of at least two faces. A combinatorial subdivision of a planar polygonal complex K is a planar polygonal complex

Ks such that each open cell—either a vertex, edge, or face—of Ks is contained in an open cell of K. That is, one can re-construct the closed cells of K by taking the union of finitely many closed cells of Ks. If Ks is a subdivision of K, we write Ks ≤ K. Of course, for any complex K we have K ≤ K, but such a “subdivision” is trivial. We will generally only consider combinatorial subdivisions Ks ≤ K that are totally non-trivial, meaning that every face of K is the union of at least two faces of Ks. In other words, we are interested in subdivisions that actually subdivide all of K. There is an important subtlety here depending on one’s perspective. It is possible that as CW complexes we have Ks ≤ K, and yet combinatorially it is the case that Ks =∼ K. Both perspectives have merit, and it will generally be clear through the context the viewpoint which one should be taking. It is in this combinatorial subdivision sense that a subdivision operator τ is actually an operator. Given a planar polygonal complex K, and a subdivision operator τ, we define the complex τK to be the new planar polygonal complex obtained by subdividing each closed n-gon face of K according to the seed τn. Note that this process is indeed unambiguous—because τ is rotationally invariant and K is oriented, there is a unique way to subdivide each face of K, and further, because τ is edge-compatible, the subdivision of neighboring faces agree on their shared edge(s). If τ is a totally non-trivial operator, then so too will be the subdivision τK. Because subdivision operators provide a rule for subdividing faces of every polygonal type, a subdivision operator τ can be applied to any complex K to produce the subdivision τK. As such, this process can be repeated as many times as one wishes, so we inductively define the complex τ mK = τ(τ m−1K), with τ 0K = K, m = 0, 1, ···.1

1K need not be a planar polygonal complex; τ acts as an operator on any polygonal complex. Hence, in the same m m m way that τ K is deﬁned, one can deﬁne from τ a new operator τ as the collection of seeds {τ (τn): n =3, 4, 5, ···}.

31 Defining exotic subdivision operators can be difficult. Indeed, it requires a description of infinitely many seeds, and furthermore that they must all mesh nicely together according to edge- compatibility. Next we describe so-called simple subdivision operators that essentially subdivide every combinatorial polygon according to the same rule.

The combinatorial n-gon ∆n is acted on by an appropriate rotation group; let t be a triangular fundamental domain of that action. Prescribing a “subdivision” of t then prescribes a subdivision of the entirity of ∆n under the action of the rotation group. See Figure 4.1. Moreover, since every combinatorial polygon, regardless of polygonal type, admits the same triangular fundamental domain, the subdivision of t prescribes a subdivision of every combinatorial polygon. Hence, we say that a subdivision operator τ is simple if all of the seeds τn are rotationally symmetric and agree on triangles forming fundamental domains of the action of an appropriate rotation group on the seed. Figure 4.1 shows the fundamental triangles for several simple subdivision operators, and demonstrates either the corresponding 4-gon or 5-gon seeds, but of course the fundamental triangle prescribes an n-gon seed for every n ≥ 3. Note that for any planar polygonal complex K, if τ is the star subdivision operator, then τK is a planar 3-gon complex, and that τK = K⋆, where K⋆ is the star subdivision deﬁned in Section 2.3. Furthermore, note that of these eight illustrated operators, only the twisted pentagonal operator lacks dihedral symmetry, a property that we will soon see makes it signiﬁcantly more unwieldy than its dihedrally symmetric counterparts.

4.2 Hierarchies

Hierarchical structures appear naturally in the classical study of substitutive tilings, and they too appear here in the combinatorial setting, which should come as little surprise to the reader as subdivision operators are essentially just the combinatorial analogues of these geometric substitution rules. There are three types of hierarchies we will consider in this section, ranging from the simple combinatorial hierarchies, to the far more esoteric fractal hierarchies.

4.2.1 Combinatorial Hierarchies

Deﬁnition 4.1. The planar polygonal complex K is said to exhibit a combinatorial hierarchy if there is a bi-inﬁnite sequence {Kn : n ∈ Z}—called a combinatorial hierarchy for K—of planar polygonal complexes such that the following three conditions hold:

32 Figure 4.1: Seeds of several simple subdivision operators. The left-hand column shows the triangular fundamental domain of the operator τ, and the right-hand column shows the resultant seed τ5 for the ﬁrst four operators, and the seed τ4 for the last four operators.

33 face core edge core

vertex core

Figure 4.2: The cores of Z.

1. K0 = K;

2. Kn+1 ≤ Kn, ∀n ∈ Z (i.e. Kn+1 is a combinatorial subdivision of Kn);

3. Kn+1 ∼ Kn, ∀n ∈ Z (i.e. Kn+1 is locally isomorphic to Kn).

For a given complex K we will consider only hierarchies {Kn} such that there is a subdivision operator denoted by σn for every n so that σnKn = Kn+1. We will call the collection σ = {σn : m n ∈ Z} the subdivision sequence for the hierarchy {Kn}. If m ≥ n, let σn denote the operator for Kn that applies the operators σn,...,σm−1 successively to obtain Km. Note that, under this n+1 n notation, we have σn = σn, and σn is the identity.

Let F be a finite subcomplex of the complex K = K0. We say that the subcomplex E of Kn 0 (n ≤ 0) engulfs F if F is a subcomplex of σnE, which in turn is a subcomplex of K0. However, we are rarely interested in engulfing by arbitrary subcomplexes, but rather by cores. A core of K is a subcomplex that comes in three flavors: either the collection of faces that meet at a vertex of K, known as a vertex core, two faces that meet along a common edge, an edge core, or just a single face of K, a face core. Figure 4.2 illustrates all possible cores of the square grid complex Z. Notice that because Z consists of only 4-gon faces, and has constant degree 4 at every vertex, there is only one possible core of each type.

Deﬁnition 4.2. The combinatorial hierarchy {Kn} for K is expansive if, for every ﬁnite subcomplex F of K, there exists an integer n ≤ 0 such that F is engulfed by a core of Kn, and hence by a core of Km for any m ≤ n.

34 We have yet to introduce a notion of distance between faces in our complexes, and we finally do so here, using the skinny path metric. That is, define the skinny path distance δK (f,g) between any two faces f and g in K to be the smallest integer k such that there exists a chain of faces {f = f0,f1,...,fk = g} of K with fi ∩ fi+1 6= ∅, where i = 0,...,k − 1. Further, if F is a subcomplex of K, the skinny path diameter of F is the largest skinny path distance (in F ) between any two faces of K that meet F . The concept of an expansive hierarchy is often too weak for our type problem ambitions in the next section. In particular, the number of steps backwards in the hierarchy one must take to engulf isomorphic subcomplexes may be wildly different. The next definition provides a strengthening of the idea of expansivity, requiring that the number of backwards steps depend only on the diameter of the subcomplex.

Deﬁnition 4.3. Let φ : N → R be a non-decreasing function. We say that the combinatorial hierarchy is φ-expansive if there is a positive constant M such that, for any integer m>M, any ﬁnite subcomplex F of K of δK -diameter at most φ(m) is engulfed by a core of K−m. In the case where {Kn} is φ-expansive for some non-decreasing function φ, we say that the hierarchy is strongly expansive.

We introduce yet further terminology, depending on the form of the function φ. If φ(m)= λm for some λ > 0, we say that the hierarchy is linearly expansive. If φ(m) = λm for some λ > 1, we say that the hierarchy is exponentially expansive. In either case, it is of course still strongly expansive.

Example 4.1. The following two examples will illustrate expansive and non-expansive hierarchies. Unfortunately, they both have the property that all the complexes in the hierarchy are isomorphic (rather than merely locally isomorphic), as more complicated hierarchies require the expansion complexes of Chapter 5 to construct.

i. The most simple of hierarchies is perhaps illustrated by the square grid complex Z introduced

in Example 2.1. Equip Z with a constant subdivision sequence, where every σn is the quad

subdivision seen in Figure 4.1. Then Z has as a combinatorial hierarchy the sequence {Zn},

where each Zn is isomorphic to Z0 = Z. This hierarchy is of course strongly expansive.

ii. The discrete hyperbolic plane complex H studied in Chapter 3.2 admits a more interesting

hierarchy. Let H0 = H, and deﬁne the subdivision operator σ0 to be such that σ0H0 =

35 H1 = H + i. That is, using the notation of Chapter 3.2, σ0 subdivides each face of H with

vertices vn,k,v2n,k−1,v2n+1,k−1,v2n+2,k−1,vn+1,k by adding new vertices vn+1/2,k,v2n+1/2,k−1,

and v2n+3/2,k−1 to the midpoints of each of its horizontal edges, and adding a vertical edge

[vn+1/2,k,v2n+1,k−1] that partitions the face down the middle into two pentagons. We then can inductively deﬁne σn to be such that σnHn = Hn+1 = Hn + i. This gives us the combinatorial

hierarchy {Hn}, where again we have Hn =∼ H for every n. This hierarchy is not expansive; every core of H is sandwiched between two horizontal levels, and as one passes backwards through the hierarchy, the cores grow only horizontally, and never vertically. Hence, if one considers a subcomplex of H that spans across even three horizontal levels, no core will ever engulf it.

Our interest in fact lies not with such general combinatorial hierarchies, where we may have complicated subdivision sequences that allow for a different subdivision operator at every step in the hierarchy. Rather, we want hierarchies that are generated by a single subdivision operator τ so that our generic subdivision operator σn = τ for every n ∈ Z. A subdivision operator τ is said to manifest a combinatorial hierarchy for the planar polygonal complex K if K exhibits a combinatorial hierarchy {Kn} with the property that τKn = −1 Kn+1, ∀n ∈ Z. On this sequence we can also define the inverse operator τ to be an operator such −1 −1 that its application to Kn yields Kn−1. That is, τ Kn = Kn−1. We call τ an amalgamation operator for {Kn}. Strong expansivity of hierarchies is necessary in proving our type problem results later in this section, so we introduce now what will be a sufficient condition for strong expansivity. Let v be a o o o vertex of the combinatorial n-gon ∆n. The set ∠v = {v}∪ d ∪ e ∪ ∆n , where d and e are the edges of ∆n that are incident with v, is called the open angle of ∆n at v.

Deﬁnition 4.4. We say that the subdivision operator τ is strictly shrinking if, for every n, every closed face of the seed τn is contained in an open angle of ∆n. Moreover, we say that τ is shrinking if for some k ≥ 1, the operator τ k obtained from τ by iterating it k times is strictly shrinking.

