On Hyperbolic once-punctured-torus bundles II. Fractal tessellations of the plane
Abstract
We describe fractal tessellations of the complex plane that arise naturally from Cannon-Thurston maps associated to complete, hyperbolic, once-punctured-torus bundles. We determine the symmetry groups of these tessellations.
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ON HYPERBOLIC ONCE-PUNCTURED-TORUS BUNDLES II. FRACTAL TESSELLATIONS OF THE PLANE. JAMES W. CANNON∗AND WARREN DICKS† To our wives, Ardyth and Elena Abstract. We describe fractal tessellations of the complex plane that arise naturally from Cannon-Thurston maps associated to complete, hyperbolic, once-punctured-torus bundles. We determine the symmetry groups of these tessellations. 1. Outline This section gives an overview of the article. More details, including definitions, will be given in the appropriate sections. Let hA, B, Cidenote the group with presentations hA, B, Ci=hA, B, C, D |A2=B2=C2=ABCD = 1i =hA, B, C |A2=B2=C2= 1i. We call the conjugates of D(= CBA)distinguished elements of hA, B, Ci. Let AuthA, B, Cidenote the group of automorphisms of hA, B, Ci. We specify an element Fof AuthA, B, Ciby the triple (F(A), F(B), F(C)). We will be using the three automorphisms R:= (A, BCB, B), L := (B, BAB, C), M := (C, B, A), and this article is mainly about Rand L. It is known that AuthA, B, Ci stabilizes the set consisting of the distinguished elements and their inverses; Date: May 11, 2005. 2000 Mathematics Subject Classification. Mathematics Subject Classification: 57M50, 57M05, 20E05, 20F28. Key words and phrases. Cannon-Thurston map, hyperbolic once-punctured-torus bundle, fractal tessellation, planar symmetry group. ∗Partially supported by the NSF (USA) through Grants DMS-9803868 and DMS-10104030. †Research funded by the DGI (Spain) through Projects BFM2000-0354 and BFM2003-06613. 1
2 JAMES W. CANNON∗AND WARREN DICKS† see, for example, [5, Notation 2.3]. Let Aut+hA, B, Cidenote the subgroup of AuthA, B, Ciwhich stabilizes the set of distinguished elements of hA, B, Ci. Then AuthA, B, Ci= Aut+hA, B, CiohMi. Let ˆ Cdenote the Riemann sphere C∪{∞}, and, for any subfield Kof C, let ˆ Kdenote the subset K∪{∞} of ˆ C. Let Aut+(ˆ C) denote the group of all conformal transformations of ˆ C. Let Rdenote the set of all discrete, faithful representations ρ:hA, B, Ci→ Aut+(ˆ C) such that ρ(D) is parabolic. Recall that the limit set of ρ, denoted Λ(ρ), is the closure in ˆ Cof the ρ(hA, B, Ci)-orbit of the point fixed by ρ(D). Let Rˆ Cand Rˆ Rdenote the sets of elements of Rwith limit sets ˆ Cand ˆ R, respectively. In a natural way, Rand Rˆ Care Aut+hA, B, Ci×Aut+(ˆ C)-sets. Suppose that we have two representations, ρˆ R∈Rˆ Rand ρˆ C∈Rˆ C, fixed for the remainder of this section. McMullen [8, Theorem 1.1] showed that there exists a (unique) (surjective) continuous hA, B, Ci-map from Λ(ρˆ R) to Λ(ρˆ C), sometimes denoted CT = CT(ρˆ R, ρˆ C): Λ(ρˆ R)³Λ(ρˆ C), and called the Cannon-Thurston map associated to (ρˆ R, ρˆ C). Thus CT: ˆ R³ ˆ Cis a sphere-filling curve that inherits a cyclic ordering, or orientation, from ˆ R. Results of Thurston [10] and Bonahon [2] give a surjective map b:Rˆ C→ IrRLSeqs, where IrRLSeqs denotes the set of all irrational bi-infinite sequences in the two-letter alphabet {R, L}; see Definitions 3.1 below. Moreover, binduces another surjective map Aut+(ˆ C)\Rˆ C/Aut+hA, B, Ci³IrRLSeqs, and Minsky [9] showed that the latter map is bijective. For example, ρˆ Carises from (the representation of the fiber of) a complete, hyperbolic, once-punctured-torus bundle if, and only if, there exists a (semigroup) word Fin {R, L}such that Fis not a power of a single letter, and b(ρˆ C) = Q Z F, the (irrational) bi-infinite R-L-sequence obtained by concatenating infinitely many copies of Fin both directions. In this event, some element of the Aut+hA, B, Ci-orbit of ρˆ Cextends (uniquely) to a discrete, faithful representation, in Aut+(ˆ C), of the subgroup hA, B, C, Fiof Aut+hA, B, Ci. In Definitions 9.2 below, we recall what it means for ρˆ Cto be ‘given by a C-T-model’, as in [4]. It is conjectured that this always holds, and Bowditch [3] proved that ρˆ Cis given by a C-T-model if ρˆ Carises from a complete, hyperbolic, once-punctured-torus bundle, or, more generally, if
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 3 there is a finite bound on the length of power-of-a-single-letter subwords of b(ρˆ C)∈IrRLSeqs. Now suppose further that ρˆ Cis given by a C-T-model, and let qˆ Cbe a point in ˆ C. The number of elements in the preimage CT−1{qˆ C}is one, two, or countably infinite, and, in this last case, CT−1{qˆ C}consists of the ‘head’ and ‘feet’ of a ‘double spider’. Let Idenote the set of connected components of ˆ R−CT−1{qˆ C}; then Iconsists of one open interval, two open intervals, or a bi-infinite sequence of open intervals. Clearly, CT induces a continuous, surjective map ˆ R−CT−1{qˆ C}³ˆ C−{qˆ C}.Let CT(I) := {CT(I)|I∈I},(1.0.1) a set consisting of (one, two or a bi-infinite sequence of) subsets of the oriented plane ˆ C−{qˆ C}. We will see that CT(I) does not depend on the choice of ρˆ Rwithin Rˆ R. If CT−1{qˆ C}has exactly one element, then the single element of CT(I) is the plane ˆ C− {qˆ C}. If CT−1{qˆ C}has exactly two elements, then the two elements of CT(I) arise from complementary Jordan domains in ˆ Cwhose common Jordan-curve boundary contains qˆ C. Our interest lies in the third case. Here there is a (unique) distinguished element D0of hA, B, Cisuch that ρˆ C(D0) fixes qˆ C. Then CT(I) is a hD0i-set of subsets of the oriented hD0i-plane ˆ C−{qˆ C}. Here, we call each element of CT(I) a column, and we arrange the plane so that the columns lie in an up-and-down direction and D0shifts the columns from left to right. By reversing the cyclic ordering of the sphere-filling curve if necessary, we may assume that CT fills in the set of columns from left to right. We then color the cut-points and the interior of each column; we use the color ‘gray’ if CT fills in the column from bottom to top, and ‘white’ if CT fills in the column from top to bottom. (We like to imagine that the sphere-filling travelling point switches back and forth between its two ink cartridges each time it hits qˆ C.) We show, in Definitions 9.4 below, that each column is the concatenation of a bi-infinite sequence of Jordan domains. Together, all the resulting Jordan domains can be used as closed two-cells to endow ˆ C−{qˆ C}with the structure of an oriented, colored Z-CW-complex, denoted CW(ρˆ C, qˆ C). The coloring can be used to recover the columns. In Proposition 9.5 below, we will show that each of the following determines the others: the Aut+(ˆ C)-orbit of the oriented, colored Z-CW-complex CW(ρˆ C, qˆ C); the isomorphism class of the oriented, colored Z-CW-complex CW(ρˆ C, qˆ C); the irrational bi-infinite R-L-sequence b(ρˆ C).
4 JAMES W. CANNON∗AND WARREN DICKS† In fact, (the isomorphism class of) the oriented, colored Z-CW-complex CW(ρˆ C, qˆ C) determines b(ρˆ C) visually, giving an alternative definition of b(ρˆ C) whenever ρˆ Cis given by a C-T-model. The final part, Sections 11 to 14, is devoted to the study of the symmetry group of CW(ρˆ C, qˆ C). We will be particularly interested in the case where ρˆ Carises from a complete, hyperbolic, once-punctured-torus bundle, or, equivalently, CW(ρˆ C, qˆ C) has a conformal Z2-action; see Proposition 13.6 and Section 10. 1.1. Examples. (i) Suppose b(ρˆ C) = Q Z RLLRRRLLLL =Q Z R1L2R3L4. Figure 1.1.1, which will be repeated as Figure 14.13.1, depicts (1.0.1) and CW(ρˆ C, qˆ C) for this case. The elements of (1.0.1) appear as colored columns, and ρˆ C(D0) shifts the columns two steps to the right. The sphere-filling curve CT draws the tessellation (completed with the point qˆ C, common to all the columns) from left to right (gradually), alternately proceeding upward (gradually), producing a gray column, and downwards (gradually), producing a white column. We consider such a figure to be normalized. Figure 1.1.1. Part of the tessellation for b(ρˆ C) = Q Z RLLRRRLLLL. The cut-points of the white (resp. gray) columns are white (resp. gray) zero-cells. By a gray-white path, we mean the left border of a white column,
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 5 or, equivalently, the right border of a gray column; it will be thought of as an upward path given by a bi-infinite sequence of zero-cells. A white-gray path is defined dually. From Figure 1.1.1, we see the following. The bi-infinite zero-cell-color sequence of any gray-white path is Q Z [white, gray, gray, white, white, white, gray, gray, gray, gray,]; on replacing ‘white’ with Rand ‘gray’ with L, we recover the bi-infinite R-L-sequence Q Z RLLRRRLLLL. A white-gray path gives the same bi-infinite zero-cell-color sequence. Each gray zero-cell provides one term in the bi-infinite zero-cell-color sequence of each border of a gray column. After the two bi-infinite zero-cell-color sequences are identified in a canonical way, the gray zero-cell provides two terms; these are consecutive in the gray subsequence, and the term corresponding to the gray-white path occurs after the term corresponding to the white-gray path. Similarly, the two borders of a white column intersect in their white zero-cells, and the white-gray path lags one white step behind the gray-white path. (ii) Suppose b(ρˆ C) = Q Z LLLLRRRLLR. A normalized figure for this case can be obtained by rotating Figure 1.1.1 through 180◦. Using permutation notation for this example, we say the change from (i) to (ii) is (forwards, backwards)(left, right)(up, down), by which we mean that, in passing from (i) to (ii), b(ρˆ C) is read backwards, while, in the figure, up and down are interchanged, and left and right are interchanged. (iii) Suppose b(ρˆ C) = Q Z LRRLLLRRRR. A normalized figure for this case can be obtained by reflecting Figure 1.1.1 about a vertical line and interchanging the two colors. The change from (i) to (iii) is then (L,R)(left, right)(gray, white), that is, in b(ρˆ C), the letters Land Rare interchanged, and, in the figure, left and right are interchanged, and gray and white are interchanged. (iv) Suppose b(ρˆ C) = Q Z LLLLRRRLLR. A normalized figure for this case can be obtained by reflecting Figure 1.1.1 about a horizontal line and interchanging the two colors. The change from
6 JAMES W. CANNON∗AND WARREN DICKS† (i) to (iv) is then (forwards, backwards)(L,R)(up, down)(gray, white). ¤ 2. General Notation In this section we record some of our conventions. 2.1. Notation. For any metric space X, we write Isom(X) to denote the group of all isometries of X. If Xis oriented, we write Isom+(X) to denote the group of all orientation-preserving isometries of X. Let Ndenote {0,1,2, . . .}, the set of finite cardinals. For any n∈N, we give Rnthe usual Euclidean metric, and give Znthe induced metric. There is a natural embedding Isom(Zn)→Isom(Rn), and, at the risk of some confusion, we often view Isom(Zn) as a subgroup of Isom(Rn). Each element of Isom(R) is of the form az +b, where zrepresents the identity map, while a∈ {1,−1}and b∈Rrepresent constant maps. Here az+blies in Isom(Z) if and only if b∈Z. Notice that Isom(R) = Isom+(R)o h−zi. Each element of Isom(R2) is of the form (x, y)7→ (ax +by +e, cx +dy +f),(2.1.1) for some a, b, c, d, e, f ∈Rsuch that a2+b2= 1 and (c, d) = ±(−b, a). We will denote (2.1.1) by (ax +by +e, cx +dy +f); in this notation, xand ycan be thought of as projections onto the first and second coordinates, respectively. The map (2.1.1) lies in Isom(Z2) if and only if all six coefficients are integers. ¤ 2.2. Notation. Let Xand Ybe sets. Let YXdenote the set of all maps from Xto Y. We let Sym(X), or Sym X, denote the set of all permutations of X. Then Sym(X)×Sym(Y) acts on YXin the natural way, that is, any (α, β)∈Sym(X)×Sym(Y) acting on any γ∈YXgives β·γ·α−1, where the dot denotes composition. Any subgroup Gof Sym(X)×Sym(Y) acts on YX, and we may consider the set G\YXof the G-orbits in YX. We denote the sets of Sym(X)-orbits, Sym(Y)-orbits, and Sym(X)×Sym(Y)-orbits, as YX /Sym(X), Sym(Y)\YX, and Sym(Y)\YX /Sym(X), respectively. Let us now restrict to the case where X=Z, where we can view Sym(Z) as containing Isom(Z). An Isom+(Z)-orbit in YZwill be called a bi-infinite sequence in Y.¤
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 7 2.3. Notation. Recall the notation at the beginning of Section 1. Let us write P:= LR−1L=R−1LR−1= (C, CBC, CBABC), Q:= LR−1= (B, C, CBABC). Then L=Q−1Pand R=Q−2P. Also P2=Q3=D. The relation P2=Q3presents the subgroup hP, Qi=hR, Liof AuthA, B, Ci. Also, PRP−1=L−1and PLP −1=R−1. For each W∈ hA, B, Ci, we write h|W|i to denote the smallest normal subgroup of hA, B, Cicontaining W. We view hA, B, Cias a subgroup of AuthA, B, Ciby identifying each W∈ hA, B, Ciwith the automorphism (WAW−1, W BW−1, WBW −1) of hA, B, Ci.¤ 2.4. Notation. Consider the complex projective line, or Riemann sphere, ˆ C=C∪{∞}. Let Aut(ˆ C) denote the group of all the transformations of ˆ Cwhich are conformal or anticonformal. We specify an element of Aut(ˆ C) by a (non-unique) expression of the form az+b cz+dor a¯z+b c¯z+d, with a, b, c, d ∈C, ad 6=bc, were zdenotes the identity map on ˆ Cand zdenotes complex conjugation on ˆ C. Alternatively, we can think of zas a generic element of ˆ C. Recall that Aut+(ˆ C) is the index-two subgroup of Aut(ˆ C) consisting of the conformal transformations of ˆ C. Thus Aut(ˆ C) = Aut+(ˆ C)ohzi, There are natural identifications Aut+(ˆ C) = PSL2(C) = PGL2(C), through the action of GL2(C) on ˆ Cby M¨obius transformations. An element of Aut+(ˆ C) is said to be parabolic if it has a unique fixed point in ˆ C. A subset of Aut+(ˆ C) (= PSL2(C)) is discrete if its preimage in SL2(C) is discrete with respect to the usual metric on C4⊇SL2(C). ¤ 2.5. Definitions. By a representation of hA, B, Ciin Aut+(ˆ C), we mean a homomorphism from hA, B, Cito Aut+(ˆ C). There is a natural action of Aut(ˆ C) on Aut+(ˆ C) by left conjugation. Thus, by Notation 2.2, the set of all representations of hA, B, Ciin Aut+(ˆ C) is acted on by the group AuthA, B, Ci × Aut(ˆ C), and hence by all of its subgroups. We will be particularly interested in the orbits under the actions of AuthA, B, Ci×Aut(ˆ C), Aut+hA, B, Ci×Aut+(ˆ C), Aut(ˆ C), and Aut+(ˆ C). Two representations of hA, B, Ciin Aut+(ˆ C) are equivalent if they are in the same Aut+(ˆ C)-orbit.
8 JAMES W. CANNON∗AND WARREN DICKS† The Aut+hA, B, Ci-orbit of a representation ρof hA, B, Ciin Aut+(ˆ C) can be codified as a pair consisting of the subgroup ρ(hA, B, Ci) of Aut+(ˆ C) together with the image under ρof the set of distinguished elements of hA, B, Ci. We say that ρis faithful if it is injective, and that ρis discrete if ρ(hA, B, Ci) is discrete in Aut+(ˆ C). The limit set of ρ, Λ(ρ), is the smallest nonempty, closed ρ(hA, B, Ci)-invariant subset of ˆ C. If ρ(D) is parabolic, and qdenotes the point fixed by ρ(D), then Λ(ρ) is the ˆ C-closure of the ρ(hA, B, Ci)-orbit of q. Recall that R=R(hA, B, Ci, D) denotes the set of all discrete, faithful representations ρof hA, B, Ciin Aut+(ˆ C) such that ρ(D) is parabolic. Also, Rˆ Cand Rˆ Rdenote the sets of elements of Rwith limit sets ˆ Cand ˆ R(= R∪{∞}), respectively. Both Rˆ Cand Rare closed under the AuthA, B, Ci×Aut(ˆ C)-action. ¤ 2.6. Notation. We use boldface to denote elements of R2. We write 0:= (0,0), x:= (1,0), y:= (0,1). If (v0,v1, . . . , vn) is a sequence in R2, we denote by hull[v0,v1, . . . , vn] the convex hull of {v0,v1, . . . , vn}in R2.¤ 3. The principal concepts Our objective in this article is to derive certain conclusions in the situation where we are given a pair (ρˆ C, qˆ C)∈Rˆ C׈ Csuch that qˆ Cis a parabolic point of ρˆ C(hA, B, Ci), that is, qˆ Cis fixed by a (unique) distinguished element of ρˆ C(hA, B, Ci). Up to Aut+(ˆ C)-orbits, this data can be codified using the following concepts, which provide a language that is useful for expressing our results. 3.1. Definitions. An element fof {R, L}Zwill be called an R-L-map; we will write n7→ fn, rather than n7→ f(n), and we think of fas a formal infinite product, Q n∈Z fn. We say that fis an irrational R-L-map if, as ntends to infinity, neither fnnor f−nis eventually constant. We let IrRLMaps denote the set of all irrational R-L-maps. Let f∈IrRLMaps. We define the successor map of fas the (bijective) map succf:Z→Z, n 7→ succf(n) := min{m∈Z|m > n and fm=fn}. The inverse map is denoted predf. As in Notation 2.2, Isom(Z)×Sym{R, L}acts on IrRLMaps. The Isom+(Z)-orbit of fwill be denoted [f].
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 9 In Section 1, we denoted IrRLMaps /Isom+(Z) by IrRLSeqs, and called the elements irrational bi-infinite R-Lsequences.¤ 3.2. Definitions. Consider any map F:Z→Aut+hA, B, Ci,n7→ Fn. We define the quotient map of Fas f= quotient(F): Z→Aut+hA, B, Ci, n 7→ fn:= F−1 n−1Fn. We say that Fis an irrational AuthA, B, Ci-map if f∈IrRLMaps; in particular, the image of fis {R, L}. We let IrAutABCMaps denote the set of irrational AuthA, B, Ci-maps. Suppose that F ∈ IrAutABCMaps, and let f= quotient(F). We write DF:= F0(D), a distinguished element of hA, B, Ci. It can be shown that Fn(D) = DF, for all n∈Z, since Fn(D) = Fn−1fn(D) = Fn−1(D). We define the residue map of Fto be the map F0= residue(F): Z→Aut+hA, B, Ci, n7→ F0 n:= Fn−1MfnM=(Fn−1Lif fn=R, Fn−1Rif fn=L. As in Notation 2.2, Isom(Z)×Aut+hA, B, Ciacts on IrAutABCMaps, where we understand that Aut+hA, B, Ciacts on itself by left multiplication. We can identify the Aut+hA, B, Ci-orbit of Fwith f, and get an identification Aut+hA, B, Ci\IrAutABCMaps ∼ →IrRLMaps .(3.2.1) We can further identify the Isom+(Z)×Aut+hA, B, Ci-orbit of Fwith [f], the Isom+(Z)-orbit of f.¤ The main purpose of this article is to investigate IrAutABCMaps, and construct models that correspond to the statements that were given in Section 1. Suppose that F ∈ IrAutABCMaps and let f= quotient(F). In Section 4, we associate to fsimple CW-complexes CW0(f) and CW(f). Suppose that ρˆ R∈Rˆ R, and let us use ρˆ Rto endow ˆ Rwith the structure of an hA, B, Ci-space. In Sections 5 and 6, we recall how to construct a model S2=S2(F) of ˆ C, with a distinguished point q2 S, and a continuous surjective hA, B, Ci-map CT: ˆ R³S2. In Section 7, we describe a CW-structure CW(F), on S2−{q2 S}, determined by CT and q2 Svia the method described in Section 1; we then show that CW(F) is isomorphic to CW(f). In Section 8, we recall the deep relationship between IrAutABCMaps and Rˆ C, and, in Section 9, we deduce the statements made in Section 1 about CW(ρˆ C, qˆ C).
16 JAMES W. CANNON∗AND WARREN DICKS† F A B C D F(x, y) (1 −x, −y) (1 −x, 1−y) (−x, 1−y) (−x, −y) F R L M F(x, y) (x+y, y) (x, x +y) (y, x) Table 5.3.1. The action of AuthA, B, Cion R2. By restriction, hA, B, Ciacts on R2, and here the action is by multiplication by ±1 followed by translation by an element of Z2. The kernel of the AuthA, B, Ci-action on R2is h|D2|i, and hA, B, Ci/h|D2|i acts properly discontinuously on R2. The quotient orbifold is of the form S0,0,4(2) , a sphere with four double points. The action of hA, B, Ci/h|D2|i determines a planar symmetry group of type p 2 of index four in Isom(Z2). We take hull[0,x,y] as a fundamental domain, and get a triangulation of R2; see Figure 5.3.1. (The labelling is discussed in Definitions 5.5.) AAAAA C B C B C B C B C B C AAAAA C B C B C B C B C B C AAAAA C B C B C B C B C B C AAAAA Figure 5.3.1. The hA, B, Ci/h|D2|i-tessellation of R2 The quotient orbifold of R2−Z2is of the form S0,0,3(2)+1, a once-punctured sphere with three double points. (We digress to remark that the specified action of AuthA, B, Cion R2−Z2determines an isomorphism from AuthA, B, Ci/hA, B, Cito the mapping class group of S0,0,3(2)+1, that is, the group of all admissible
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 17 self-homeomorphisms of S0,0,3(2)+1 modulo isotopy. The specified action chooses a self-homeomorphism in each isotopy class.) Using Notation 5.1, we find that hA, B, Ci\C+and hA, B, Ci\(R2− Z2) are non-canonically homeomorphic orbifolds, of the form S0,0,3(2)+1. Lifting to the branched coverings, we find that h|D2|i\C+and R2− Z2are hA, B, Ci-homeomorphic spaces. It follows that there exists an hA, B, Ci-homeomorphism h|D2|i\(C+∪P)∼ →R2,(5.3.1) where R2is now viewed as an orbifold in which the elements of Z2are infinite-cyclic points. We want to convert the hA, B, Ci-map (5.3.1) into an AuthA, B, Ci-map. It can be shown that h|D2|i\P∼ →Z2is already an AuthA, B, Ci-map of the form h|D2|i\P≃ h|D2|i\hA, B, Ci/hDi ≃ h|D2|i\hBC, BAi/hD2i ≃ ≃ h|D2|i\hBC, BAi ≃ Z2. It is now straightforward to give C+∪Pthe structure of an AuthA, B, Cispace in such a way that (5.3.1) is an AuthA, B, Ci-map. It can then be shown that C+∪ˆ Ris an AuthA, B, Ci-space. Using complex conjugation, we can make all of ˆ Cinto an AuthA, B, Cispace, and we get h|D2|i\(C−∪P)∼ →R2.(5.3.2) We will assume throughout that we have fixed AuthA, B, Ci-homeomor- phisms (5.3.1) and (5.3.2), and that (5.3.2) is the complex conjugate of (5.3.1). ¤
18 JAMES W. CANNON∗AND WARREN DICKS† 5.4. Definition. By the Farey representation we mean the element ρ0∈Rˆ R given by Table 5.4.1. W A B C D ρ0(W)−1 z z−1 2z−1 z−1 2z−1z+ 3 Table 5.4.1. The Farey action of hA, B, Cion ˆ C. Thus ρ0(D) is parabolic with fixed point ∞. The set of parabolic points of ρ0(hA, B, Ci) is then ˆ Q(= Q∪{∞}). There exists an orientation-preserving self-homeomorphism αof ˆ Cwhich transforms the hA, B, Ci-action via ρˆ Rinto the hA, B, Ci-action via ρ0, that is, ρ0and ρˆ Rare ‘topologically conjugate’. If we temporarily view ρ0and ρˆ R as homomorphisms from hA, B, Cito the group of all self-homeomorphisms of ˆ C, then ρˆ Rfollowed by conjugation by αgives ρ0. One can construct the restrictions of αto C+,C−, and ˆ R, separately, using the observations in Definitions 5.2 and Notations 5.3. It follows that, for our purposes, there would be no loss of generality in assuming that ρˆ R=ρ0and, hence, P=ˆ Qand qˆ R=∞.¤ We now pick out some special points of ˆ R. 5.5. Definitions. Consider the hA, B, Ci-action on R2. We label the closed one-cell hull[0,x] with the letter A; notice that this one-cell is rotated 180◦by A. We label hull[x,y] and hull[y,0] with the letters Band C, respectively. We now tessellate R2with the images under hA, B, Ciof the labelled triangle hull[0,x,y]. Thus the horizontal lines are labelled with A’s, the diagonal lines with B’s, and the vertical lines with C’s, as in Figure 5.3.1. We think of each zero-cell in Z2as an infinitesimally small circle divided into twelve parts, one part in each of the incident one-cells and two-cells. We will be particularly interested in the two parts which subtend angles of 90◦; we will call these quarters of the point. The base quarter-point is the quarter of 0that lies in hull[0,x,y]. For any W∈ hA, B, Ci, there is a (possibly self-intersecting) path in R2−Z2joining the base quarter-point to its image under W, such that Wis recovered by reading off the labels on the one-cells crossed by the path; we say that the path reads W. For example, BCBC and BCA carry the base quarter-point to different quarters of 2x. Conversely, any path in R2−Z2joining the base quarter-point to a quarter-point, and crossing only finitely many one-cells, arises in this way.