Referring to the subdivision operators illustrated in Figure 4.1, the reader will observe that they are all strictly shrinking except for the star, delta, and hex operators. The hex operator is shrinking, however, as iterating it twice yields a strictly shrinking operator.

36 There is an important observation to be made regarding how subdivisions aﬀect distances. Notice that if L is a subdivision of K, then for any two faces f ′ and g′ of L, which are contained in ′ ′ the respective faces f and g of K, we have δL(f ,g ) ≥ δK (f,g). We write δL ≥ δK . Although the inequality need not be strict, the observation, in an intuitive sense, is that as one passes through a combinatorial hierarchy {Kn}, distances tend to increase as n → ∞, and distances tend to decrease as n → −∞. That is, for a combinatorial hierarchy {Kn} and the corresponding sequence of skinny path metrics δKn we have δKn ≥ δKn+1 for every n. We would ideally like this inequality to be eventually strict (excepting, possibly, when two faces are adjacent), and it turns out that this can be guaranteed simply by requiring that the subdivision operator τ be shrinking. This is used in [3] to prove the following:

Theorem 4.1. Let τ be a subdivision operator that manifests a combinatorial hierarchy for the planar polygonal complex K. If τ is shrinking, then the hierarchy {Kn} is exponentially expansive.

That is, {Kn} is strongly expansive.

There are two properties that are almost always required of classical hierarchical tilings— repetitivity, and finite local complexity. Thus far, we have imposed neither of these. The next theorem shows that if we enforce these two properties, then a combinatorial hierarchy is not just a property of an individual complex K, but rather of its entire local isomorphism class (K). We refer the reader to [3] for the proof. Let K be a planar polygonal complex, and let D ⊂ K be a finite combinatorial disk. Further- more, let size(D) denote the number of faces of D, and let size(∂D) denote the number of edges in the simple closed edge path ∂D that is the combinatorial boundary of D. We say that K satisfies a θ-isoperimetric inequality if there exists a positive function θ : N → R such that

size(D) ≤ θ(size(∂D)) for every ﬁnite combinatorial disk D ⊂ K.

Definition 4.5. A planar polygonal complex K is said to have finite local complexity, abbrevi- ated as FLC, if K has bounded degree and K satisfies a θ-isoperimetric inequality for some positive function θ : N → R.

37 Theorem 4.2. Let K be a combinatorially repetitive, planar polygonal complex that has FLC. If the subdivision operator τ manifests a combinatorial hierarchy for K, then that same operator τ manifests a combinatorial hierarchy for any planar polygonal complex L that is locally isomorphic to K.

4.2.2 Conformal Hierarchies

When one thinks of subdividing a tiling, there are two ways to envision doing this. The ﬁrst is a subdivision at the combinatorial level. That is, given a planar polygonal complex K, we can always construct its conformal tiling TK . If, in addition, we have an inﬁnite subdivision operator

τ, one can subdivide K to get τK, and then construct the new tiling TτK . We call this process, and the tiling TτK , a combinatorial subdivision of TK . We can always do this to produce a new tiling from an old, but the resultant tiling may have little in common geometrically with the original. On the other hand, one may subdivide a conformal tiling T at the geometric level of the tiles themselves. A geometric subdivision of T is a tiling S (of the same geometry as T ) such that each tile of S is contained in a tile of T . Let KT denote the planar polygonal complex associated with T (i.e. KT reﬂects the combinatorics of T , in that we have the tiling TKT = T ). Hence, if S is a geometric subdivision of K we have that the planar polygonal complex KS determined by S is a combinatorial subdivision of KT . Of course, we are rarely interested in arbitrary subdivisions; we say that a geometric subdivision S of T is a conformal subdivision of T if S itself is a conformal tiling. In other words, conformal subdivisions are those that mesh perfectly with combinatorial subdivisions. The reverse viewpoint is also important here—instead of thinking of obtaining S from T by conformally subdividing the tiles of T , one can also think about obtaining T from S by amalgamating collections of tiles of S into larger “supertiles” in a conformally consistent manner. Hence, if S is a conformal subdivision of T , we also say that T is a conformal amalgamation of S.

If τ is a subdivision operator, let τT denote any geometric subdivision of T such that KτT =

τKT . If it is possible to choose τT such it is a conformal subdivision of T , then the operator τ is said to be conformal. For the remainder of this section, if τ is known to be conformal, the notation τT should always be understood to be the (unique) conformal subdivision of T in the pattern of τ.

38 Conformal subdivision operators are in fact quite common, as the next theorem shows.

Theorem 4.3. If τ is a dihedrally symmetric simple inﬁnite subdivision operator, then it is conformal.

We refer the reader to [3] for the details of the proof, but we would like to briefly sketch the argument, as it is applicable to the next section. To prove this, one equips each seed τn with ⋆ ⋆ the star-equilateral structure, to obtain a conformally regular model n-gon |tn|eq. That |tn|eq is conformally regular follows from Theorem 2.3 because it admits a rotational symmetry. Hence, for ⋆ each tile t of a conformal tiling TK , there exists a homeomorphism ht : |tn|eq → t that is conformal on the interior, and identifies the vertices. By applying this map to every tile t, we obtain a tiling of the same geometry that is conformally regular, and reflective on the subtiles of each t. It remains to be seen that this tiling is also reflective across subedges of the original tiles, but it can be shown that this follows because the subdivision is applied in a simple, dihedrally symmetric pattern. Conformal hierarchies are hierarchies of tilings (not merely combinatorics), in which one can pass through the hierarchy by iterated conformal subdivision/amalgamation.

Deﬁnition 4.6. The conformal tiling T exhibits a conformal hierarchy if there is a bi-inﬁnite sequence {Tn : n ∈ Z}—called a conformal hierarchy for T —of conformal tilings indexed by the integers such that the following three conditions hold:

1. T0 = T ;

2. Tn+1 is a conformal subdivision of Tn, ∀n ∈ Z;

3. Tn+1 ∼ Tn, ∀n ∈ Z (i.e. KTn+1 is locally isomorphic to KTn ).

Examples of conformal hierarchies abound. The square grid tiling Z that we have considered several times already is a trivial example of a tiling that admits a conformal hierarchy, simply by applying the quad subdivision operator to conformally subdivide each tile into four more equal squares, and amalgamating patches of four tiles that meet at a vertex into a single supertile. The result of either of these operations is clearly still a conformal tiling, as it is just a rescaling of the original tiling Z. The Bowers-Stephenson pentagonal tiling shown in Figure 2.1 is another tiling that admits a conformal hierarchy. While we don’t prove this at the moment, a visual inspection of the tiling—which we urge the reader to do—strongly suggests that one can conformally subdivide

39 and amalgamate tiles in the pattern of the pentagonal subdivision operator, resulting in yet another conformal tiling that is locally isomorphic to the original. Indeed, it is this very image that motivates the entire concept of a conformal hierarchy. A subdivision operator τ is said to manifest a conformal hierarchy for the tiling T if T exhibits a conformal hierarchy {Tn} with the property that τTn = Tn+1, for every n ∈ Z. In particular, this means that τ must be a conformal subdivision operator in order to manifest a conformal hierarchy. The next theorem, due to [3], shows more:

Theorem 4.4. Let τ be a dihedrally symmetric simple inﬁnite subdivision operator that manifests a combinatorial hierarchy {Kn} for the planar polygonal complex K. Then τ is a conformal subdi- ∼ vision operator, and manifests a corresponding conformal hierarchy {Tn}, where KTn = Kn for all n ∈ Z.

By the deﬁnition of conformal hierarchies it is clear that a conformal hierarchy {Tn} is simply another layer of structure on top of an underlying combinatorial hierarchy {Kn = KTn }. We say that {Tn} is a conformal realization of {Kn}. Hence, many of the concepts from combinatorial hierarchies can be recovered in the setting of conformal hierarchies simply by identifying combinatorial faces of K with their corresponding tiles in TK . For example, if c is a core of K, we say that its corresponding patch of tiles |c| in TK is a core of TK . Similarly, we say that the core

|c| of TK engulfs a compact set D if D ⊂ |c|. Hence, we say that a conformal hierarchy {Tn} is (strongly) expansive if it is a conformal realization of a (strongly) expansive combinatorial hierarchy {Kn}. As one might hope, it can be shown that expansive conformal hierarchies have the property that any compact subset of the plane/disk (depending on the tiling’s type) can be engulfed by a core of some Tn (n ≤ 0). That is, compact subsets can be engulfed by cores of tiles simply by amalgamating the tiles in the hierarchy enough times.

4.2.3 Fractal Hierarchies

Recall that a tile is a closed Jordan domain with finitely many distinguished boundary points, and a tiling is a collection of tiles with disjoint interiors that meet only along full edges or along vertices, and whose union is the whole space, either C or D. Until now, we have often flippantly referred to conformal tilings as just tilings, because these were the only types of tilings we were interested in. In this section we will have reason to consider arbitrary (locally finite) tilings, so

40 we will be careful to mention explicitly whether a tiling is conformal or not. We introduce a new type of hierarchy of tilings in this section—a fractal hierarchy—that is not nearly as restrictive as conformal hierarchies, but still seeks to preserve the properties of conformal hierarchies that are used in answering the type problem. The ﬁrst of these properties is that conformal hierarchies are sequences of conformal tilings where subdivision and amalgamation can be performed in situ on the conformal tilings themselves, so that the tiles can be geometrically subdivided, yielding the next conformal tiling in the hierarchy. But it is not always the case that combinatorial subdivisions and geometric subdivisions have to mesh together nicely. That is, if τ is an operator, in general it is not the case that there exists a geometric subdivision τTK such that τTK and the conformal tiling TτK will be conformally equivalent.

Deﬁnition 4.7. Let {Jn : n ∈ Z} be a bi-inﬁnite sequence of tilings of G ∈ {C, D} such that the sequence {Kn = KJn : n ∈ Z} is a combinatorial hierarchy of planar polygonal complexes.

The sequence {Jn} admits geometric subdivisions if, for every n ∈ Z, there exists a geometric subdivision σnJn of Jn such that σnJn = Jn+1.