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 19 Consider any irrational-sloped straight line in R2, consider its inverse image in C+∪Punder (5.3.1), and consider a connected component of this inverse image. If the original straight line is disjoint from Z2, then the connected component is an open arc in C+whose closure in ˆ Cadds two points of ˆ Rto the arc. Any such point can be expressed in the form cρˆ R(W) such that W(∈E) is read by a path in R2−Z2that starts at the base quarter-point, eventually reaches the straight line, and travels along the straight line to infinity in the appropriate direction. If the original straight line meets Z2(in a single point), then the above connected component is called a footless spider, and it consists of a point, called the head, in P, and a single infinite-cyclic orbit of open arcs, called the legs, in C+. The closure in ˆ Cof each leg is a closed arc with one endpoint being the head and the other endpoint lying in ˆ R−P, and called a foot. The footless spider together with all the feet is called a spider. Similar considerations apply to C−∪P. Notice that the preimages in C−∪ Pare obtained by complex conjugation, and the resulting compactification would give the same points. By a double spider, we mean the union, in ˆ C, of two spiders with a common head (in P). ¤ 6. The model S2(F)of ˆ C(ρˆ C, qˆ C) 6.1. Notation. Let F ∈ IrAutABCMaps, and let f= quotient(F). ¤ 6.2. Notation. Suppose that Notation 6.1 holds, and that F0= 1. Thus DF=D. Recall that 0= (0,0), x= (1,0), y= (0,1). For x, y ∈R,xx+yy= (x, y). For v= (x, y)∈R2, we define |v|=px2+y2. For v∈R2− {0}, we write hv:= {tv|t∈[0,∞[}. Subsets of R2of this form are called radial lines. A radial line is said to be rational, resp. irrational, if its slope is rational, resp. irrational. If h−,h+are radial lines, we define [h−, h+] := S θ∈[θ−,θ+] h(θ) for any θ−, θ+∈Rsuch that 0 ≤θ+−θ−<2π,h(θ−) = h−and h+=h(θ+), where we are writing h(θ) for hcos(θ)x+sin(θ)y. Let n∈Z. We will study the action of Fnon R2. We write xn=Fn(x), yn=Fn(y). Notice that Fnlies in hR, Li, hence Fnacts linearly on R2, hence Fn permutes the set of radial lines. Thus, for all v∈R2−{0},Fn(hv) = hFn(v), and Fn[hx, hy]=[hxn, hyn]. Since Fncarries half-planes to half-planes,
20 JAMES W. CANNON∗AND WARREN DICKS† and the quadrant [hx, hy] is strictly contained in a half-plane, we see that [hxn, hyn] is strictly contained in a half-plane. Thus, for any v∈[hxn, hyn], there exist non-negative real numbers p,q, such that v=pxn+qyn.¤ The following gives standard facts, usually expressed in terms of the M¨obius action of PSL2(Z) on ˆ R. 6.3. Lemma. Suppose that Notation 6.1 holds, and that F0= 1. (i) (Fn[hx, hy]|n∈Z)is a strictly decreasing chain of subsets of R2, and its intersection, T n∈ZFn[hx, hy], is an irrational radial line. (ii) (Fn[h−y, hx]|n∈Z)is a strictly increasing chain of subsets of R2, and its intersection, T n∈ZFn[h−y, hx], is an irrational radial line. Proof. (i). Let nbe a variable ranging over Z. Since Fn[hx, hy]∈ {Fn−1R[hx, hy],Fn−1L[hx, hy]}, and R[hx, hy] = [hx, hx+y]⊂[hx, hy], L[hx, hy] = [hx+y, hy]⊂[hx, hy], it follows that Fn[hx, hy]⊂ Fn−1[hx, hy]. This proves that the sequence is strictly decreasing. Let nbe a variable ranging over N. Let θndenote the angle between hxnand hyn. By the foregoing, (θn| n∈N) is a strictly decreasing sequence. Since F0= 1, θ0=π 2. It follows that θn∈¤0,π 2¤. The area of hull[0,xn,yn,xn+yn] is (sin θn)|xn||yn|, and the area of hull[0,x,y,x+y] is 1. Since hull[0,xn,yn,xn+yn] = Fn(hull[0,x,y,x+y]), and Fnpreserves area, we see that (sin θn)|xn||yn|= 1. Therefore (sin θn)−2=|xn|2|yn|2∈N−{0}. Hence the map N→N−{0}, n 7→ (sin θn)−2=|xn|2|yn|2, is a strictly increasing sequence in a discrete set. Therefore, lim n→∞(sin θn)−2= lim n→∞ |xn|2|yn|2=∞. Thus lim n→∞ θn= 0. Hence the intersection of the chain is a radial line; let us denote it by h. To see that his irrational, let us suppose that his rational. Then there exists some v∈h∩(Z2−{0}). Now v∈[hxn, hyn], and hence there exist non-negative real numbers pn,qnsuch that v=pnxn+qnyn. Since we are
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 21 in the quadrant [hx, hy], we see that |v|≥|pnxn|and |v|≥|qnyn|, and thus |v|2≥pnqn|xn||yn|. Hence, lim n→∞ pnqn= 0. Also (pn, qn) = pnx+qny=F−1 n(pnxn+qnyn) = F−1 n(v)∈Z2. Thus, for some N∈N,pNqN= 0. If pN= 0, then h=hv=hqNyN=hyN, and yn=yNfor all n≥N, and fn=Lfor all n > N. This contradicts the fact that F ∈ IrAutABCMaps. A similar argument holds if qN= 0. Hence (i) holds. (ii). Let H ∈ IrAutABCMaps be given by n7→ PF−nP−1. By applying (i) with Hin place of F, we deduce (ii). ¤ Let us briefly rework the definitions from [5, Section 3], without giving proofs. 6.4. Definitions. Suppose that Notation 6.1 holds, and that F0= 1. Thus DF=D. Let h−=h−(F) := T n∈ZFn[h−y, hx], and h+=h+(F) := T n∈ZFn[hx, hy]. By Lemma 6.3, h−and h+are irrational radial lines. For each m∈Z, we define h2m−1(F) := Dmh−and h2m(F) := Dmh+. Then (hm(F)|m∈Z) consists of four radial lines repeating cyclically. Let hDih−and hDih+denote the (whole) lines containing h+and h−, respectively. The vector-space decomposition R2=hDih+⊕ hDih−gives vector-space retractions π+:R2³hDih+, π−:R2³hDih−, such that π+(hDih−) = π−(hDih+) = {0}. Since hA, B, Cipermutes the fibers of π+, we can view hDih+as an hA, B, Ci-quotient space of R2. A similar statement holds for hDih−. We now apply Notation 5.3. Let Π−:C+∪P³hDih−denote the composition C+∪P³h|D2|i\(C+∪P)∼ →R2π− ³hDih−. Then Π−is a surjective hA, B, Ci-map, and each fiber of Π−has countably many components. Each component of each Π−-fiber over π−(R2−Z2) is an open arc, and taking its closure in ˆ Cadds two endpoints, in ˆ R−P. Each component of each Π−-fiber over π−(Z2) is a footless spider, and taking its closure in ˆ Cadds the feet, in ˆ R−P. See Definitions 5.5. Let L+denote the set consisting of the components of the fibers of Π−; thus, L+partitions C+∪Pinto footless spiders and open arcs.
22 JAMES W. CANNON∗AND WARREN DICKS† On applying complex conjugation and replacing π−with π+, we get a map Π+, which is a composition of the following form: C−∪P³h|D2|i\(C−∪P)∼ →R2π+ ³hDih+. As before, we get a partition L−of C−∪Pinto footless spiders and open arcs. Let L=L(F) denote the set consisting of the following: all the double spiders formed from elements of L+∪L−, one double spider for each element of P; all the closed arcs which are closures of open arcs belonging to L+∪L−; all the one-element subsets of ˆ Rthat do not lie in any of the foregoing double spiders and closed arcs. Thus Lis a partition of ˆ C, and it is closed under the hA, B, Ci-action. We now define an oriented hA, B, Ci-space S2=S2(F). As a set, S2is just L. As a topological space, S2is the quotient space of ˆ Cobtained by collapsing each element of Lto a point. By a theorem of R. L. Moore, S2 is a two-sphere; see, for example, [5, Appendix]. Moreover, S2inherits an orientation from ˆ Csince the collapsing respects the two-sidedness of ˆ C. The unique element of Lfixed by D, denoted s=s(F), is the double spider in Lwith head qˆ R. We denote the corresponding element of S2by qS2. For m∈Z∪ {−∞,∞}, we define Wm=Wm(F) as follows. For each irrational radial line hin [h−y, h−x], let W(h) denote the element of Eread by h, as in Definitions 5.5. For each m∈Z, we define W2m−1:= DmW(h−),W2m:= DmW(h+), and we set W−∞ := D−∞,W∞:= D∞. For m∈Z∪{±∞}, we define wm=wm(F), as follows. For each m∈Z, we let wm:= cρˆ R(Wm). We set w±∞ := cρˆ R(W−∞) = cρˆ R(W+∞) = qˆ R.Then the head of the double spider sis w±∞ =qˆ R, and the set of feet of sis {wm|m∈Z}.The cyclic indexing of s∩ˆ R={wm|m∈Z∪ {±∞}} is compatible with the cyclic ordering of ˆ R, since ρˆ R(D) shifts the points of ˆ R−{qˆ R}cyclically upward. ¤
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 23 We now remove the restriction F0= 1. 6.5. Definitions. Suppose that Notation 6.1 holds. Definitions 6.4 apply to give a partition L(F−1 0F) of ˆ C, closed under the action of hA, B, Ci. By Notation 5.3, we have an AuthA, B, Ci-action on ˆ Cby continuous maps. On applying F0to ˆ C, we get a partition L= L(F) := F0(L(F−1 0F)) of ˆ C, closed under the action of hA, B, Ci. As in Definitions 6.4, S2=S2(F) is defined to be the quotient space of ˆ Cobtained by collapsing each element of Lto a point. Up to hA, B, Ci-homeomor- phism, S2will be independent of the choice of ρˆ Rwithin Rˆ R. We denote the quotient map by κ=κF:ˆ C³S2. Thus S2is an hA, B, Ci-space and κis an hA, B, Ci-map. Let ρS2denote the implicit homomorphism from hA, B, Cito the group of all self-homeomorphisms of S2. The action of F0on ˆ Cinduces a homeomorphism F0:S2(F−1 0F)→ S2(F) such that F0(Wp) = F0(W)F0(p), for all W∈ hA, B, Ciand all p∈S2(F−1 0F). In particular, S2(F) is an oriented sphere. We define qS2=qS2(F):= F0(qS2(F−1 0F)), the unique point of S2fixed by DF. We define s=s(F) := F0(s(F−1 0F)) = κ−1{qS2}, the unique element of L(F) fixed by DF. For each m∈Z∪{−∞,∞}, we let Wm(F) := F0(Wm(F−1 0F)). For each m∈Z∪{±∞}, we let wm(F) := F0(wm(F−1 0F)). These give the head and feet of the double spider s, as before. Let CT = CT(ρˆ R, ρS2) denote the composition ˆ R,→ˆ Cκ ³S2. Each element of Lcontains an element of ˆ R, so CT is a surjective hA, B, Ci-map. Let cρS2denote the composition E cρˆ R ³ˆ R,→ˆ Cκ ³S2. Again, this is a surjective hA, B, Cimap: it is the composite E cρˆ R ³ˆ RCT ³S2.¤ 7. The model CW(F)of CW(ρˆ C, qˆ C) 7.1. Notation. Let F ∈ IrAutABCMaps. Let f= quotient(F) and F0= residue(F). Let L=L(F), S2=S2(F), κ=κF:ˆ C³S2, CT = CT(ρˆ R, ρS2): ˆ R³ S2,cρS2=\ρS2(F):E³S2,qS2=qS2(F)and s=s(F) = κ−1{qS2}. Let sdenote the image of sunder complex conjugation; thus s∪sis a quadruple spider. For each m∈Z∪{−∞,+∞}, let Wm=Wm(F). For each m∈Z∪{±∞}, let wm=wm(F). For each m∈Z, let `mdenote the closed leg of sincident to the foot wm; thus `m⊆Csign(−1)m∪ˆ R, where sign(−1)mis + if mis even, and −if mis odd. Let `mdenote the image of `munder complex conjugation. It
24 JAMES W. CANNON∗AND WARREN DICKS† is not difficult to see that `m∪`mis a Jordan curve in ˆ C. Recall that κ(`m) = κ(s) = qS2; we write ∂m:= κ(`m∪`m) = κ(`m). In this section, we will see that {∂m|m∈Z}is a set of arcs which etch a CW-structure, denoted CW(F), on the plane S2−{qS2}. When F0= 1, we will further understand that h−=h−(F), h+= h+(F), and hm=hm(F) for each m∈Z, as in Definitions 6.4. Notice that `m−{wm} maps homeomorphically to hmunder the chosen map Csign(−1)m∪P³h|D2|i\(Csign(−1)m∪P)∼ →R2. ¤ 7.2. Remarks. Given elements u,v,w, of ˆ R, we write u≺v≺wif u6=v, v6=wand u,v,ware in cyclic order, that is, vis reached before win travelling cyclically starting from u. Given elements u,w, of ˆ R, we write ]u, w[ := {v∈ˆ R|u≺v≺w},and [u, w] := ]u, w[∪{u, w}. The former is called an open cyclic interval in ˆ R, while the latter is a closed cyclic interval in ˆ R. Notice that CT−1{qS2}={wm|m∈Z∪{±∞}}; and ˆ R−CT−1{qS2}=[ m∈Z ]wm−1, wm[, a disjoint union of open cyclic intervals in ˆ R. In this section, we study the subsets CT( ]wm−1, wm[ ) of S2.¤ We will want to speak of cyclic intervals in E, as suggested by Figure 5.2.1. 7.3. Definition. We can carefully pull back the cyclic ordering of ˆ Ralong cρˆ R:E³ˆ Rto induce a cyclic ordering on E, described algebraically as follows. First we define a total order <on E. The six two-letter, reduced words in hA, B, Ciare ordered as: AB < AC < BC < BA < CA < CB. We then endow Ewith the resulting lexicographic order on infinite reduced products of two-letter words. Thus two distinct ends W,V, are compared by finding the least n∈Nsuch that W−1 2nW2n+2 6=V−1 2nV2n+2, and then comparing this pair of two-letter words. This gives a total order on E, and we make it into a cyclic ordering in the obvious way. Given elements U,V, W, we write U ≺ V ≺ W if U 6=V,V 6=W, and U,V,Ware in cyclic order. For U,W ∈ E, we write ]U,W[ := {V ∈ E| U ≺ V ≺ W},and [U,W] := ]U,W[∪{U,W}. Subsets of Eof the form [U,W] will be called closed cyclic intervals in E.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 25 We define [A] := {W ∈ E| W|1=A}, and similarly for [B] and [C]. Thus [A]=[ABC∞, ACB∞],[B] = [BCA∞, BAC∞],and [C] = [CAB∞, CBA∞]. It can be shown that the AuthA, B, Ci-action on Ecarries closed cyclic intervals to closed cyclic intervals. We recall that A[A] = [B]∪[C], B[B] = [C]∪[A], C[C] = [A]∪[B]. Further aspects of this action are recorded in Table 7.3.1 using [5, Lemma 7.1]; recall the definition of Pfrom Notation 2.3. Moreover, Land Rrespect the total ordering of E, and Mreverses the total ordering of E.¤ 7.4. Proposition. Suppose that Notation 7.1 holds. Then the following hold. (i) (Fn[B]|n∈Z)is a strictly decreasing chain of closed cyclic intervals in E. (ii) T n∈ZFn[B] = {W0}. (iii) S n∈ZFn[B] = ]W−1,W1[. Proof. We may replace Fwith F−1 0·F, and thus assume that F0= 1. In particular, DF=D. (i). From Table 7.3.1, we see that R[B] = B[C]⊂[B] and L[B] = B[A]⊂ [B]. For each n∈Z, we see that Fn[B] = Fn−1fn[B]⊂ Fn−1[B]. Hence (i) holds. (ii). We first claim that T n∈ZFn[B] contains at most one point. Let S={n∈N|fn=L, fn+1 =R}. Then Sis infinite, by the definition of IrAutABCMaps. Consider any n0∈ S. To prove the claim, we may replace Fwith F−1 n0·F ·(z+n0), and thus assume that 0 ∈S. For any a, b ∈N−{0}, RaLb(BC) = ((BC)aBA)bBC, RaLb(BA) = (BC)aBA, and RaLb([B]) ⊆(BC)a[B]. For any n∈S, we find that Fn([B]) ⊆Wn[B], for some Wn∈ hA, B, Ci ending in BC. Moreover, W0= 1 and the length of Wnincreases strictly with n. By (i), all the Wn,n∈S, are initial segments of a single W ∈ E, and T n∈ZFn[B]⊆ {W}, as desired.
32 JAMES W. CANNON∗AND WARREN DICKS† CqS2to BqS2; in particular the arc does not contain qS2. Here, h0is oriented away from the origin. The boundary of cρS2(B[A]) does not contain qS2, and we now wish to show that qS2does not lie in the interior of cρS2(B[A]). Since F1=R, it is not difficult to show that qS2lies in the boundary of cρS2(B[C]), and hence, by [5, Theorem 5.6(i)], qS2does not lie in the interior of cρS2(B[A]), as desired. (ii)(3). If f0=L, then F−1=L−1. Since L−1(x) = −x−yand L−1(y) = y, it follows that h−1passes through hull[−x−y,−y]. Hence the line parallel to h−1through x=A0passes through hull[y,x+y]. It follows that AqS2lies on ∂−[BA]S2. (ii)(4). If f2=L, then F2=RL. Since RL(x) = 2x+yand RL(y) = x+y, it follows that h0passes through hull[x+y,2x+y]. Hence the line parallel to h0through −x=CB0passes through hull[y,x+y]. It follows that CBqS2lies on ∂+[BA]S2.¤ 7.11. Lemma. Suppose that Notation 7.1 holds and that (m, n)∈Z2. (i) If fn=L, then the following hold: (1) Fn(AC)DF=Fn−1(A)and Dm FFsuccf(n)(AC)DF= Dm FFn−1(B). (2) qS2<Fn(AC)qS2<Fn+1(AC)qS2≤ Fn−1(B)qS2< qS2in ∂−1. (3) qS2<Fn(AC)qS2≤ Fn−2(B)qS2<Fn−1(B)qS2< qS2in ∂0. (4) The union of the closed segments of ∂−1and ∂0from Fn(AC)qS2to Fn−1(B)qS2is a Jordan curve in S2;the Jordan domain in S2− {qS2}bounded by this Jordan curve is cρS2(F0 n[B]). (ii) If fn=R, then the following hold: (1) DFFn(AC) = Fn−1(C)and Dm+1 FFsuccf(n)(AC) = Dm FFn−1(B). (2) qS2< DFFn(AC)qS2≤ Fn−2(B)qS2<Fn−1(B)qS2< qS2in ∂0. (3) qS2< DFFn(AC)qS2< DFFn+1(AC)qS2≤ Fn−1(B)qS2< qS2in ∂1. (4) The union of the closed segments of ∂0and ∂1from DFFn(AC)qS2to Fn−1(B)qS2is a Jordan curve in S2;the Jordan domain in S2−{qS2}bounded by this Jordan curve is cρS2(F0 n[B]). Proof. We will prove (i); the proof of (ii) is similar. Table 7.3.1 will be useful. (1). Let n0= succf(n).
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 33 Here Fn0=Fn−1LRn0−nL. Now Fn(AC)DF=Fn(AC ·D) = Fn(ABA) = FnL−1(A) = Fn−1(A). Hence Fn0(AC)DF=Fn0−1(A) = Fn−1LRn0−n(A) = Fn−1L(A) = Fn−1(B), and (1) follows. (2)–(4). By replacing Fwith F ·(z+n−1), we may assume that n= 1. Now cρS2(F0 1[B]) = cρS2(F0R[B]) = cρS2(F0B[C]), and F1(AC) = F0(A)D−1 Fby (1). Also F2(AC) = F0Lf2(AC) = (F0(B)D−1 Fif f2=L, F0(AB) if f2=R, and F−1(B) = F0f−1 0(B) = (F0(A) if f0=L, F0(C) if f0=R. If F0= 1, we get the desired results by Lemma 7.10(i). Hence the results hold for F−1 0F, and transforming by F0gives the general case. ¤ Recall the construction of CW(f) in Definitions 4.2. 7.12. Theorem. Suppose that Notation 7.1 holds. There exists a Z-homeomorphism |CW(f)|∼ →S2−{qS2}which, for each (m, n)∈Z2, carries the (m, n)th closed two-cell of CW(f)homeomorphically to cρS2(Dm FF0 n[B]). Proof. We first map R2=¯¯CW0(f)¯¯to S2−{qS2}as follows. Let (m, n)∈Z2, and let n+= succf(n). We send the zero-cell (2m, n) to Dm FFn−1(B)qS2. We send the zero-cell (2m−1, n) to Dm FFn(AC)qS2. We map the vertical one-cell out of (−1, n) homeomorphically to the increasing segment of ∂−1from Fn(AC)qS2to Fn+1(AC)qS2; see Lemma 7.11(i)(2) and (ii)(3). We extend by the Z-action to map the vertical one-cell out of (2m−1, n) homeomorphically to the increasing segment of ∂2m−1from Dm FFn(AC)qS2to Dm FFn+1(AC)qS2. We map the vertical one-cell out of (0, n) homeomorphically to the increasing segment of ∂0from Fn−1(B)qS2to Fn(B)qS2; see Lemma 7.11(i)(3) and (ii)(2). We extend by the Z-action to map the vertical one-cell out of (2m, n) homeomorphically to the increasing segment of ∂2mfrom Dm FFn−1(B)qS2to Dm FFn(B)qS2. If fn=L, we collapse the closed slanted one-cell between (2m, n) and (2m−1, n+) to the point Dm FFn−1(B)qS2=Dm FFn+(AC)qS2; see Lemma 7.11(i)(1). If fn=R, we collapse the closed slanted one-cell between (2m, n) and (2m+ 1, n+) to the point Dm FFn−1(B)qS2=Dm+1 FFn+(AC)qS2; see Lemma 7.11(ii)(2).
34 JAMES W. CANNON∗AND WARREN DICKS† By Lemma 7.11(i), if fn=L, then cρS2(F0 n[B]) is a Jordan domain, and its left boundary is the increasing segment of ∂−1from Fn(AC)qS2 to Fn−1(B)qS2=Fn+(AC)qS2, and its right boundary is the increasing segment of ∂0between the same two points. It follows that ∂−1∩∂0= {Fn(AC)qS2|n∈Z,fn=L}, and that {cρS2(F0 n[B]) |n∈Z,fn=L} is a bi-infinite sequence of Jordan domains with left boundary ∂−1and right boundary ∂0. Similarly, it can be shown that ∂0∩∂1={DFFn(AC)qS2|n∈Z,fn=R} and {cρS2(F0 n[B]) |n∈Z,fn=R} is a bi-infinite sequence of Jordan domains with left boundary ∂0and right boundary ∂1. It is not difficult to show that ∂m∩∂kis empty if |m−k| ≥ 2. We map the (0, n)th closed two-cell of CW0(f) to the Jordan domain cρS2(F0 n[B]) in a manner that extends the map on the boundary and induces a homeomorphism after collapsing the two closed slanted one-cells. We extend by the Z-action in the natural way. Since the collapsings in CW0(f) give CW(f), we have the desired Z-homeo- morphism. ¤ In CW(f), the one-cells and zero-cells can be recovered from the closed two-cells by intersection. 7.13. Definition. Suppose that Notation 7.1 holds. In a natural way, Theorem 7.12 gives an oriented, colored Z-CW-com- plex, which will be denoted CW(F). The following hold, where (m, n) is a variable ranging over Z2. (i) The underlying oriented space is S2−{qS2}. (ii) The set of zero-cells of CW(F) is CW0(F) := {Dm FFn(B)qS2|(m, n)∈Z2}= ={Dm FFn(AC)qS2|(m, n)∈Z2}, and both of the indicated maps from Z2are bijective. If fn=L, then Dm FFsuccf(n)(AC)qS2=Dm FFn−1(B)qS2. If fn=R, then Dm FFn−1(B)qS2=Dm+1 FFsuccf(n)(AC)qS2. (iii) The family of zero cells in ∂2m, in increasing order, is (Dm FFn(B)qS2|n∈Z).