As we have seen, conformal hierarchies admit geometric subdivisions, and this is the ﬁrst property on which we wish to model our fractal hierarchies. In other words, we would like fractal hierarchies to be sequences of tilings such that one may pass through the hierarchy by carrying out geometrically correct subdivisions and amalgamations in situ on the tilings. The second feature of conformal hierarchies that we wish to mimic is the following: Theorem 2.2 tells us that combinatorial equivalence of patches in conformal tilings guarantees that these patches will in fact be conformally equivalent. But because conformal hierarchies provide us with a way to conformally subdivide and amalgamate the conformal tiling itself, this property extends across

′ ∼ ′ the entire hierarchy. That is, if L and L are patches of tiles of Tn and Tm such that KL = KL , then L and L′ are conformally equivalent. We would like this to also be a feature that the tilings in fractal hierarchies exhibit.

Deﬁnition 4.8. A bi-inﬁnite sequence of tilings {Jn : n ∈ Z} of G ∈ {C, D} is conformally ′ unique on patches if, for any two patches L and L of Jp and Jq, if the complexes KL and KL′ are combinatorially equivalent, then L and L′ are conformally equivalent.

41 In Section 4.3 we will see that these two properties turn out to be the key features of conformal hierarchies used in answering the type problem for conformal tilings which admit conformal hierarchies. By ensuring that fractal hierarchies have the same properties, we can use similar methods to answer the type problem in this setting as well. For now, this motivates our deﬁnition of a fractal hierarchy.

Definition 4.9. The conformal tiling T exhibits a fractal hierarchy if there exists a bi-infinite sequence {Jn : n ∈ Z}—called a fractal hierarchy for T —of tilings indexed by the integers such that the bi-infinite sequence {Kn = KJn : n ∈ Z} is a combinatorial hierarchy of planar polygonal complexes, and such that the following four conditions hold:

1. T = TK0 ;

2. T and J0 are both tilings of G ∈ {C, D};

3. {Jn} admits geometric subdivisions;

4. {Jn} is conformally unique on patches.

Note that this deﬁnition implies several facts about {Jn}. First, because {Jn} admits geometric subdivisions, and because J0 is a tiling of G, it follows that Jn is also a tiling of G, for every n.

Secondly, because {Kn = KJn } is a combinatorial hierarchy, we have Jn+1 ∼ Jn for every n— i.e. KJn+1 is locally isomorphic to KJn . Finally, because T = TK0 , it follows that T and J0 are ∼ combinatorially equivalent. That is, KT = KJ0 . The following theorem—whose proof is trivial—observes that fractal hierarchies are indeed generalizations of conformal hierarchies. On the other hand, we will soon see that exhibiting a conformal hierarchy is in fact strictly stronger than admitting a fractal hierarchy. There exist conformal tilings that admit fractal hierarchies but do not admit conformal hierarchies; the results of Chapter 5 provide straightforward techniques for constructing such tilings.

Theorem 4.5. If T is a conformal tiling that exhibits a conformal hierarchy {Tn}, then T exhibits a fractal hierarchy {Jn}. Moreover, {Tn} itself is a fractal hierarchy for T .

A subdivision operator τ is said to manifest a fractal hierarchy for the conformal tiling

T if T exhibits a fractal hierarchy {Jn} with the property that τKJn = KJn+1 for every n ∈ Z.

42 Alternatively, because fractal hierarchies admit geometric subdivisions, there exists a geometric subdivision τJn such that τJn = Jn+1, for every n ∈ Z.

Because a fractal hierarchy {Jn} for T always comes equipped with an underlying combinatorial hierarchy {Kn}, this induces a bi-inﬁnite sequence of conformal tilings {Tn = TKn }. We call the fractal hierarchy {Jn} a fractal realization of either {Tn} or {Kn}. As we had with conformal hierarchies, once again in the setting of fractal hierarchies many of the concepts from combinatorial hierarchies reappear simply by identifying combinatorial faces of K with their corresponding tiles in JK . For example, if c is a core of K, we say that its corresponding patch of tiles |c| in JK is a core of JK . Similarly, we say that the core |c| of JK engulfs a compact set D if D ⊂|c|. Hence, we say that a fractal hierarchy {Jn} is (strongly) expansive if it is a fractal realization of a

(strongly) expansive combinatorial hierarchy {Kn}.

Theorem 4.6. If the fractal hierarchy {Jn} for the conformal tiling T of G ∈ {C, D} is expansive, then every compact subset D ⊂ G is engulfed by a core of one of the the tilings Jn for some value of n ≤ 0.

Proof. Since J0 is a locally ﬁnite tiling of G and D is compact, only ﬁnitely many tiles of J0 meet

D and there is a ﬁnite subcomplex F of KJ0 = K0 such that |JF | ⊃ D. Since the hierarchy is expansive, there is an integer m ≤ 0 such that F is engulfed by a core of KJm = Km. This means 0 that there is a core c of Km such that the subcomplex σmc of K0 contains the complex F . Since F

0 0 is a subcomplex of σmc, |JF | is a subset of |Jσmc|. But because {Jn} is a fractal hierarchy, J0 is a

0 0 a geometric subdivision of Jm. That is, we have |c| = |Jσmc|. Thus, D ⊂ |JF |⊂|Jσmc| = |c|, and D is engulfed by |c|.

We have yet to motivate the name fractal hierarchy, nor is it clear how—other than by taking conformal hierarchies—one can observe these fractal hierarchies. The idea comes from an experimental observation ﬁrst made by Cannon, Floyd, and Parry for certain tilings. They observed that, for some example conformal tilings, combinatorial subdivision by rotationally symmetric operators, although not conformal at the level of the tiling, resulted in new tilings that ﬁxed the vertices of the original tiles in place. It is this observation that we will use to build the so called fractal realizations of appropriate conformal tilings.

43 Let S and T be two tilings of G ∈ {C, D} such that KS =∼ KT , i.e. S and T are combinatorially equivalent. If there exists an orientation-preserving isometry of G that identifies the vertices of S with the vertices of T , then we say that S and T agree on vertices. In the case that two tilings agree on vertices, we will always assume that the tilings are positioned so that no such rigid transformation need actually be applied. In other words, if S and T agree on vertices, then the only difference between them is that the edges of the individual tiles may trace out different paths.

Let {Tn : n ∈ Z} be any sequence of tilings (not necessarily conformal) such that the sequence

{Kn = KTn } is a combinatorial hierarchy of planar polygonal complexes. Given a tiling Tn+1 in ′ the sequence, the individual tiles can be amalgamated in the pattern of σn to produce a tiling Tn ′ that is combinatorially equivalent to Tn. If Tn and Tn agree on vertices for every n ∈ Z then we say that the sequence of tilings {Tn} has ﬁxed vertices. For example, every conformal and fractal hierarchy will have ﬁxed vertices because they admit geometric subdivisions.

Now suppose {Tn} is a sequence of conformal tilings that has fixed vertices, and such that the sequence {Kn = KTn } is a combinatorial hierarchy of planar polygonal complexes. We would like to extract a sort of limit structure from this sequence. Let e0 be an edge of a tile t in T0, with end- point vertices v and w. The edge e0 is an analytic arc in G ∈ {C, D} from v to w. Per the previous paragraph, we have seen that we can amalgamate the tiles of T1 in the pattern of σ0 to produce a ′ tiling T0 that is combinatorially equivalent to T0, and because the sequence {Tn} has fixed vertices, ′ we have the corresponding edge e1 in T0 with end-point vertices v and w. The edge e1 consists of the edges of actual conformal tiles in T1, and hence will in general be a piecewise analytic arc, with a finite number of turning angles at the points a1,...,am, which are all vertices of tiles in T1. We 2 can repeat this process again. That is, amalgamate the tiles of T2 in the pattern of σ0 to produce ′′ a tiling T0 that is combinatorially equivalent to T0, and we will have the corresponding edge e2 with end-point vertices v and w. The edge e2 is once again a piecewise analytic arc, with a finite number of turning angles now not only at the points a1,...,am, but also at finitely many points inbetween each successive pair in the sequence v,a1,...,am,w. Continuing ad infinitum, we call this process, and the infinite sequence {en : n = 0, 1, 2, · · ·} of piecewise analytic arcs obtained, the fractal transition of the edge e. Of course, instead of focusing on just individual edges, one may look at an individual tile t under this process, the obtained sequence of Jordan domains being the

44 fractal transition of t, or even still at the entire tiling T0, the obtained sequence of tilings being the fractal transition of T0. −→ −→ If {en} converges in Hausdorﬀ distance to an arc e , we call e the fractal limit of the edge e. Given a sequence of conformal tilings {Tn} that has ﬁxed vertices, and such that the sequence −→ {Kn = KTn } is a combinatorial hierarchy, let the tiling Jn be the tiling that agrees on vertices with Tn, and whose edges are the fractal limits of the edges of Tn, assuming that they exist. We −→ call Jn the fractal limit of Tn. Perhaps the most important observation here is that, by this −→ process, we are guaranteed that the sequence {Jn : n ∈ Z} of fractal limit tilings will necessarily admit geometric subdivisions. Figures 4.3–4.5 demonstrates the fractal transition process, and the resulting fractal limit of a conformal tiling.

Admittedly, a problem arises. Showing that {en} converges in Hausdorﬀ distance to an arc is not easy in general. Indeed, we believe that it always does, and experimental evidence such as the tiling shown in Figure 4.5 strongly suggests that this is the case. Regardless, we will suggest two open conditions on tilings that will imply convergence of {en} to an arc, with the understanding that Theorem 4.7 is true under the assumption that these two conditions are true.

Let {Tn} be a sequence of conformal tilings that has ﬁxed vertices, and such that the sequence

{Kn = KTn } is a combinatorial hierarchy of planar polygonal complexes.

(⋆) Edge Containment – Let e′ be the ﬁrst stage in the fractal transition of an edge e. For every ′ n ∈ Z, and for every edge e of Tn, the edge e is contained in the union of the two vertex cores of the endpoint vertices of e.

(⋆⋆) Geometrically Shrinking – Let t be a tile of T0, and let δn(t) be the largest diameter of the

tiles of the nth subdivision of t. For every tile t ∈ T0, δn(t) → 0 as n → ∞.

Theorem 4.7. Let τ be a rotationally symmetric inﬁnite subdivision operator that manifests a combinatorial hierarchy {Kn} for the planar p-gon complex K with bounded degree. Then τ man- ∼ ifests a corresponding fractal hierarchy {Jn}, where KJn = Kn for all n ∈ Z. In particular, the sequence of conformal tilings {TKn } admits ﬁxed points, and each Jn in the fractal hierarchy is the fractal limit of the conformal tiling TKn .

Proof. Let τ be a rotationally symmetric inﬁnite subdivision operator that manifests a combinatorial hierarchy {Kn} for the planar p-gon complex K with bounded degree. The proof is divided into three parts.

45 Figure 4.3: The twisted pentagonal tiling P.

Figure 4.4: The ﬁrst four stages of the fractal transition of single tile of P.

−→ Figure 4.5: The fractal limit P of the twisted pentagonal tiling P.

46 PART 1: Fixed Vertices.

In this part of the proof we show that the sequence of conformal tilings {Tn = TKn : n ∈ Z} has fixed vertices. To do this, we first construct a branched covering space K˜ that will be used repeatedly throughout this proof. The construction of K˜ is straightforward: because K has bounded degree, we can find an integer r 6= p such that the degree of each vertex of K divides r, and such that 1/2 + 1/p + 1/r < 1. Let K˜ be the planar p-gon complex of constant degree r, and let f : K˜ → K be a branched covering map of K, branching only over the vertices of K, such that if v is a vertex of K of degree q, the degree of f over v is r/q. Of course, we have chosen the orientation of K˜ to be such that it is preserved under the map f. The first observation we will make is that the triangle group ∆(2,p,r) generates, after amalgamating around vertices of degree 2p, a conformal tiling of the disk D by regular hyperbolic p-gons with interior angles 2π/r. This follows because r is chosen so that 1/2 + 1/p + 1/r < 1. In particular, since the conformal tiling TK˜ is combinatorially equivalent to this amalgamated tiling by the triangle group ∆(2,p,r), Theorem 2.1 implies that TK˜ is also a tiling of D by regular hyperbolic p-gons with interior angles 2π/r. Our covering complex K˜ has constant degree and only p-gon faces, implying that rotations by 2π/r around every vertex of K˜ are automorphisms of K˜ . Let G be the group generated by these ˜ rotations by 2π/r, with a generator for each vertex of K. Considering now a tile t of TK˜ , the action ˜ of G on K descends to conformal rotations of the tiling TK˜ , and because t is a regular hyperbolic p-gon we have that its vertex set V is stabilized by a cyclic subgroup of G of order p. Now, consider the combinatorial subdivision τK˜ . Because τ is rotationally symmetric, it follows that G still acts as a group of automorphisms on τK˜ . In particular, these again descend to conformal rotations on the conformal tiling TτK˜ . Hence, in TτK˜ the set V is still stabilized by a cyclic subgroup of G of order p, and it follows that the set V are the vertices of a regular hyperbolic p-gon t′. Moreover, since the vertices in V are all individually stabilized by a cyclic subgroup of G of order r, the interior angles at the vertices of this p-gon t′ are 2π/r, implying that both t′ and the tile t are in fact congruent regular hyperbolic p-gons. From this observation, it follows that a geometric amalgamation of TτK˜ in the pattern of τ produces a tiling of D that agrees on vertices with TK˜ .

47 Recall that f : K˜ → K is a branched covering map of K, branching only over the vertices of K. ′ This induces a branched covering map f : TK˜ → TK that branches only over the vertices of the ′ tiles of TK . In particular, f is conformal away from these branch points. By considering the image of the tiles of TτK˜ , the observation of the above paragraph—that a geometric amalgamation of

TτK˜ in the pattern of τ produces a tiling that agrees on vertices with TK˜ —implies that there exists an analogous geometric amalgamation of Tτ K in the pattern of τ produces a tiling that agrees on vertices with TK . One can repeat this argument indeﬁnitely. That is, using our branched covering space K˜ again, one can take a branched covering of the complex τK˜ to show that a geometric amalgamation of ′ Tτ 2K˜ in the pattern of τ produces a tiling of D that agrees on vertices with TτK˜ , and under f this again shows that a geometric amalgamation of Tτ 2K in the pattern of τ produces a tiling that agrees

n on vertices with TτK . Repeating ad inﬁnitum, it follows that the sequence {Tn = TKn = Tτ K } has ﬁxed vertices. PART 2: The Fractal Limit.

Because {Tn} has ﬁxed vertices, for every edge, tile, and tiling in the sequence we can construct their fractal transitions. In this part of the proof we show that these fractal transitions converge, so that the corresponding fractal limits exist.

Let e be an edge of the tiling T0 with end-points v and w, and let {en} be its fractal transition.

Let Bn be the union of all the tiles of Tn that meet en. Each Bn is a continuum, and condition (⋆) guarantees that we have a nested sequence of continua B0 ⊃ B1 ⊃ B2 ⊃···. It is a standard result ∞ that B = ∩i=0Bi is also a continuum, and condition (⋆⋆) guarantees that removal of any point of B (other than the end-points v and w) leaves a path-disconnected space remaining. It follows that

B is necessarily an arc, implying that the fractal transition sequence {en} converges in Hausdorﬀ distance to its fractal limit −→e . We should also note that tiles converge to their fractal limits as well. Indeed, if t is a tile, then every tile tn in its fractal transition {tn} is conformally regular by Theorem 2.3, precisely because it is rotationally symmetric. Let fn : t → tn be a conformal homeomorphism that extends continuously to the boundary. There exists a subsequence which converges uniformly on compact sets to a conformal map f onto a domain Ω. In particular, by the Carath´eodory kernel theorem,

Un = fn(t) converges to its kernel. But conditions (⋆) and (⋆⋆) imply that the kernel is precisely

48 the domain whose boundary is given by the fractal limits of the edges of t, which we have shown converge to arcs. −→ Hence, for every tiling Tn in the sequence, there exists a corresponding fractal limit tiling Tn. −→ We claim that the sequence {Tn : n ∈ Z} is a fractal hierarchy for T . Indeed, properties (1)–(3) of being a fractal hierarchy are satisﬁed, and property (4) is all that remains to be shown. That is, −→ we need to show that {Tn} is conformally unique on patches. PART 3: Conformally Unique on Patches. −→ ˜ Let TK˜ be the fractal limit of our covering space K, which exists by part 2 of the proof. Let L ′ −→ −→ and L be two combinatorially equivalent patches of Kn and Km respectively, and let TL and TL′ −−→ −−→ be their corresponding patches in the fractal limit tilings TKn and TKm . We wish to show that the ′ −→ −→ combinatorial isomorphism φ : L → L induces a conformal homeomorphism between TL and TL′ .

Let be be the union of the two closed faces that meet along an interior edge e of L, and let te −→ −−→ ′ ′ be the union of the two corresponding tiles of TL ⊂ TKn . Let be and te be the analogous objects −→ −−→ ′ ′ ′ ′ in L and TL ⊂ TKm , respectively, for the corresponding edge e of L .

Finally, lete ˜ be any lift of e to K˜ , and let f˜1 and f˜2 be the two faces of K˜ that are incident ˜ ˜ ˜ ˜ ˜ −→ ˜ alonge ˜. Then be = f1 ∪ f2 is a lift of be to K. Let te be the tiles of TK˜ corresponding to be. We have that t˜e is a lift of te. Similarly, we can lift be′ to a pair of adjacent faces b˜e′ in K˜ , giving a lift of te′ to a pair of adjacent tiles t˜e′ . The group G of rotations of K˜ acts transitively on the vertices of K˜ , so there exists an element ge of Aut(K˜ ) such that g(b˜e) = b˜e′ . In particular, ge can be chosen to preserve the combinatorial ′ −1 isomorphism between be and be—i.e. so that the map ψe = f ◦ ge ◦ f (where f is the branched covering map) commutes with φ|be . This map ψe then induces a conformal homeomorphism Ψe : te → te′ . −→ All that remains to be shown is that Ψe can be extended to all of TL, and we do this by building a chart for every interior edge of L in such a way that charts of overlapping edge cores agree on overlaps. That is, let s 6= e be another edge of the face f1 such that s is on the interior of L. By the same process we can lift bs to a subcomplex b˜s of K˜ consisting of two adjacent faces. There exists an element gs ∈ G, such that gs(b˜s) and b˜e agree on lifts of f1, and in particular on lifts of −1 e and h. It follows that the map ψs = f ◦ ge ◦ gs ◦ f commutes with φ|bs , and this map induces a conformal homeomorphism Ψs : ts → ts′ that agrees on overlaps with Ψe. Continuing in this

49 manner, choosing appropriate charts for each edge core, gives rise to a conformal homeomorphism −→ −→ Ψ: TL → TL′ .

4.3 Type

In proving that various tilings are parabolic we will rely heavily on a characterization of parabolic surfaces that arises from the theory of quasiconformal mappings. This characterization—stated in Theorem 4.8—avers that a non-compact, simply connected Riemann surface S is parabolic if and only if every compact subset of S can be separated from inﬁnity by annuli that do not become arbitrarily thin in modulus. We review the necessary background material below. A ring domain C is a doubly connected subset of the sphere C. That is, it is an open topological annulus in the plane. For each ring domain C there is a conformal homeomorphism mapping C to an actual round annulus A(r, R)= {z ∈ C : 0 ≤ r< |z|

Theorem 4.8. A non-compact, simply connected Riemann surface S is parabolic if and only if there exists a constant µ> 0 such that, for every compact subset E of S, there is a ring domain C separating E from the ideal boundary of S such that Mod(C)≥ µ.

The proof of this result is straightforward, and for the details we defer to the standard source, Lehto and Virtanen [8], but it follows from two observations. The ﬁrst is that a simply connected, non-compact Rieman surface S is parabolic if and only if S \ D, where D is a closed disk in S, is conformally equivalent to a ring domain C with Mod(C)= ∞. The second is the superadditivity of the modulus—i.e. if C1,C2,... is a sequence of pairwise disjoint ring domains that are subdomains

50 of the ring domain C, with every Cn separating the boundary components of C from one another, then ∞ X Mod(Ci) ≤ Mod(C) i=1 Returning to the problem of determining the type of conformal tilings, it is shown in [3] that expansive conformal hierarchies of conformal tilings with bounded degree are necessarily hierarchies of parabolic tilings. Moreover, if that hierarchy is additionally strongly expansive, then parabolicity is a property not just of the tilings in the hierarchy, but of their entire local isomorphism class. We summarize the results below.

Theorem 4.9. If T is a conformal tiling of bounded degree that exhibits an expansive conformal hierarchy, then T is parabolic and tiles the plane C.