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 35 The family of zero cells in ∂2m−1, in increasing order, is (Dm FFn(AC)qS2|n∈Z). ∂2m−1∩∂2m={Dm FFn(B)qS2|n∈Z,fn+1 =L}= ={Dm FFn(AC)qS2|n∈Z,fn=L}, ∂2m∩∂2m+1 ={Dm FFn(B)qS2|n∈Z,fn+1 =R}= ={Dm+1 FFn(AC)qS2|n∈Z,fn=R}. (iv) The set of open one-cells of CW(F) is the set of components of S m∈Z ∂m−CW0(F). Thus, we get the closed one-cells of CW(F) by taking the closed segment of ∂2m−1between Dm FFn(AC)qS2 and Dm FFn+1(AC)qS2, and the closed segment of ∂2mbetween Dm FFn(B)qS2and Dm FFn+1(B)qS2. (v) The set of closed two-cells of CW(F) is {cρS2(Dm FF0 n[B]) |(m, n)∈ Z2}, and the indicated map from Z2is bijective. (vi) For the Z-action on CW(F), the distinguished generator acts as DF, that is, ρS2(DF). ¤ 7.14. Remarks. Let (m, n) be a variable ranging over Z2. (i) In ˆ R, ([Fn(A)w±∞,Fn(C)w±∞]|n∈Z) is a decreasing sequence of closed cyclic intervals, with intersection {w0}and union ]w−1, w1[ . At each stage we divide the interval at Fn(B)w±∞, {cρˆ R(F0 n[B]) |n∈Z,fn=L}covers ]w−1, w0[, and {cρˆ R(F0 −n[B]) |n∈Z,fn=R}covers ]w0, w1[. By Lemma 7.11, cρˆ R(F0 n[B]) = [F0 n(A)w±∞,F0 n(C)w±∞] =([Fpredf(n)−1(B)w±∞,Fn−1(B)w±∞] if fn=L, [Fn−1(B)w±∞,Fpredf(n)−1(B)w±∞] if fn=R, =([Fn(AC)w±∞,Fsuccf(n)(AC)w±∞] if fn=L, [DFFsuccf(n)(AC)w±∞, DFFn(AC)w±∞] if fn=R. (ii) In S2−{qS2}, CT( ]wm−1, wm[ ) is the mth column of CW(F), and we have seen that this is the union of a bi-infinite sequence of Jordan domains, with each domain overlapping the next in a single point. The two-cells forming the even/gray/up columns correspond to fn=L, and the two-cells forming the odd/white/down columns correspond to fn=R. We have also seen that if the oriented plane S2− {qS2}is arranged so that DFmoves points from left to right, then, in the limit, the points Dm FFn(B)qS2move from left to right as mincreases, and from bottom to
36 JAMES W. CANNON∗AND WARREN DICKS† top as nincreases. This information plays the role of a frame for S2−{qS2}. ¤ 8. The sphere ˆ C(ρˆ C, qˆ C) 8.1. Definitions. Let ρˆ C∈Rˆ Cand let qˆ Cbe a parabolic pointof ρˆ C(hA, B, Ci). We say that (ρˆ C, qˆ C) is a parabolic pair in Rˆ C׈ C. Let ˆ C(ρˆ C, qˆ C) denote ˆ C viewed as an hA, B, Ci-space via ρˆ C, and having specified base point qˆ C. Let F ∈ IrAutABCMaps, and recall that S2(F) has a specified base point, qS2(F). We say that Fcodifies (ρˆ C, qˆ C) if there exists a (unique) (surjective) continuous, orientation-preserving, base-point-preserving, hA, B, Ci-map γ=γ(F, ρˆ C): S2(F)→ˆ C(ρˆ C, qˆ C). If this holds, then qˆ Cis the unique element of ˆ Cfixed by ρˆ C(DF). ¤ 8.2. Definitions. Codification, as defined above, specifies a subset of IrAutABCMaps ×Rˆ C׈ C, and we call this the codification pairing relation between IrAutABCMaps and Rˆ C׈ C. Recall that, as in Notation 2.2, Isom(Z)×Aut+hA, B, Ciacts on IrAutABCMaps, where we understand that Aut+hA, B, Ciacts on itself by left multiplication. We can also let Aut+hA, B, Ciacts on itself by right multiplication by inverses, and we call the resulting action of Isom(Z)×Aut+hA, B, Cithe right-hand action. Let us say that two elements Fand Gof IrAutABCMaps are equivalent if they are in the same hDi×Isom+(Z)-orbit for the right-hand action. Suppose the latter condition holds. There exists (a, b)∈Z2such that, G=F ·(z+b)·(D)a, that is, {Dm FFn|(m, n)∈Z2}={Dm GGn|(m, n)∈Z2}. It is not difficult to see that S2(F) = S2(G) and qS2(F)=qS2(G). Thus F and Gcodify exactly the same elements of Rˆ C׈ C. Recall that, as in Notation 2.2, Aut+hA, B, Ci×Aut+(ˆ C) acts on Rˆ C; we let Aut+hA, B, Ci×Aut+(ˆ C) act on ˆ C, with Aut+hA, B, Ciacting trivially and Aut+(ˆ C) acting in the canonical way. Then Aut+hA, B, Ci×Aut+(ˆ C) acts on the direct product Rˆ C׈ C. Let (ρ1, q1) and (ρ2, q2) be parabolic elements of Rˆ C׈ C. We say that (ρ1, q1) and (ρ2, q2) are equivalent if they are in the same Aut+(ˆ C)-orbit. Suppose that this holds. There exists σ∈Aut+(ˆ C) such that iσ·ρ1=ρ2and σ(q1) = q2. Here σinduces an orientation-preserving, base-point-preserving,
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 37 hA, B, Ci-homeomorphism ˆ C(ρ1, q1)∼ →ˆ C(ρ2, q2). Thus (ρ1, q1) and (ρ2, q2) are codified by exactly the same elements of IrAutABCMaps. Hence the codification pairing relation can be collapsed to a pairing relation on the equivalence classes of the foregoing equivalence relations. It is not difficult to show that the codification pairing relation is closed under the Aut+hA, B, Ci-action, that is, if Fcodifies (ρˆ C, qˆ C), and F∈ Aut+hA, B, Ci, then F·F codifies (ρˆ C·F−1, qˆ C). ¤ We will now quote various results in the literature to obtain the following; it is stated in a somewhat unusual form, suited to our purposes. 8.3. Theorem. (Thurston [10], Bonahon [2], Minsky [9], McMullen [8]) The codification pairing relation induces a bijective correspondence between the set IrAutABCMaps /(hDi×Isom+(Z)) of right-hand-action (hDi×Isom+(Z))-orbits in IrAutABCMaps and the set of Aut+(ˆ C)-orbits of parabolic pairs in Rˆ C׈ C. This correspondence respects the Aut+hA, B, Ci-action. Proof. Let IrAutABCMapsD:= {F ∈ IrAutABCMaps |DF=D}. By considering hA, B, Ci-orbits in the codification pairing relation, we see that we may assume that F ∈ IrAutABCMapsD, and that (ρˆ C, qˆ C)∈Rˆ C׈ C is such that qˆ Cis the fixed point of ρˆ C(D). For F ∈ IrAutABCMapsDand ρˆ C∈Rˆ C, let us say that Fcodifies ρˆ Cif there exists a (unique) (surjective) continuous, orientation-preserving hA, B, Ci-map γ=γ(F, ρˆ C): S2(F)→ˆ C(ρˆ C),(8.3.1) where ˆ C(ρˆ C) denotes ˆ Cviewed as an hA, B, Ci-space via ρˆ C. It now suffices to show that this new codification pairing relation gives a bijective correspondence hDi\IrAutABCMapsD/Isom+(Z)∼ →Aut+(ˆ C)\Rˆ C.(8.3.2) Thurston and Bonahon [2] associated, to each element of Rˆ C, an ordered pair of distinct irrational numbers, called the ending invariant. Thurston [10] showed that any pair of distinct irrational numbers is the ending invariant of some element of Rˆ C. Minsky [9] showed that two elements of Rˆ Clie in the same Aut+(ˆ C)-orbit whenever they have the same ending invariant. (Let us emphasize that we are recording only those aspects of the general theory which we will be using.) Suppose that F ∈ IrAutABCMapsD.
38 JAMES W. CANNON∗AND WARREN DICKS† Let h+=T n∈ZFn[hx, hy] and h−=T n∈ZFn[h−y, hx]. It follows, from Lemma 6.3, that h+and h−are non-collinear irrational radial lines. Reading off these lines in the triangulated plane, we obtain W(h+) and W(h−) in E, from which we can construct {Wm(F)|m∈Z∪{−∞,∞}}.(8.3.3) Let µ−and µ+denote the inverses of the slopes of h−and h+, respectively; thus, (µ−, µ+) is a pair of distinct irrational numbers. Consider any ρˆ C∈Rˆ Cwith ending invariant (µ−, µ+). McMullen [8] showed that cρˆ C:E→ˆ Cis constant on (8.3.3). It can be shown that S2(F) is the universal Hausdorff hA, B, Ci-space having a continuous hA, B, Ci-map from Ewhich collapses (8.3.3) to a point. It follows that there exists a continuous hA, B, Ci-map as in (8.3.1), and it can be shown that this map is orientation-preserving. Hence Fcodifies ρˆ C. Now consider any ρˆ C∈Rˆ Cthat is codified by F. Then cρˆ C:E→ˆ C carries (8.3.3) to the point qˆ C∈ˆ Cfixed by ρˆ C(D), and McMullen [8] also showed that there are exactly four irrational radial lines (forming two irrational lines) that give rise to elements of cρˆ C −1{qˆ C}. Thus the ending invariant of ρˆ Cmust be (µ−, µ+). Thus we have a well-defined Aut+hA, B, Ci-map as in (8.3.2). We now proceed in the reverse direction. Consider any ρˆ C∈Rˆ C. Let (µ−, µ+) be the ending invariant of ρˆ C. There exist two radial lines h−, h+with slopes 1 µ−,1 µ+, respectively, such that h+⊆[h−,−h−]. The pair (h−, h+) is then unique up to multiplying by −1. By a standard argument, recalled in Review 8.4 below, there exists F ∈ IrAutABCMapsDsuch that \ n∈ZFn[h−y, hx] = h−and \ n∈ZFn[hx, hy] = h+,(8.3.4) and the hD2i × Isom+(Z)-orbit of Fis unique. It is easy to see that this information gives the inverse of (8.3.2). ¤ 8.4. Review. For the convenience of the reader, we recall the standard argument referred to in the last paragraph of the above proof. (i). We begin by proving the existence of F. Consider first the case where h+⊆[hx, hy] and h−⊆[h−y, hx]. We remark that this amounts to µ−<0< µ+. Here we construct Frecursively in both directions, as follows. We set F0= 1.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 39 Suppose we are given n≥0 and Fnsuch that h+⊆ Fn[hx, hy]. As in the proof of Lemma 6.3, Fn[hx, hy] = Fn[hx, hx+y]∪Fn[hx+y, hy] = FnR[hx, hy]∪FnL[hx, hy] and FnR[hx, hy]∩FnL[hx, hy] = Fn[hx, hx+y]∩Fn[hx+y, hy] = Fn(hx+y). The latter radial line has rational slope and hence is not h+. We choose Fn+1 to be the unique element of {FnR, FnL}such that h+⊆ Fn+1[hx, hy]. Similarly, if we are given n≤0 and Fnsuch that h−⊆ Fn[h−y, hx], we can choose Fn−1to be the unique element of {FnR−1,FnL−1}such that h−⊆ Fn−1[h−y, hx]. This defines F ∈ IrAutABCMapsD, and (8.3.4) holds by Lemma 6.3. We next weaken the restraints on h+and h−to the assumption that there exists F∈Aut+hA, B, Cisuch that F(D) = D, h+⊆F[hx, hy], h−6⊆ F[hx, hy],−h−6⊆ F[hx, hy]. Here F−1(h+)⊆[hx, hy], F−1(h−)6⊆ [hx, hy],−F−1(h−)6⊆[hx, hy], and F−1(h+)⊆[F−1(h−),−F−1(h−)]. It follows that F−1(h−)⊆[−hy, hx]. By the preceding case, there exists G ∈ IrAutABCMapsDsuch that T n∈ZGn[hx, hy] = F−1(h+) and T n∈ZGn[h−y, hx] = F−1(h−). Now (8.3.4) holds with F=F·G. We now remove the restraints on h+,h−. Since Pacts on R2as a ninety-degree (counter-clockwise) rotation, there exists F∈ hPisuch that h+⊆F[hx, hy]. Choose any irrational radial line h⊆F[−hy, hx]. Then h6⊆ F[hx, hy], −h6⊆ F[hx, hy], and h+⊆[h, −h]. By the preceding case, applied to (h, h+), there exists G ∈ IrAutABCMapsDsuch that T n∈ZGn[hx, hy] = h+. Thus, for a sufficiently large n∈Z, h+⊆ Gn[hx, hy], h−6⊆ Gn[h−y, hx],−h−6⊆ Gn[h−y, hx]. By the preceding case again, now applied to (h−, h+), (8.3.4) holds for some Fin IrAutABCMapsD. (ii). We now prove the uniqueness of the hD2i×Isom+(Z)-orbit of F. Thus, suppose that Fand Gare elements of IrAutABCMapsDsuch that \ n∈ZFn[hx, hy] = \ n∈ZGn[hx, hy] = h+ and \ n∈ZFn[h−y, hx] = \ n∈ZGn[h−y, hx] = h−.