Theorem 4.10. Let T be a conformal tiling of bounded degree that exhibits a strongly expansive ′ conformal hierarchy. Then any conformal tiling T whose complex KT ′ is locally isomorphic to KT is parabolic and tiles the plane C.

Corollary 4.11. Let τ be a shrinking, dihedrally symmetric simple subdivision operator that manifests a combinatorial hierarchy for a planar polygonal complex K of bounded degree. Then τ manifests a conformal hierarchy for TK , and TK is necessarily parabolic and tiles the plane C. Moreover, conformal type is constantly parabolic across the local isomorphism class (K).

These follow by showing that one can separate compact subsets of the conformal tiling from in- ﬁnity by one of ﬁnitely many conformally equivalent ring domains, hence satisfying the hypotheses of Theorem 4.8. That one can do this follows from the fact that combinatorially equivalent patches of tilings in a conformal hierarchy are necessarily conformally equivalent; that is, conformal hierarchies are conformally unique on patches. Because this is a property shared by fractal hierarchies, we use the same method of proof in the analogous theorem below.

Theorem 4.12. If T is a conformal tiling of bounded degree that exhibits an expansive fractal hierarchy, then T is parabolic and tiles the plane C.

Proof. Let T = TK be a conformal tiling of bounded degree, so that its planar n-gon complex K has bounded degree, and let c be a core of K. Furthermore, let B(c) be the subcomplex of K that

51 consists of the faces of the unbounded complementary domain of c that meet K. We will call B(c) the combinatorial collar of the core c. Let {Jn} be a fractal hierarchy for T . In J0 we would like to view the corresponding geometric patch |B(c)| in as an annulus separating |c| from infinity, but the situation need not be so simple. That is, the bounded complementary domain of |B(c)| may consist of several different components, and not just one. Nevertheless, in what follows we will extract an annulus from |B(c)| in a unique way. Since B(c) is a finite subcomplex of K, it follows that |B(c)| is at most finitely connected, and hence by Koebe’s General Uniformization Theorem there exists a conformal homeomorphism Γ : |B(c)| → U(c), where U(c) is a circle domain—i.e. the complement of a finite number of closed round disks in the plane. Let D(c) be the disk in the complement that contains the core |c|. By applying an appropriate Möbius transformation we may center the disk D(c) at origin, and we may map another of the disks of the complementary domain to the exterior of the disk D. Therefore, we may assume that U(c) is a circle domain contained in D with the core c contained in the complementary disk D(c) that is centered at the origin. There is a largest annulus A(c) ⊂ U(c) that encircles D(c) and separates D(c) from the rest of the boundary of U(c). Call R(c) = Γ−1(A(c)) ⊂ |B(c)| the standard collar of the core |c|. Now, because K has bounded degree, there are only finitely many different combinatorial types of cores, and moreover, again because K has bounded degree, this implies that there exist only

finitely many different combinatorial collars B1,...,Bm. In other words, if c is a core of K, we must have that its collar B(c) is equivalent to one of B1,...,Bm. Finally, because {Jn} is a fractal hierarchy, the patch |B(c)| of |c| is conformally equivalent to one of finitely many domains

|B1|,..., |Bm|, and therefore we have that the standard collar R(c) for |c| is conformally equivalent to one of R1,...,Rm. ′ Let c be any core of a planar polygonal complex Kp in the combinatorial hierarchy {Kn = KJn }. Then c′ must be combinatorially equivalent to one of the ﬁnitely many that appear in K, and its ′ ′ combinatorial collar B(c ) must be one of B1,...,Bm, precisely because B(c ) embeds in K as a combinatorial collar of a core c of K. Because {Jn} is a fractal hierarchy, we have that the ′ ′ geometric realization |B(c )| in Jp is conformally equivalent to |B(c)| in J0, implying that |B(c )| ′ is conformally equivalent to one of |B1|,..., |Bm|, and therefore that the standard collar R(c ) for ′ |c | is conformally equivalent to one of the list R1,...,Rm.

52 Let µ > 0 be the smallest modulus of a ring domain from the list R1,...,Rm. Let D be a compact subset of the underlying space G of J0. Because the hierarchy is expansive, by Theorem 4.6 there exists a number of amalgamations n ≥ 0 neccessary such that D is engulfed by a core

|c| of the tiling J−n. Hence, D is separated from inﬁnity by the ring domain R(c), and by the observations of the previous paragraph we have that R(c) ≥ µ > 0. The hypotheses of Theorem

4.8 have thus been satisﬁed, completing the proof that J0—and hence T —is parabolic and tiles the plane C.

If the fractal hierarchy is strongly expansive, then parabolicity is a property not just of any one tiling in the hierarchy, but of its entire local isomorphism class.

Theorem 4.13. Let T be a conformal tiling of bounded degree that exhibits a strongly expansive ′ fractal hierarchy. Then any conformal tiling T whose complex KT ′ is locally isomorphic to KT is parabolic, and tiles the plane C.

Because fractal hierarchies admit geometric subdivisions and are conformally unique on patches, the proof is an immediate generalization of the proof for conformal hierarchies as in Theorem 4.10 given by Bowers and Stephenson. As such, due to its length, and its use of a technical lemma not discussed in this paper, we do not rehash the argument here, instead referring the reader to [3] for the details. The next corollary combines the results of this chapter, and gives combinatorial conditions that guarantee constancy of type across local isomorphism classes.

Corollary 4.14. Let τ be a shrinking, rotationally symmetric subdivision operator that manifests a combinatorial hierarchy for a planar n-gon complex K of bounded degree. If conditions (⋆) and

(⋆⋆) are satisﬁed, then τ manifests a fractal hierarchy for TK , and TK is necessarily parabolic and tiles the plane C. Moreover, conformal type is constantly parabolic across the local isomorphism class (K).

Proof. Theorem 4.7 implies that the conformal tiling T0 = TK exhibits a fractal hierarchy {Jn} ∼ for which KJn = Kn for all n ∈ Z, where {Kn} is a combinatorial hierarchy for K manifested by

τ. Theorem 4.1 implies that the hierarchy {Kn}, and hence the fractal hierarchy {Jn}, is strongly expansive. Theorem 4.12 implies that TK is parabolic, tiling the plane C. Finally, Theorem 4.13

53 implies that type is constantly parabolic across the local isomorphism class (K), ﬁnishing the proof.

54 CHAPTER 5

EXPANSION COMPLEXES

The planar polygonal complexes from which we build our conformal tilings are typically constructed as an expansion complex for an associated finite subdivision operator. In this chapter we define finite subdivision operators, we introduce expansion complexes, and we show that expansion complexes have many of the desirable properties of planar polygonal complexes that are studied in the previous chapters, including repetitivity, FLC, admitting combinatorial hierarchies, etc. In particular, we show that expansion complexes for appropriate rotationally symmetric finite subdivision operators admit a fractal heirarchy, and are therefore parabolic, generalizing the similar result of Bowers and Stephenson in [3] for dihedrally symmetric finite subdivision operators. We then close the chapter, and indeed the paper, with a short addendum demonstrating how one can build the discrete hyperbolic plane complex as an expansion complex, albeit using more general finite subdivision operators than those allowed here.

5.1 Finite Subdivision Operators

A finite subdivision operator τ is a finite collection of seeds τn such that each τn is a regular, oriented, locally-finite CW-decomposition of the combinatorial n-gon ∆n. That is, τ is the collection {τn : n ≥ 3,n ∈ Ω}, where Ω is some finite subset of Z. Furthermore, we require that for every seed τn ∈ τ, if f is an m-gon face of τn, we have m ∈ Ω. This is of course exactly the same as an infinite subdivision operator, with the only difference being that an infinite subdivision operator has a seed for every n, while a finite subdivision operator has a seed for only finitely many values of n, and that these seeds contain faces from a finite list of polygonal types. If the finite subdivision operator τ is given by τ = {τn : n = n1,...,nk}, we say that τ isa(n1,...,nk)-finite subdivision operator. Alternatively, we may simply say that τ is a finite subdivision operator with k face types. We have all the same notions in this setting as we had before with infinite subdivision operators, defined analogously. To recall, these are rotational symmetry, edge-compatibility, and total non-

55 triviality. Again, as before, unless otherwise stated a ﬁnite subdivision operator τ will be assumed to have all three of these properties.

As with the infinite subdivision operators, a (n1,...,nk)-finite subdivision operator τ can be used to provide a totally non-trivial subdivision τK of a complex K, where τK is given simply by partitioning the faces of K according to their polygonal type, and subdividing the faces of each partition of K according to the appropriate seeds of τ. There is of course a key difference here between finite subdivision operators and infinite subdivision operators: An infinite subdivision operator τ defines a subdivision τK for any and every complex K, meanwhile a (n1,...,nk)-finite subdivision operator τ defines a subdivision τK for only planar (n1,...,nk)-gon complexes. In this case, since τK is again a (n1,...,nk)-gon complex, we can iterate this process to define the complex τ nK as before.

Lastly, let τ be a (n1,...,nk)-ﬁnite subdivision operator. The substitution matrix M for τ is the k × k matrix M deﬁned by

Mij = the number of faces of polygonal type ni in τnj

N If M is primitive—that is, if there exists an N such that [M ]ij > 0 for each i,j ∈ {1,...,k}— then we say that the finite subdivision operator τ is primitive. Note that the entry Mij represents N the number of faces of type ni in the τ subdivision of the j-gon ∆j, and hence the entry [M ]ij is N the number of faces of type ni in the iterated τ subdivision of ∆j. The motivation behind this definition is to provide a condition that guarantees that if any face type is successively subdivided enough times, it will eventually contain all possible face types. Primitivity is almost always imposed on the substitution rules of classical aperiodic tilings, and we will likewise generally restrict our attention to only primitive finite subdivision operators. Of course, given a finite subdivision operator, it can potentially be very difficult to check for primitivity. We avoid this in practice by constructing finite subdivision operators whose seeds always have the following properties:

1. τni contains at least one face of type ni+1, for 1 ≤ i

2. τnk contains at least one face of type n1;

3. τn1 contains at least one face of type n1.

56 This gives a substitution matrix M such that M11 ≥ 1, M1k ≥ 1, and every entry on the subdiagonal is also positive. Clearly such a matrix is primitive.