40 JAMES W. CANNON∗AND WARREN DICKS† There exists a sufficiently large N∈Nsuch that GN[hx, hy]⊆ F0[hx, hy]. Let us set F=F−1 0GN. Since F(D) = D, it is not difficult to use Notation 2.3 to show that there exists m∈Zsuch that PmFis a monoid word in either {R, L}or {R−1, L−1}. Recall that Rand Lcontract [hx, hy] onto proper subsets of itself, while Pacts as ninety-degree rotation. Since F[hx, hy]⊆[hx, hy], we see that m∈4Z, and PmFis a monoid word in {R, L}. Thus Pm∈ hD2i, and, by replacing Gwith Pm·G, we may assume that m= 0. Hence Fis a monoid word in {R, L}; let ddenote the length of this word. Since F0F[hx, hy] = Gn[hx, hy]⊇h+, we see that Fis the unique monoid word in {R, L}of length dsuch that F0F[hx, hy]⊇h+, that is, F0F=Fd. By replacing Gwith G · (z−d+N), we may assume that N=d. Now Gd=GN=F0F=Fd. It follows by induction that, for each n∈N,Gd+n=Fd+nand Gd−n=Fd−n. Hence F=G, as desired. ¤ 8.5. Remark. It is not difficult to deduce that the ordered pair (W−1(F),W0(F)) in E2determines the Isom+(Z)-orbit of F. Review 8.4 deals with the case where DF=D. It remains to show that for any irrational radial lines h,h0in [h−y, hy] and elements W,W0of ABC, if W·W(h) = W0·W(h0) in Ethen W=W0 and h=h0. By left multiplying the equation W· W(h) = W0· W(h0) by a suitable element of hA, B, Ci, we may assume that Wis completely cancelled in W·W(h), and that W0is completely cancelled in W0·W(h0). The two sides of the equation can then be used to construct piecewise linear paths in R2 which must be equal or be separated by an element of Z2. The result now follows. ¤ 9. The complex CW(ρˆ C, qˆ C) 9.1. Definitions. Theorem 8.3, or its proof, gives the bijective Aut+ hA, B, Ci-map of (8.3.2). On dividing out by the Aut+hA, B, Ci-action, and then using (3.2.1), we get a bijective map IrRLMaps /Isom+(Z)∼ →Aut+(ˆ C)\Rˆ C/Aut+hA, B, Ci.(9.1.1) The inverse determines a map b:Rˆ C→IrRLMaps /Isom+(Z). in a natural way. The inverse of (9.1.1) was discussed in Section 1, where the domain of (9.1.1) was denoted IrRLSeqs. Minsky [9] gives (9.1.1) explicitly, in that, by generalizing a construction of Jørgensen [6], he associates a sophisticated model with each element of IrRLMaps /Isom+(Z); see [9, Figure 3]. ¤ We now consider the case where we can identify S2=ˆ Cand cρS2=cρˆ C.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 41 9.2. Definitions. Let ρˆ C∈Rˆ C. Let qˆ Cbe a parabolic point for ρˆ C(hA, B, Ci). Choose some F ∈ IrAutABCMaps that codifies (ρˆ C, qˆ C), and let γdenote the associated map γ(F, ρˆ C): S2(F)→ˆ C(ρˆ C, qˆ C). It is conjectured that γis a homeomorphism. If γis a homeomorphism, we say that ρˆ Cis given by a C-T-model. It can be seen that ρˆ Cbeing given by a C-T-model depends only on the bi-infinite sequence b(ρˆ C), that is, the Isom+(Z)-orbit of quotient(F). ¤ 9.3. Theorem. (Bowditch [3]) If ρˆ C∈Rˆ Cand there is a finite bound on the length of power-of-a-single-letter subwords of b(ρˆ C), then ρˆ Cis given by a C-T-model. ¤ 9.4. Definitions. Let ρˆ C∈Rˆ Cand suppose that ρˆ Cis given by a C-T-model. Let qˆ Cbe a parabolic point for ρˆ C(hA, B, Ci). Choose some F ∈ IrAutABCMaps which codifies (ρˆ C, qˆ C). Let f= quotient(F) and let γ=γ(F, ρˆ C): S2(F)∼ →ˆ C(ρˆ C, qˆ C). Let (m, n) be a variable ranging over Z2. The set of all elements of IrAutABCMaps which codify (ρˆ C, qˆ C) is {Dm F·F ·(z+n)|(m, n)∈Z2}, and this set can be reconstructed from the set of values {Dm FFn|(m, n)∈Z2} ⊆ Aut+hA, B, Ci. Thus the latter set captures the Aut+(ˆ C)-orbit of (ρˆ C, qˆ C). Now γcarries the CW-structure CW(F) on S2(F)− {qS2(F)}to a CW-structure, denoted CW(ρˆ C, qˆ C), on ˆ C−{qˆ C}. All the information obtained about CW(F) in Section 7 passes over to CW(ρˆ C, qˆ C). Let us mention all of the following. The set of closed two-cells of CW(ρˆ C, qˆ C) is {cρˆ C(Dm FF0 n[B]) |(m, n)∈ Z2}; see Figure 9.6.1. By Theorem 7.12, there is an isomorphism CW(f)∼ →CW(ρˆ C, qˆ C), of oriented, colored Z-CW-complexes, which carries the (m, n)th closed two-cell of CW(f) homeomorphically to cρˆ C(Dm FF0 n[B]). Let CT = CT(ρˆ R, ρˆ C): ˆ R³ˆ C. For each m∈Z∪ {±∞}, let wm= wm(F). Then the following hold. (i) CT−1{qˆ C}={wm|m∈Z∪{±∞}}. (ii) ˆ R−CT−1{qˆ C}=S m∈Z ]wm−1, wm[ . The terms in the union are open cyclic intervals in ˆ R, and are pairwise disjoint.
48 JAMES W. CANNON∗AND WARREN DICKS† Since σcarries (0, n) to (−1, n), σcarries the unique slanted one-cell incident to (0, n) to the unique slanted one-cell incident to (−1, n). Hence σcarries (signf(n),succf(n)) to (−signf(n)−1,predf(n)). Thus σmust act like (−x−1, y−succf(n)+predf(n)) near {signf(n)}×R. Hence succf(n)−predf(n) is constant on both f−1{R}and f−1{L}; let us denote the constant values by dRand dL, respectively. Clearly dR≥2. Also dRcannot be 2 since fis irrational. If dR= 3, then [f] = Q Z RRL and dL= 6. If dR= 4 then [f] = Q Z RL or [f] = Q Z RRLL, and dL= 4. If dR≥5 then dL≤4. It now follows that [f]∈ {Q Z RL, Q Z RLL, Q Z RRL, Q Z RRLL}. In each of these cases, σexists and is (uniquely) realized by a map which is affine on each column, namely, for each m∈Zand (x, y)∈R2, σ((x, y)) = ((−x−1, dLx+y−(dL+dR)m) if x∈[2m−1,2m], (−x−1,−dRx+y+ (dL+dR)m) if x∈[2m, 2m+ 1]. It can be shown that σ2((x, y)) = (x, y −dL). Suppose that we are in one of the above cases, and let (m, n)∈Z2. By using σtogether with half-turns and translations, we find that there is a CW-automorphism νm,n of CW0(f) such that νm,n maps each column to itself affinely, and νm,n carries the (m, n)th closed two-cell to itself, interchanging its two slanted one-cells. Then νm,n is unique. It is color-preserving, orientation-reversing, Z-action preserving, and is not an isometry. It is left to the reader to verify Remark 4.5. ¤ 12. AuthA, B, Ci×Aut(ˆ C)-actions We now define actions of AuthA, B, Ci×Aut(ˆ C) on IrAutABCMaps and IrRLMaps. 12.1. Definitions. Let Aut−(ˆ C) := Aut(ˆ C)−Aut+(ˆ C), Aut−hA, B, Ci:= AuthA, B, Ci−Aut+hA, B, Ci. Let F ∈ IrAutABCMaps. The transpose of F, denoted transpose(F), is the element of IrAutABCMaps given by n7→ F−n−1P−1.We let Aut(ˆ C) act on IrAutABCMaps with Aut+(ˆ C) acting trivially, and Aut−(ˆ C) acting by transposing. It can be shown that (3.2.1) respects the Aut(ˆ C)-actions. We already have Aut+hA, B, Ciacting on IrAutABCMaps.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 49 For any Gin Aut−hA, B, Ci, define GFto be the element of IrAutABCMaps given by n7→ (GF)n:= GF−n−1P−1M. Again, it can be shown that (3.2.1) respects the AuthA, B, Ci-actions. All these concepts pass to Isom+(Z)-orbits. Let f∈IrRLMaps. Define the transpose of fas transpose(f) := (R, L)·f·(−z); define fread backwards by backwards(f) := f·(−z); and define the R-L-interchange of f as interchange(f) := (R, L)·f. We let AuthA, B, Ciact on IrRLMaps with Aut+hA, B, Ciacting trivially, and with Aut−hA, B, Ciacting by transposing. We let AuthA, B, Ci act on IrRLMaps with Aut+hA, B, Ciacting trivially, and Aut−hA, B, Ci acting by reading backwards. All these definitions pass to Isom+(Z)-orbits. ¤ 12.2. Remarks. Let us record the behavior of the action of AuthA, B, Ci× Aut(ˆ C). Let n∈Z,F,G ∈ IrAutABCMaps. Write f= quotient(F), g= quotient(G), F0= residue(F), G0= residue(G). Where we write F(S2(F)), we are thinking of S2(F) as a set of subsets of E.¤ 13. The symmetry group of CW(F) 13.1. Notation. Let F ∈ IrAutABCMaps. Let f= quotient(F)∈IrRLMaps. Let b=b(F) = [f]∈IrRLMaps /Isom+(Z). ¤ 13.2. Remarks. Suppose that Notation 13.1 holds. Observe that CW0(f) is an Isom(CW0(f))-CW-complex. In a natural way, CW(f) then becomes an Isom(CW0(f))-CW-complex. By Theorem 7.12 and Definition 7.13, we have an isomorphism CW(f)∼ →CW(F) (13.2.1) of colored, oriented Z-CW-complexes. We can use (13.2.1) to make CW(F) into an Isom(CW0(f))-CW-complex, and (x+ 2, y) then has the same action as DF. Later we will adjust (13.2.1) to alter the action of Isom(CW0(f)) on CW(F). We will not change the action on the set of cells. Notice that the permutation of the cells is determined by the permutation of the two-cells, since every one-cell and every zero-cell is an intersection of a pair of closed two-cells. ¤
50 JAMES W. CANNON∗AND WARREN DICKS† GFFGFtranspose(F) F∈Aut+hA, B, CiG∈Aut−hA, B, Ci GnFFnGF−n−1P−1MF−n−1P−1 DGF(DF)G(D−1 F)D−1 F gnfnf−nMf−nM g f backwards(f) transpose(f) Gn[B]F(Fn[B]) G(F−n−1(AB[A]∪A[C])) F−n−1(AB[A]∪A[C]) Wm(G)F·Wm(F)G·W−m−1(F)Wm−1(F) s(G)F(s(F)) G(s(F)) s(F) S2(G)F(S2(F)) G(S2(F)) S2(F) with the opposite orientation G0 n[B]F(F0 n[B]) G(F0 −n[B]) if L=gn(= f−n), G(D−1 F(F0 −n[B])) if R=gn(= f−n) F0 −n[B] if R=gn(6=f−n), D−1 F(F0 −n[B]) if L=gn(6=f−n) Table 12.2.1. The action of AuthA, B, Ci×Aut(ˆ C).
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 51 13.3. Definitions. Suppose that Notation 13.1 holds. Recall that, if G∈AuthA, B, Ci, then G(S2(F)) = S2(GF) as partitions of ˆ C, and as unoriented spaces. Also, G(qS2(F)) = qS2(GF)as points, and as subsets of ˆ C. We define Aut(hA, B, Ci,S2(F)) := {G∈AuthA, B, Ci | G(S2(F)) = S2(F)}. We then define the symmetry group of CW(F), denoted Aut(hA, B, Ci,CW(F)), as the set consisting of those Gin Aut(hA, B, Ci,S2(F)) which act as CW-automorphisms of CW(F). Recall that the underlying space of CW(F) is the oriented Z-plane S2(F)−{qS2(F)}. Hence, if G∈Aut(hA, B, Ci,S2(F)) then G∈AuthA, B, Ci and Gfixes qS2(F).¤ Over the course of this section we will see the converse. We will then see that Aut(hA, B, Ci,CW(F)) and Isom(CW0(f)) coincide when viewed as groups of permutations of the cells of CW(F). Also, we will describe the elements of Aut(hA, B, Ci,CW(F)) explicitly in terms of F. We begin by describing the G∈AuthA, B, Ciwhich act on the plane S2(F)−{qS2(F)}. 13.4. Lemma. Suppose that Notation 13.1 holds. Let G∈AuthA, B, Ci. The following are equivalent: (a) G(S2(F)) = S2(F)and G(DF) = D±1 F. (b) G(S2(F)) = S2(F)and G(qS2(F)) = qS2(F). (c) G(qS2(F)) = qS2(F)as subsets of ˆ C. (d) There exists (m, n)∈Z2such that either G·F =Dm F·F·(z+n)or transpose(G·F) = Dm F·F·(z+n). Proof. It is clear that (a) implies (b), and that (b) implies (c). It follows, from Remark 8.5, that the Isom+(Z)-orbit of Fis determined by the ordered pair (W−1(F),W0(F)) ∈E2. This can be used to show that (c) implies (d). If (d) holds then S2(GF) = S2(F) as unoriented spaces, and (a) holds. ¤ We have already considered the horizontal-translation symmetries, and we observed the following. 13.5. Proposition. Suppose that Notation 13.1 holds. Then DF∈Aut(hA, B, Ci,CW(F)), and the resulting permutation of cells of CW(F)is the same as that for the horizontal translation (x+2, y)∈ Isom(CW0(f)).¤
52 JAMES W. CANNON∗AND WARREN DICKS† We now consider the case where fis periodic and CW0(f), CW(F) have vertical-translation symmetries. Examples 14.9–14.14 illustrate this. 13.6. Proposition. Suppose that Notation 13.1 holds. Let d∈Z. The following are equivalent: (a) (x, y +d)∈Isom(CW0(f)). (b) f·(z+d) = f. (c) There exists F∈Aut+hA, B, Cisuch that F·F =F ·(z+d). If Fis as in (c), then F=FdF−1 0,F∈Aut(hA, B, Ci,CW(F)), and the resulting permutation of cells of CW(F)is the same as that for the vertical translation (x, y +d)∈Isom(CW0(f)). Proof. The second column of Table 12.2.1 will be useful. Notice that (a) and (b) are equivalent by Corollary 11.3. On taking quotients, we see that (c) implies (b). Now suppose that (b) holds. Let F:= FdF−1 0and G:= F·F. Now quotient(G) = quotient(F·F) = quotient(F) = f=f·(z+d). Also, G0=FF0=Fd. It follows that G=F ·(z+d), and (c) holds. By Lemma 13.4, F(S2(F)) = S2(F). Hence F\ρS2(F)F−1=\ρS2(FF)= \ρS2(F). For all (m, n)∈Z2, in E, we have F−1(Dm FF0 n[B]) = Dm F(F−1(F0 n[B])) = Dm F(F−1(G0 n−d[B])) = Dm FF0 n−d[B]. A similar equation holds in S2(F) since \ρS2(F)commutes with the action of F−1. Hence, F−1acts as a self-homeomorphism of S2(F) which permutes the two-cells of CW(F) in the same way that (x, y −d) does. The result now follows. ¤ We next consider the situation where backwards(b) = band CW0(f), CW(F) have half-turn symmetries. Example 14.9 illustrates the case where d= 0, and Example 14.11 illustrates the case where d= 1. 13.7. Proposition. Suppose that Notation 13.1 holds. Let d∈Z. The following are equivalent: (a) (−x−1,−y+d)∈Isom(CW0(f)). (b) f·(−z+d) = f. (c) backwards(f) = f·(z+d). (d) There exists G∈Aut−hA, B, Cisuch that G·F =F ·(z+d).
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 53 If Gis as in (d), then G=FdMPF−1 −1,G∈Aut(hA, B, Ci,CW(F)), and the resulting permutation of cells of CW(F)is the same as that for the half-turn (−x−1,−y+d)∈Isom(CW0(f)). Proof. The third column of Table 12.2.1 will be useful. Notice that (a) and (b) are equivalent by Corollary 11.3, and clearly (b) and (c) are equivalent. On taking quotients, we see that (d) implies (c). Now suppose that (c) holds. Let G:= FdMP F−1 −1, and G:= G·F. Now quotient(G) = quotient(G·F) = backwards(f) = f·(z+d). Also, G0=GF−1P−1M=Fd. It follows that G=F ·(z+d), and (d) holds. For each (m, n)∈Z2, in E, G−1(Dm FF0 n[B]) = D−m F(G−1(F0 n[B])) = D−m F(G−1(G0 n−d[B])) =(D−m FF0 −n+d[B] if L=f−n+d(= fn), D−m−1 FF0 −n+d[B] if R=f−n+d(= fn). It is now straightforward to use Theorem 7.12 to show that G−1acts as a self-homeomorphism of S2(F) which induces the same permutation of the two-cells of CW(F) that (−x−1,−y+d) does. The result now follows. ¤ We next consider the situation where transpose(b) = band CW0(f), CW(ρˆ C, p) have horizontal-glide-reflection symmetries. Example 14.12 illustrates the case d= 3. 13.8. Proposition. Suppose that Notation 13.1 holds. Let d∈Z. The following are equivalent: (a) (x+ 1,−y+d)∈Isom(CW0(f)). (b) (R, L)·f·(−z+d) = f. (c) transpose(f) = f·(z+d). (d) There exists F∈Aut+hA, B, Cisuch that transpose(F·F) = F · (z+d). If Fis as in (d), then dis odd, F=FdPF−1 −1,F∈Aut(hA, B, Ci,CW(F)), and the resulting permutation of cells of CW(F)is the same as that for the horizontal glide-reflection (x+ 1,−y+d)∈Isom(CW0(f)). Proof. The second and fourth columns of Table 12.2.1 will be useful. Notice that (a) and (b) are equivalent by Corollary 11.3, and clearly (b) and (c) are equivalent.
54 JAMES W. CANNON∗AND WARREN DICKS† On taking quotients, we see that (d) implies (c). Now suppose that (c) holds. Here, for all n∈Z,Mf−nM=fn+d, that is, fn6=MfnM=f−n+d. Notice that dmust be odd. Let F:= FdPF−1 −1and G:= transpose(FF). Now quotient(G) = quotient(transpose(FF)) = transpose(f) = f·(z+d). Also G0=FF−1P−1=Fd. It follows that G=F ·(z+d) and (d) holds. For each (m, n)∈Z2, F−1(Dm FF0 n([B])) = Dm F(F−1(F0 n([B]))) = Dm F(F−1(G0 n−d([B]))) =(Dm FF0 −n+d([B])) if R=fn(6=f−n+d), Dm−1 FF0 −n+d([B])) if L=fn(6=f−n+d). As in the previous proof, one can show that F−1acts as a self-homeo- morphism of S2(F) which permutes the two-cells of CW(F) in the same way that (x−1,−y+d) does. The result now follows. ¤ We next consider the situation where interchange(b) = band CW0(f), CW(F) have vertical-glide-reflection symmetries. Example 14.14 illustrates the case d= 6. 13.9. Proposition. Suppose that Notation 13.1 holds. Let d∈Z. The following are equivalent: (a) (−x, y +d)∈Isom(R2,CW0(f)). (b) (R, L)·f·(z+d) = f. (c) interchange(f) = f·(z+d). (d) There exists G∈Aut−hA, B, Cisuch that transpose(G·F) = F · (z+d). If Gis as in (d), then d6= 0,G=FdMF−1 0,G∈Aut(hA, B, Ci,CW(F)), and the resulting permutation of cells of CW(F)is the same as that for the vertical glide-reflection (−x, y +d)∈Isom(CW0(f)). Proof. The last two columns of Table 12.2.1 will be useful. Notice that (a) and (b) are equivalent by Corollary 11.3, and clearly (b) and (c) are equivalent. On taking quotients, we see that (d) implies (c). Now suppose that (c) holds. Here, for all n∈Z,Mfn+dM=fn, that is, fn6=MfnM=fn+d. Hence d6= 0. Let G:= FdMF−1 0and G:= transpose(GF). Then quotient(G) = interchange(f) = f·(z+d). Also G0=GF0P−1MP−1=GF0M=Fd.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 55 It follows that G=F ·(z+d) and (d) holds. For each (m, n)∈Z2, G−1(Dm F(F0 n([B]))) = D−m F(G−1(F0 n([B]))) = D−m F(G−1(G0 n−d([B]))) =D−m FF0 n−d([B]). As in the previous proof, one can show that G−1acts as a self-homeomor- phism of S2(F) which permutes the two-cells of CW(F) in the same way that (−x, y −d) does. The result now follows. ¤ 13.10. Remarks. Let G∈AuthA, B, Ci. We claim that G∈Aut(hA, B, Ci,CW(F)) if and only Gsatisfies the equivalent conditions of Lemma 13.4. Clearly if G∈Aut(hA, B, Ci,CW(F)) then Gacts on the plane S2(F)− {qS2(F)}, and hence Lemma 13.4(b) holds. Conversely, if Lemma 13.4(d) holds then Propositions 13.5–13.9 together imply that G∈Aut(hA, B, Ci,CW(F)). The claim is now proved. In particular, Aut(hA, B, Ci,S2(F)) = hA, B, Ci·Aut(hA, B, Ci,CW(F)) and here hA, B, Ci ∩ Aut(hA, B, Ci,CW(F)) = hDFi.¤ We now have the main result of this section. 13.11. Theorem. Suppose that Notation 13.1 holds. Then CW0(f)and CW(f)are Isom(CW0(f))-CW-complexes, and CW(F) is an Aut(hA, B, Ci,CW(F))-CW-complex. There exists a group isomorphism Isom(CW0(f)) ∼ →Aut(hA, B, Ci,CW(F)),(13.11.1) and it is compatible with some isomorphism CW(f)∼ →CW(F) (13.11.2) of oriented, colored CW-complexes, that respects the Z2-labellings of the two-cells. Proof. We observed that CW0(f) and CW(f) are Isom(CW0(f))-CW-com- plexes, in Remarks 13.2. By definition, CW(F) is an Aut(hA, B, Ci,CW(F))-CW-complex. By Theorem 7.12 and Definition 7.13, we have an isomorphism (13.11.2) of oriented, colored Z-CW-complexes, respecting the Z2-labelling. By Propositions 13.5–13.9, Isom(CW0(f)) and Aut(hA, B, Ci,CW(F)) determine the same group of permutations of the set of cells of CW(F). Thus we have an isomorphism (13.11.1) that is compatible with (13.11.2) as a map between sets of cells, but not necessarily as a map of spaces.