5.2 Expansion Complexes

Finite subdivision operators are typically used in the context of a single starting combinatorial n-gon ∆. With an appropriate ﬁnite subdivision operator τ (i.e. one that contains an n-gon seed), we can subdivide ∆ to obtain the subdivision τ∆. Applying it successively, we have the inﬁnite sequence of subdivisions {∆j} where ∆0 = ∆ and ∆j+1 = τ∆j, for j = 0, 1, ···. An expansion complex for τ will be a limiting planar complex extracted from nested patches within the sequence

{∆j}, and therefore will in particular contain as subcomplexes inﬁnitely many copies of ∆j for every j.

Given a (n1,...,nk)-ﬁnite subdivision operator τ = {τn1 ,...,τnk }, let

···→֒ F1 ֒→ F2 ֒→··· ֒→ Fm ֒→ Fm+1 (‡) be any sequence of isomorphic embeddings of CW complexes that satisﬁes the following properties:

1. For each positive integer m, Fm is a connected subcomplex of ∆im , for some positive integer

im, with im

Each map Fm ֒→ Fm+1 is a cellular, orientation-preserving isomorphic embedding of CW .2 complexes;

3. For each m, there exists a positive integer p for which the image of Fm under the composition

;Fm ֒→··· ֒→ Fm+p is contained in the interior of Fm+p

4. For each m and each combinatorial simple closed edge path γ in Fm, there exists a positive integer q for which the image of γ under the composition Fm ֒→ ··· ֒→ Fm+p bounds a

combinatorial disc in Fm+q.

Since the chosen sequence {Fm} is a nested sequence of complexes, it has a directed limit complex

K = lim Fm. The reader should note that in the above sequence the original combinatorial n-gon −→

∆ (where n is any of n1,...,nk), called the seed of the expansion complex, is often essentially immaterial—if τ is primitive, the same limit complex can be built from a diﬀerent seed choice simply by shifting the index.

57 Deﬁnition 5.1. The CW complex K = lim Fm, the direct limit of the system (‡), is called an −→ expansion complex for the ﬁnite subdivision operator τ.

Expansion complexes defined as above are far more general than the expansion complexes studied by Canon, Floyd, and Parry in [5] and [6]. Indeed, their expansion complexes are required to come equipped with an expansion map—essentially an isomorphism between an expansion complex K and its subdivision τK—which is not a feature our expansion complexes will typically exhibit. In fact, Bowers and Stephenson show that only finitely many expansion complexes for a given finite subdivision operator can admit such an expansion map [3]. Finally, we note that we can of course obtain more general complexes than defined above by relaxing properties (3) and (4), but for our purposes we want the limit complex defining our expansion complexes to have certain topological properties that these conditions guarantee, as the following theorem shows.

Theorem 5.1. If τ is a primitive (n1,...,nk)-ﬁnite subdivision operator then the expansion complex K = lim Fm is a planar (n1,...,nk)-complex. −→

Proof. Property (3) guarantees that the topological space |K| underlying the CW complex K is a non-compact topological 2-manifold. Property (4) guarantees that |K| is simply connected. It follows that |K| is homeomorphic to the plane, so that K is a CW-decomposition of the plane.

im Finally, because each ∆im = τ ∆ is a regular CW complex, so too is K. This establishes K as a planar polygonal complex, and it follows that because τ is a primitive (n1,...,nk)-ﬁnite subdivision operator, K is a planar (n1,...,nk)-gon complex.

A τ-amalgamation of a planar polygonal complex K is a combinatorial amalgamation L of K such that τL =∼ K. We do not claim that such an amalgamation is unique. That is, there may be two complexes L1 and L2 such that L1 ≇ L2 and yet τL1 =∼ τL2 =∼ K. Even worse, even if

L1 =∼ L2, it may still be the case that they are diﬀerent amalgamations. To illustrate this, consider again the square grid complex Z described in Example 2.1; let σ be the quad subdivision operator, and let τ = σ2. Let f be the unit square face of Z whose lower left-hand vertex is the origin (0,0). Of the multiple diﬀering ways to τ-amalgamate Z, let L be an amalgamation in which the face f is a corner face of the unique face g of L containing f, and let L′ be an amalgamation in which the face f is an interior face of the unique face g of L containing

58 f. Even though of course we have L =∼ L′, no isomorphism of L to L′ can take f set-wise to iteself, and it is in this sense that we say these are two different amalgamations, i.e. L 6= L′. This issue has been resolved for classical hierarchical tilings. Boris Solomyak shows in [9] that aperiodicity alone is enough in that setting to guarantee uniqueness of amalgamations. The combinatorial analogue of aperiodicity of a complex K would appear to be plurality of K, as this guarantees that Aut(K) does not act cocompactly, and hence one might conjecture that plurality alone is sufficient for uniqueness of amalgamations. Indeed, there are only two known examples of expansion complexes that admit multiple amalgamations. These are the square grid complex Z already discussed, and the constant 6-degree triangulation of the plane, both of which are readily seen to be singular. Regardless, we do not attempt to answer this question here. The next theorem shows that an expansion complex for a primitive finite subdivision operator τ will not only always admit a τ-amalgamation, but that such an amalgamation is yet another expansion complex for τ.

Theorem 5.2. Any expansion complex K = lim Fm for a primitive (n1,...,nk)-ﬁnite subdivision −→ operator τ has a τ-amalgamation that is itself an expansion complex for τ.

The proof is an immediate generalization to the primitive setting of the proof in [3], and due to its length we do not include it. However, we would like to highlight here a few more subtleties with amalgamations that complicate the proof. Ideally, one would like to simply take a “local amalgamation” Hm of each Fm, construct a nested sequence of these, and then take the directed limit complex as an amalgamation of F . Unfortunately, the situation is far more delicate than this. Consider again the square grid complex Z, with τ again being two iterates of quad subdivision, and let c be a vertex core of Z. Let L be a τ-amalgamation of Z, and let K be the smallest subcomplex of L such that τK contains c. There are three possibilites for K—it is either a face core, an edge core, or a vertex core. Hence, if c′ is another vertex core of Z, and K′ is the smallest subcomplex of L such that τK′ contains c′, then despite the fact that c =∼ c′, there is no guarantee that K =∼ K′. Even worse, even if we have K =∼ K′, it may be the case that c is nested within K diﬀerently than c′ is within K′, so that the CW complex pairs (c,τK) and (c′,τK′) are not isomorphic as pairs. All of these problems complicate the nesting of local local amalgamations idea suggested in the previous paragraph, and we caution the reader to be aware of these subtleties.

59 Theorem 5.2 turns out to be remarkably powerful, and it has many important implications. These appear in the four corollaries that follow. The proofs are again immediate generalizations to the primitive setting of those that appear in [3], but are included regardless for completeness.

Corollary 5.3. If K and L are two locally isomorphic planar polygonal complexes, and K = lim Fm −→ is an expansion complex for τ, then L is an expansion complex for τ.

Proof. First, by identifying each Fm with its canonical embedded copy in the direct limit K, we ∞ may write K as the increasing union ∪m=1Fm. Let Cm for m ≥ 1 be a sequence of pairwise disjoint simple closed edge-paths in L such that Cm+1 separates Cm from inﬁnity and let Bm be the combinatorial disk bounded by Cm. Then Bm is contained in the interior of Bm+1. Since K is ′ locally isomorphic to L, Bm is isomorphic to a subcomplex Bm of K and there exists an index k(m) ′ such that Bm is contained in Fk(m). Let ∆ be the seed of K. We may assume by choosing k(m) sequentially that k(m)

∼ ′ jm Bm = Bm ⊂ Fk(m) ⊂ τ ∆, and properties (1) through (4) in the deﬁnition of expansion complex ∞ ′ are satisﬁed. We conclude that L = ∪ Bm =∼ lim B is an expansion complex for τ. m=1 −→ m

The above corollary shows that any complex that is locally isomorphic to an expansion complex for τ is in fact also an expansion complex for τ. The converse is clearly not true in general—for example, with an appropriate subdivision operator and diﬀering choices of seeds, one may build two diﬀerent expansion complexes, one a planar n-gon complex, and one a planar m-gon complex. Nevertheless, the converse will hold if we require that τ is primitive.

Corollary 5.4. If τ is a primitive (n1,...,nk)-ﬁnite subdivision operator, and K = lim Fm and −→ L = lim Gm are expansion complexes for τ, then K is locally isomorphic to L. −→

Proof. Let ∆p be the seed of K, say with polygonal type p. It suﬃces to verify that, for each n positive integer n, the subdivided seed τ ∆p isomorphically embeds in K. Since τ is primitive, N let N be such that τ ∆p contains all possible face types. Apply Theorem 5.2 iteratively n + N times, starting with K, to obtain a τ n+N -amalgamation K′ of K, a planar polygonal complex such that K = τ n+N K′. Since τ is primitive, τ N K′ = K′′—a τ n-amalgamation of K—is a planar ′′ n (n1,...,nk)-gon complex. Let f be any p-gon face of K and observe that τ f is a subcomplex of n K isomorphic to τ ∆p.

60 The proof of this corollary shows more. The idea can be immediately modiﬁed to show repetitivity of expansion complexes.

Corollary 5.5. If τ is a primitive (n1,...,nk)-ﬁnite subdivision operator, every expansion complex for τ is combinatorially repetitive.

Proof. Let H be any ﬁnite connected subcomplex of the expansion complex K = lim Fm for τ. −→

If K has p-gon seed ∆p then there exists a positive integer m for which H is contained in Fm,

im where we have identiﬁed Fm, a subcomplex of τ ∆p, with its canonical copy in the direct limit N ′ K. Assuming τ is primitive, let N be such that τ ∆p contains all possible face types, and let K be a τ im+N -amalgamation of K. Note that primitivity in particular implies that the subcomplex

N ′ im+N τ f contains a p-gon ∆p for every face f of K . Moreover, the subcomplex τ f contains an

im ′ isomorphic copy of τ ∆p, and therefore an isomorphic copy of H for every face f of K . Since

im+N there are at most ﬁnitely many faces in the complex τ ∆p, say ℓ faces, every face of K is ℓ-close to an isomorphic copy of H in the dK -metric. Letting n = max{n1,...,nk}, this then implies that each vertex of K is nℓ/2-close to an isomorphic copy of H. We conclude that H is quasi-dense in K, and hence K is combinatorially repetitive.

Corollary 5.6. If K = lim Fm is an expansion complex for τ, then, for all positive integers n, the −→ subdivision τ nK is an expansion complex for τ. Moreover, if τ is primitive then τ nK is locally isomorphic to K for all positive integers n.