56 JAMES W. CANNON∗AND WARREN DICKS† It remains to adjust (13.11.2). It is not difficult to see that the quotient orbifolds Isom(CW0(f))\CW(f) and Aut(hA, B, Ci,CW(F))\CW(f) are CW-complexes that are isomorphic. However, it is possible for a half-turn to fix a cell, and then the quotient orbifold has a double point in the interior of a two-cell. We alter the quotient CW-structures by making each such double point into a zero-cell, and then joining it by a new one-cell to an arbitrary zero-cell in the two-cell. We fix a CW-isomorphism between the two refined structures, and also lift the refined CW-structures back to the original spaces. It is now straightforward to adjust (13.11.2) to make it compatible with (13.11.1), since (13.11.2) is determined as a map of sets of cells, and we lift the quotient homeomorphism of cells. ¤ 13.12. Remarks. By Remarks 11.4, the following statements hold. Let CW0(F) denote the set of zero-cells of CW(F). The group of permutations of CW0(F) determined by Aut(hA, B, Ci,CW(F)) is of index at most two in the group of those permutations of CW0(F) which can be extended to color-permuting CW-automorphisms of CW(F). Moreover, the index equals two if and only if b∈ {Q Z RL, Q Z RRL, Q Z RLL, Q Z RRLL }. In each of these index-two cases, for each closed two-cell of CW(F), there is a color-preserving, orientation-reversing, Z-action-preserving, CW-automorphism νof CW(F) that carries the given closed two-cell to itself; see Figures 14.9.1, 14.10.1, and 14.11.1. ¤ 14. The symmetry group of CW(ρˆ C, qˆ C) Using the results of the previous section, we now deduce consequences for ˆ C. 14.1. Notation. Let ρˆ C∈Rˆ Cand suppose that ρˆ Cis given by a C-T-model. Choose any parabolic point qˆ Cfor ρˆ C(hA, B, Ci). Choose any F ∈ IrAutABCMaps that codifies (ρˆ C, qˆ C). Let f= quotient(F)∈IrRLMaps. Let b=b(ρˆ C) = b(F) = [f]∈IrRLMaps /Isom+(Z). ¤ 14.2. Definitions. Suppose that Notation 14.1 holds. The symmetry group of ρˆ Cis defined to be Symm(ρˆ C) := {(σ, G)∈Aut(ˆ C)×AuthA, B, Ci | iσ·ρˆ C=ρˆ C·G}, the stabilizer of ρˆ Cunder the action of Aut(ˆ C)×AuthA, B, Cion Rˆ C.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 57 The symmetry group of (ρˆ C, qˆ C) is defined to be Symm(ρˆ C, qˆ C) : = {(σ, G)∈Symm(ρˆ C)|σ(qˆ C) = qˆ C}, ={(σ, G)∈Symm(ρˆ C)|G(DF) = D±1 F}. Let Aut(ˆ C,Im ρˆ C) denote the normalizer of ρˆ C(hA, B, Ci) in Aut(ˆ C). We define the symmetry group of CW(ρˆ C, qˆ C)), denoted Aut(ˆ C,CW(ρˆ C, qˆ C)), to be the subgroup of Aut(ˆ C) consisting of the elements which act as color-permuting CW-automorphisms of CW(ρˆ C, qˆ C). ¤ 14.3. Remark. We conjecture that, in the above definition of Aut(ˆ C,CW(ρˆ C, qˆ C)), the color-permuting restriction is redundant. Even more, we conjecture that, at each zero-cell of CW(ρˆ C, qˆ C), the four cyclically-ordered incident one-cells form well-defined angles, two of 180◦ and two of 0◦, and that the two incident open two-cells that fill in the angles of 180◦have the same color as the zero-cell. ¤ We first consider what happens before a distinguished element of hA, B, Ciis chosen. Here we have rigidity related to Mostow rigidity; see also Theorem 14.8. 14.4. Lemma. Suppose that Notation 14.1 holds. Projection on the first and second coordinates give isomorphisms Aut(ˆ C,Im ρˆ C)∼ ←Symm(ρˆ C)∼ →Aut(hA, B, Ci,S2(F)), and the induced isomorphism Aut(hA, B, Ci,S2(F)) ∼ →Aut(ˆ C,Im ρˆ C) (14.4.1) is compatible with the homeomorphism γ(F, ρˆ C): S2(F)∼ →ˆ C(ρˆ C, qˆ C). Proof. It is straightforward to check that the first projection is an isomorphism, since ρˆ Cis injective. We show next that the second projection is injective. Suppose not, so the centralizer of ρˆ C(hA, B, Ci) in Aut(ˆ C) contains a non-trivial element. Lifting back, we obtain some γ6=±1 in SL2(C)ohzisuch that the preimage Sof ρˆ C(hA, B, Ci) in SL2(C) acts by conjugation on {γ, −γ}. Hence S2is a subgroup of S, of index at most two, which centralizes γ. Now S2contains a parabolic element; by conjugating and squaring, if necessary, we may assume that ( 1 1 0 1 )∈S2. Hence one of γ,−γ,γz, or −γz is of the form α= ( 1a 0 1 ) for some a∈C. Replacing γwith −γif necessary we may assume that γor γz is α. Also S2contains a non-trivial non-parabolic element β.
64 JAMES W. CANNON∗AND WARREN DICKS† 14.6. Corollary. The natural group isomorphisms Isom(CW0(f)) ∼ →Aut(hA, B, Ci,CW(F)) ∼ →Aut(ˆ C,CW(ρˆ C, qˆ C)) are compatible with the previously constructed isomorphisms CW(f)∼ →CW(F)∼ →CW(ρˆ C, qˆ C). of oriented, colored Z-CW-complexes. ¤ 14.7. Remarks. Suppose that Notation 14.1 holds. (i). By conjugating by an element of Aut(ˆ C), we may assume that ρˆ C(DF) is z+ 2. Here qˆ C=∞, and Aut(ˆ C,CW(ρˆ C, qˆ C)) behaves somewhat like Isom(CW0(f)). By considering all possible conjugation actions on hρˆ C(DF)i =hz+ 2i, we find that, in Aut(ˆ C,CW(ρˆ C, qˆ C)), the vertical translations are of the form z+b,b∈C−R; the horizontal glide-reflections are of the form z+b,b∈C−Ri; and, the vertical glide-reflections are of the form −z+b, b∈C−R. (ii) By Remarks 13.10, we see that Aut(ˆ C,Im ρˆ C) = ρˆ C(hA, B, Ci)· Aut(ˆ C,CW(ρˆ C, qˆ C)) and ρˆ C(hA, B, Ci)∩Aut(ˆ C,CW(ρˆ C, qˆ C)) = ρˆ C(hDFi). An analogous statement holds for Symm(ρˆ C). ¤ We can reformulate the work of Jørgensen [6] mentioned in Section 10. 14.8. Theorem. Suppose that Notation 14.1 holds. Then (14.4.1) is a discrete faithful representation extending ρˆ C. Proof. It remains to show that Aut(ˆ C,Im ρˆ C) is a discrete subgroup of Aut(ˆ C). Consider first the case where Isom(CW0(f)) contains a non-identity vertical translation. Let ddenote the smallest positive integer dsuch that (x, y +d)∈ Isom(CW0(f)). Then h(x+ 2, y),(x, y +d)ihas index at most four in Isom(CW0(f)), since each element of the latter is a translation, or a half-turn, or a horizontal- or a vertical-glide reflection. By Prop. 13.6, the isomorphism Isom(CW0(f))∼ →Aut(hA, B, Ci,CW(F)) carries (x, y +d) to F:= FdF−1 0=F0· d Q n=1 fn·F−1 0. Hence hF, DFihas index at most four in Aut(hA, B, Ci,CW(F)). Thus, hA, B, C, Fihas index at most four in hA, B, Ci · Aut(hA, B, Ci,CW(F)), and the latter equals Aut(hA, B, Ci,S2(F)), by Remarks 13.10.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 65 The work of Jørgensen[6] mentioned in Section 10 shows that the isomorphism in the statement of the theorem carries hBA, BC, Fito a discrete subgroup of Aut(ˆ C,Im ρˆ C); moreover, it is a subgroup of index at most eight, by the preceding discussion. Hence Aut(ˆ C,Im ρˆ C) is itself discrete. It remains to consider the case where Isom(CW0(f)) contains no nonidentity vertical translation. Then Isom(CW0(f)) contains no vertical-glide-reflection symmetry. It follows that Isom(CW0(f))/h(x+2, y)ihas order at most two, and Isom(CW0(f)) can contain a half-turn symmetry, or a horizontal-glide-reflection symmetry, or neither. (The resulting quotient orbifold is a once-punctured sphere with two double points, or a once-punctured projective plane, or a twice-punctured sphere, respectively. In other words, an open disc with two double points, or an open M¨obius band, or an open annulus, respectively.) Hence hDFihas index at most two in Aut(hA, B, Ci,CW(F)). Thus, hA, B, Cihas index at most two in hA, B, Ci·Aut(hA, B, Ci,CW(F)), and this is Aut(hA, B, Ci,S2(F)), by Remarks 13.10. Hence, ρˆ C(hA, B, Ci) is a discrete subgroup of Aut(ˆ C,Im ρˆ C) of index at most two, and Aut(ˆ C,Im ρˆ C) is itself discrete. ¤
66 JAMES W. CANNON∗AND WARREN DICKS† 14.9. Example. Let b=QZRL. Here backwards(b) = b, interchange(b) = b, and transpose(b) = b. Hence a tessellation of Ccorresponding to this case has a half-turn symmetry, has a vertical glide-reflection symmetry, and has a horizontal glide-reflection symmetry. The planar symmetry group is of type p g g. The quotient orbifold is a projective plane with two double points. Figure 14.9.1. Part of the tessellation for b(ρˆ C) = Q Z RL.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 67 14.10. Example. Let b=QZRLL. Here backwards(b) = b, interchange(b)6=b, and transpose(b)6=b. Hence a tessellation of Ccorresponding to this case has a half-turn symmetry, has no vertical glide-reflection symmetry, and has no horizontal glide-reflec- tion symmetry. The planar symmetry group is of type p 2. The quotient orbifold is a sphere with four double points. Figure 14.10.1. Part of the tessellation for b(ρˆ C) = Q Z RLL. ¤
68 JAMES W. CANNON∗AND WARREN DICKS† 14.11. Example. Let b=QZRRLL. Here backwards(b) = b, interchange(b) = b, and transpose(b) = b. Hence a tessellation of Ccorresponding to this case has a half-turn symmetry, has a vertical glide-reflection symmetry, and has a horizontal glide-re- flection symmetry. The planar symmetry group is of type p g g. The quotient orbifold is a projective plane with two double points. Figure 14.11.1. Part of the tessellation for b(ρˆ C) = Q Z RRLL.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 69 14.12. Example. Let b=QZRLRRLL. Here backwards(b)6=b, interchange(b)6=b, and transpose(b) = b. Hence a tessellation of Ccorresponding to this case has no half-turn symmetry, has no vertical glide-reflection symmetry, and has a horizontal glide-re- flection symmetry. The planar symmetry group is of type p g. The quotient orbifold is a Klein bottle. Figure 14.12.1. Part of the tessellation for b(ρˆ C) = Q Z RLRRLL.
70 JAMES W. CANNON∗AND WARREN DICKS† 14.13. Example. Let b=QZRLLRRRLLLL. Here backwards(b)6=b, interchange(b)6=b, and transpose(b)6=b. Hence a tessellation of Ccorresponding to this case has no half-turn symmetry, has no vertical glide-reflection symmetry, and has no horizontal glide-reflection symmetry. The planar symmetry group is of type p 1. The quotient orbifold is a torus. Figure 14.13.1. Part of the tessellation for b(ρˆ C) = Q Z RLLRRRLLLL.
FRACTAL TESSELLATIONS ARISING FROM HYPERBOLIC TORUS BUNDLES 71 14.14. Example. Let b=QZRLLRRRLRRLLL. Here backwards(b)6=b, interchange(b) = b, and transpose(b)6=b. Hence a tessellation of Ccorresponding to this case has no half-turn symmetry, has a vertical glide-reflection symmetry, and has no horizontal glide-reflection symmetry. The planar symmetry group is of type p g. The quotient orbifold is a Klein bottle. Figure 14.14.1. Part of the tessellation for b(ρˆ C) = Q Z RLLRRRLRRLLL. Acknowledgments We are grateful to Makoto Sakuma for several inspiring conversations. We have benefited substantially from comments by Brian Bowditch and Joan Porti. All the figures were produced using the Wolfram Research, Inc. computer software system Mathematicar, Version 4.2. References [1] R. C. Alperin, Warren Dicks and J. Porti, The boundary of the Gieseking tree in hyperbolic three-space, Topology Appl. 93 (1999), 219–259.
72 JAMES W. CANNON∗AND WARREN DICKS† [2] F. Bonahon, Bouts des vari´et´es hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71–158. [3] Brian H. Bowditch The Cannon-Thurston map for punctured-surface groups, preprint, 2002. [4] J. W. Cannon and W. P. Thurston, Group invariant Peano curves, preprint, 1989. [5] James W. Cannon and Warren Dicks, On hyperbolic once-punctured-torus bundle, Geom. Dedicata (2) 94 (2002), 141–183. Errata and addenda: http://mat.uab.es/∼dicks/Cannon.html [6] Troels Jørgensen, On pairs of once-punctured tori, pp. 183–207 in Kleinian Groups and Hyperbolic 3-Manifolds, Proceedings of the Warwick Workshop, September 2001, London Math. Soc. Lecture Notes 299, CUP, Cambridge, 2003. [7] Darryl McCullough, Automorphisms of punctured-surface bundles, pp. 179–209 in Geometry and topology (Athens, Ga., 1985), Lecture Notes in Pure and Appl. Math. 105, Dekker, New York, 1987. [8] Curtis T. McMullen, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), 35–91. [9] Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), 559–626. [10] W. Thurston, Hyperbolic structures on 3-manifolds II: surface groups and manifolds which fiber over the circle, preprint. James W. Cannon Department of Mathematics, Brigham Young University, Provo UT 84602, U.S.A. E-mail :[email protected] Warren Dicks Dept. de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), SPAIN E-mail :[email protected] Web :www: http://mat.uab.es/∼dicks Fax :++34 -93-581-2790