Proof. First observe that τK is an expansion complex for the ﬁnite subdivision operator τ. Indeed, the sequence of embeddings deﬁning the expansion complex τK is merely τFm ֒→ τFm+1, where

im im+1 τFm is a subcomplex of ττ ∆ = τ ∆, with ∆ the seed of K. This and induction then imply that, for all positive integers n, τ nK is also an expansion complex for τ. If τ is primitive, an application of Corollary 5.4 ﬁnishes the proof.

So far we have seen that expansion complexes for primitive finite subdivision operators exhibit almost all of the familiar combinatorial properties we would hope that they have. There is one property not considered thus far however—combinatorial finite local complexity. Unlike the others, the proof that expansion complexes have FLC is not an immediately corollary of Theorem 5.2, and will also require an additional mild constraint on our finite subdivision operators.

61 A(n1,...,nk)-ﬁnite subdivision operator τ is said to have bounded degree if there is a positive m constant β, called a degree bound for τ, such that for all positive integers m, each vertex of τ ∆ni m is incident to at most β edges of τ ∆ni , for i = 1,...,k. This ensures that, if K is an expansion complex for a τ, and τ has bounded degree, then K will have bounded degree, and so will every τ m-subdivision and each τ m-amalgamation of K.

Theorem 5.7. Let K be an expansion complex for a primitive (n1,...,nk)-ﬁnite subdivision operator τ. If τ is shrinking and has bounded degree, then K has ﬁnite local complexity.

Before we prove this, we will ﬁrst prove a technical lemma regarding non-simply connected cores. It is used in the proof of Theorem 5.7 to show that if a core of an expansion complex for a shrinking subdivision operator is not simply-connected, then the entire core together with its bounded complementary domain will be engulfed by a larger core after a ﬁxed number of amalgamations.

Lemma 5.8. Let K be an expansion complex for τ, c be a non-simply connected core of K, and let h be a component of the bounded complementary domain of c. If τ is shrinking such that τ k is strictly shrinking, then τ kh is not contained in any bounded complementary domain component of any core c′ contained in τ kc.

Proof. First we make some observations about the possible configurations of non-simply connected cores. Let c be a non-simply connected core of K, and h a component of the bounded complementary domain of c. If c is an edge core, then the faces f1 and f2 of c meet along finitely many edges ′ e1,...,en; c can be viewed as an edge core of any one of these edges. Let c be a non-simply connected core contained in τ kc, with k such that τ k is strictly shrinking, and suppose c′ has a bounded complementary domain component h′ that contains τ kh. If c′ is an edge core, then c′ ′ k must be an edge core of an edge e contained in the τ subdivision of one of e1,...,en, otherwise ′ k k ′ c would be contained entirely within one of τ f1 or τ f2. For the same reason, if c is a vertex core, then it must be a vertex core with central vertex v′ contained in the τ k subdivision of one of e1,...,en. We proceed with this case first—i.e. the case where c is an edge core for the edge e. Let c′ be either a non-simply connected edge or vertex core, and let v1,...,vm be the vertex path forming the ′ k boundary of h , with v1 chosen to be a vertex in τ e. Furthermore, let p 6= 1 be such that v1,...,vp

62 k k ′ k are contained in τ f1, and vp,...,vm are contained in τ f2. Since v1,...,vp bounds h ⊇ τ h, and k k k vp 6= v1 lies on the boundary of both τ f1 and τ f2, it cannot be contained in τ e, nor in any open ′ k edge incident to e. Hence, there is a face g of c contained entirely within τ f1 that meets both e k and vp, contradicting that τ is strictly shrinking. On the other hand, suppose c is a non-simply connected vertex core of K, with h a component of the bounded complementary domain of c. If c contains as a subcomplex a non-simply connected edge core C that contains h within one of its bounded complementary domain components, then we can focus solely on C and apply the arguments of the previous paragraphs. In the case where c does not contain such an edge core as a subcomplex, we will call c a strict vertex core. If c is a strict vertex core, with central vertex v with degree d, then there are ﬁnitely many vertices v1,...,vn ′ that meet all the faces f1,...,fd of c. Let c be a non-simply connected vertex core contained in τ kc, with k such that τ k is strictly shrinking, and suppose c′ has a bounded complementary domain component h′ that contains τ kh. Note that c′ cannot be an edge core because c is a strict vertex ′ ′ core. Hence, c is a vertex core, and its central vertex v must be one of v1,...,vn, otherwise it would be entirely contained in a subcomplex of c that does not separate h from inﬁnity. The case where c is strictly a vertex core follows similarly now to the edge core case. Again ′ let v1,...,vm be the vertex path forming the boundary of h , and the observation is the same as before: There must exist a vertex vp in this vertex path that does not lie on any open edge of c that is incident the central vertex v of both c and c′. Once again, this implies the existence of a face g ′ k of c contained entirely within a subdivided face τ f of c that meets both v and vp, and hence g lies outside of any open angle of f, contradicting that τ k is strictly shrinking.

Proof of Theorem 5.7. Let τ be a rotationally symmetric, shrinking (n1,...,nk)-finite subdivision operator with bounded degree that subdivides the p-gon ∆, p ∈ {n1,...,nk}, into l faces, and subdivides each edge of ∆ into e edges. For each k we have size(τ k∆) ≤ kM kk ≤ kMkk, where k k k kMk is the L1,1 norm of the substitution matrix M for τ, and size(∂τ ∆) = pe . Hence τ ∆ satisfies a g-isoperimetric inequality for every k with g defined by g(t)= kMkt. Let K be an expansion complex for τ, and let D be a finite simply connected subcomplex of K. Since τ is shrinking, the combinatorial hierarchy for K is strongly expansive, and hence there exists some m such that m amalgamations of K is enough to engulf D by a core of τ −mK. This m

63 is dependent only on the diameter of D; we wish to ﬁnd an m′ that is instead dependent only the boundary length size(∂D).

Let Pr be a simple closed edge path of length r in K, and let B(Pr) be the subcomplex of K consisting of all closed faces of K that meet Pr and are separated from inﬁnity by Pr. We call B(Pr) the interior combinatorial collar of Pr. Because K has bounded degree, for ﬁxed r there are at most

ﬁnitely many possible interior collars for any Pr. That is, there exists ﬁnitely many interior collars

B1(Pr),...,BN(r)(Pr) such that the interior combinatorial collar of any loop Pr is combinatorially equivalent to one from this list. Let mr(i) be the minimum number of amalgamations of K required

−mr(i) to engulf the collar Bi(Pr) by a core of τ K, and let Mr = max(mr(i)).

If D is a ﬁnite simply connected subcomplex of K such that size(∂D) = r, then ∂D = Pr for some loop Pr. Hence, associated to D we have the interior combinatorial collar B(Pr) ∈ {Bi(Pr)}. Suppose that all the cores of K are simply connected. We make the following two obeservations.

First, since the interior collar B(Pr) for D separates D from infinity, and because B(Pr) is engulfed by a core of τ −Mr K, it follows that the entire subcomplex D is also engulfed. Secondly, because K has bounded degree it has only finitely many cores, so there exists some minimum positive integer C such that τ C ∆ contains every core as a subcomplex. Combining the previous two observations it follows that D can be embedded in τ C+Mr ∆, so we define f(r) = size(∂τ C+Mr ∆). Finally, we have size(D) ≤ size(τ C+Mr ∆) ≤ g(size(∂τ C+Mr ∆)) = g(f(size(∂D))) showing that K satisfies a θ-isoperimetric inequality, with θ = g ◦ f. We now consider the case where K has non-simply connected cores. By the above, we still have

−Mr that the interior collar B(Pr) for the subcomplex D is engulfed by a core c of τ K. If c engulfs all of D, we are done, otherwise we have at most ﬁnitely many bounded complementary domain

Mr components h1,...,hn of c such that τ hi is a subcomplex of D for each i = 1,...,n. Let k be such that τ k is strictly shrinking. We amalgamate K an additional k times to engulf c by another core c′ of τ −(Mr+k)K. The claim is that c′ engulfs all of D. Indeed, if it did not, then c′ has a component h′ of its bounded complementary domain such that τ Mr+kh′ is a subcomplex of D. In k ′ k ′ particular, τ h must be contained in one of the domains hi of c, so τ c contains the core c whose k ′ complementary domain component hi contains τ h as a subcomplex. Lemma 5.8 shows that this cannot occur. Hence, D is engulfed by a core of τ −(Mr+k)K, and by the previous paragraph D can

64 τ

Figure 5.1: A ﬁnite subdivision operator τ that generates expansion complexes with non- simply connected cores.

be embedded in τ C+Mr+k∆, so we deﬁne g(r) = size(∂τ C+Mr+k∆). Once again, we have

size(D) ≤ size(τ C+Mr+k∆) ≤ (size(∂τ C+Mr+k∆)) = (f ′(size(∂D))) showing that K satisﬁes a θ-isoperimetric inequality, with θ = g ◦ f ′. Since τ has bounded degree, so does the K, and hence together with the θ-isoperimetric inequality we have constructed, it follows that K has ﬁnite local complexity.

Expansion complexes with non-simply connected cores are somewhat pathological. Indeed, they have never appeared in the expansion complexes that have arisen naturally through our study of these objects. In fact, one might hope that an expansion complex for a ﬁnite subdivision operator with all of the most desirable properties—i.e. primitive, bounded degree, (strictly) shrinking, and dihedrally symmetric—would always have only simply connected cores. This is not true, as Figure 5.1 demonstrates a ﬁnite subdivision operator with all of these properties, yet still has non- simply connected edges cores. Note that this operator is trivially primitive, as it contains only a single seed.

65 5.3 Hierarchies and Type of Expansion Complexes

The hierarchies of Section 4.2 are very strong properties to impose on planar polygonal complexes. It is diﬃcult at ﬁrst to see how one might construct a planar polygonal complex K together with an appropriate subdivision operator τ that manifests a combinatorial hierarchy for K. Fortu- nately, expansion complexes for primitive subdivision operators always admit such hierarchies, as the following theorem shows. Indeed, although inspired by classical hierarchical tilings, the concepts of combinatorial, conformal, and fractal hierarchies are ultimately modeled on the observed nature of expansion complexes and their corresponding conformal tilings.

Theorem 5.9. Every expansion complex K for a primitive ﬁnite subdivision operator τ exhibits a combinatorial hierarchy manifested by τ.

Proof. Let K be an expansion complex for a primitive ﬁnite subdivision operator τ. Recall that to build a combinatorial hierarchy for K manifested by τ, we need to construct a sequence {Kn} such that for every n we have Kn+1 = τKn, and Kn+1 ∼ Kn. n To do this, we let K0 = K. Next, for each positive integer n we let Kn = τ K, and, using

Theorem 5.2 we inductively deﬁne K−n such that K−n = τK−(n+1). By our construction we have

Kn+1 = τKn for every n. Secondly, Corollary 5.6 and Theorem 5.2 imply that each Kn is an expansion complex for τ, and hence by Corollary 5.4 we have Kn+1 ∼ Kn for every n. It follows that {Kn} is a combinatorial hierarchy manifested by τ.

Addding to the hypotheses of Theorem 5.9, if τ is additionally a conformal subdivision operator, then the tiling TK will exhibit a conformal hierarchy manifested by τ. Moreover, if τ is shrinking— so that the manifested hiearchy is strongly expansive—and has bounded degree, then Theorem 4.9 tells us that TK tiles the plane C, and Theorem 4.10 tells us that the entire local isomorphism class (K) will be in fact be parabolic. This gives the following theorem.

Theorem 5.10. Let τ be a primitive, simple, shrinking, dihedrally symmetric ﬁnite subdivision operator with bounded degree. Then τ manifests a combinatorial hierarchy for any expansion complex

K for τ, and manifests a conformal hierarchy for the conformal tiling TK . Moreover, TK tiles the plane C, and conformal type is constantly parabolic across the local isomorphism class (K).

66 The above are key results of Bowers and Stephenson in [3], as answering the type problem was also one of the primary motivators of their paper. By moving away from conformal hierarchies and considering expansion complexes that exhibit only fractal hierarchies instead, the results of Chapter 4 allow us to generalize these theorems by weakening the dihedral symmetry of τ to mere rotational symmetry. Of course, we remind the reader that the following are true only if it is the case that conditions (⋆) and (⋆⋆) are satisﬁed.

Theorem 5.11. Every expansion complex K for a rotationally symmetric ﬁnite subdivision operator τ with one face type and bounded degree exhibits a fractal hierarchy manifested by τ.

Proof. Because τ has only one face type it is trivially seen to be primitive, and by Theorem 5.9 it follows that any expansion complex K for τ will exhibit a combinatorial hierarchy manifested by τ. Secondly, K is a p-gon complex because τ has only one face type, and because τ is additionally rotationally symmetric, Theorem 4.7 tells us that τ manifests a fractal hierarchy {Jn} for K.

Finally, we conclude by observing that Theorems 5.11 and 4.13, and the results of Section 4.3 combine to provide combinatorial properties on τ that will guarantee that expansion complexes for τ are parabolic.1

Corollary 5.12. Let τ be a shrinking, rotationally symmetric ﬁnite subdivision operator with one face type and bounded degree. Then τ manifests a combinatorial hierarchy for any expansion complex K for τ, and manifests a fractal hierarchy for the conformal tiling TK . Moreover, TK tiles the plane C, and conformal type is constantly parabolic across the local isomorphism class (K).

Proof. Theorem 5.9 implies that any expansion complex K for τ will exhibit a combinatorial hierarchy manifested by τ, and Theorem 5.11 implies that τ manifests a fractal hierarchy for the conformal tiling TK . Theorem 4.12 then shows that TK is parabolic and tiles the plane C. Because τ is primitive, Corollary 5.5 tells us that the expansion complex K is combinatorially repetitive. Furthermore, because τ is primitive, shrinking, and has bounded degree, Theorem 5.7 tells us that K has FLC. Hence, by Theorem 4.2, every element in the local isomorphism class (K) exhibits a combinatorial hierarchy manifested by τ. The previous paragraph applies here to tell us that K′ is therefore also parabolic.

1In fact, we do not need to invoke the full strength of Theorem 4.13 in the proof, as broad properties of expansion complexes uncovered in this section, together with Theorem 4.2 provides the same result.

67 2 τ 1 1 1 1

v vv v

τ 2 2 2

Figure 5.2: The ﬁnite subdivision operator τ.

5.3.1 Addendum. The Discrete Hyperbolic Plane as an Expansion Complex

We wish to show that the discrete hyperbolic plane complex H studied in Section 3.2 can be realized as an expansion complex, and that the conformal tiling TH is hyperbolic. Before we begin, however, one immediate problem jumps to the forefront. Corollary 5.12 con- firms that most of the friendly subdivision operators we have looked at thus far will generate parabolic expansion complexes, so in order to obtain hyperbolic expansion complexes we will have to relax many of the nice conditions we impose on our operators. In [10], Bedaride and Hilion show that in the classical setting there are no primitive substitutive2 hyperbolic tilings, and hence primitivity is a prime candidate for the first condition we will relax. However, in the example that follows we in fact dispose of far more than just primitivity—our operator will lack rotational symmetry, total non-triviality, and we will allow different labelings of faces of the same n-gon type.

Let τ be the ﬁnite subdivision operator consisting of {τ1,τ2}, where each τi is a decomposition of the 5-gon ∆i. It is illustrated in Figure 5.2. The rule τ1 subdivides the 5-gon ∆1 into two ∆1

5-gons and one ∆2 5-gon, while the rule τ2 is the trivial decomposition of the 5-gon ∆2 into another

∆2 5-gon. Note that because τ1 is not rotationally symmetric, each 5-gon ∆i comes equipped with a distinguished vertex v so that the subdivision τ1 can be applied in a unique way.

2A substitutive tiling is one generated by a subsitution rule, the classical geometric analog of a combinatorial ﬁnite subdivision operator.

68 2 4 6 τ ∆1 τ ∆1 τ ∆1

.Figure 5.3: The embeddings F1 ֒→ F2 ֒→ F3

n To build the expansion complex we will think of each τ ∆1 as a collection of horizontal levels (as we did with H), where the “uppermost” level is the one that consists of the single face f0. Consider 2m the set {Fm | m ∈ N} where each Fm is the subcomplex of τ ∆1 consisting of all closed faces except f0. For each m there is a unique orientation-preserving cellular embedding Fm ֒→ Fm+1 such that the image of Fm is contained in the interior of Fm+1. This gives an inﬁnite sequence of embeddings F1 ֒→ F2 ֒→ F3 ֒→···, whose direct limit lim Fm is an expansion complex for τ, and is −→ readily seen to be isomorphic to the discrete hyperbolic place complex H. See Figure 5.3.

Theorem 5.13. The conformal tiling TH is hyperbolic.

Proof. We will prove this statement by constructing a quasiconformal homeomorphism between the tiling TH and the upper-half plane H, and because no such map exists from C to H it will follow that TH must be a hyperbolic tiling of the disk D.

To do this, we will map the tiles of TH to the tiles of a tiling T of H by euclidean squares. As squares, the tiles of T will not meet full-edge to full-edge, and thus we will describe each “square” tile more formally as a pentagon, using 5 vertices and 5 edges instead. The vertices of the tiles of T are given by the points (n2k−1, 2k),n,k ∈ Z. We describe the edges using two sets. The horizontal edges are given by [(n2k−1, 2k), ((n + 1)2k−1, 2k)] : n,k ∈ Z, and the vertical edges are given by

[(n2k−1, 2k), (n2k−1, 2k−1)] : n,k ∈ Z.

69 Figure 5.4: The square tiling T (left) and the triangular tiling T ⋆ (right) of H.

Our interest is in fact not with T , but with the geometric star subdivision T ⋆ of T , obtained by connecting by straight (euclidean) line segments each vertex of every tile of T to that tile’s barycenter. This gives a tiling of H by two classes of congruent triangles. See Figure 5.4. The piecewise linear surface |H⋆|eq can be mapped onto the triangular tiling T ⋆ by a homeomorphism φ that is linear on each triangular face of |H⋆|eq. The map φ is clearly quasiconformal, and hence we can extend φ to a quasiconformal homeomorphism f : TH → H merely by lifting TH to |H⋆|eq, showing that TH must be hyperbolic.

This proof actually shows more. One can build a similar square tiling of the upper half plane H H in the pattern of any representative s of ( ), and repeat this argument to show that THs is also hyperbolic, giving the following result:

Theorem 5.14. Conformal type is constantly hyperbolic across the local isomorphism class (H).

70 BIBLIOGRAPHY

[1] Philip L. Bowers and Kenneth Stephenson. A “regular” pentagonal tiling of the plane. Conformal Geometry and Dynamics 1 58-86, 1997.

[2] Philip L. Bowers and Kenneth Stephenson. Conformal Tilings I: Foundations, theory, and practice. preprint, 2016.

[3] Philip L. Bowers and Kenneth Stephenson. Conformal Tilings II: Local isomorphism, hierarchy, and conformal type. preprint, 2016.

[4] J.W. Cannon, W.J. Floyd, and W.R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics 5 153-196, 2001.

[5] J.W. Cannon, W.J. Floyd, and W.R. Parry. Expansion complexes for ﬁnite subdivision rules I. Conformal Geometry and Dynamics 10 63-99, 2006.

[6] J.W. Cannon, W.J. Floyd, and W.R. Parry. Expansion complexes for ﬁnite subdivision rules II. Conformal Geometry and Dynamics 10 326-354, 2006.

[7] Maria Ramirez-Solano. Non-commutative geometrical aspects and topological invariants of a conformally regular pentagonal tiling of the plane. Ph.D. Thesis, 2013.

[8] Olli Lehto, K.I. Virtanen. Quasiconformal mappings in the plane, Die Grundlehren der Mathematischen Wissenschaften. vol. 126, Springer-Verlag, 1973.

[9] Boris Solomyak. Nonperiodicity implies unique composition for self-similar translationally ﬁnite tilings. Discrete & Computational Geometry 20 265-279, 1998.

[10] Nicolas Bedaride, Arnaud Hilion. Geometric realizations of two dimensional substitutive tilings. arXiv:1101.3905 [math.GT]

71 BIOGRAPHICAL SKETCH

The author was born in Nassau, Bahamas in 1987, where he attended St. Andrew’s School from grades 1–12, ﬁnally graduating in 2004. He then attended Ramapo College of New Jersey in Mah- wah, New Jersey, USA, ultimately receiving Bachelor’s degrees in both Physics and Mathematics in 2008. The following year, he began his Ph.D. studies at Florida State University in Tallahassee, Florida, USA, and was educated—and re-educated—until this document was ﬁnally completed.