Growth series for expansion complexes
aa r X i v : . [ m a t h . D S ] D ec GROWTH SERIES FOR EXPANSION COMPLEXES
J. W. CANNON, W. J. FLOYD, AND W. R. PARRY
Abstract.
This paper is concerned with growth series for expansioncomplexes for finite subdivision rules. Suppose X is an expansion com-plex for a finite subdivision rule R with bounded valence and meshapproaching 0, and let S be a seed for X . One can define a growthseries for ( X, S ) by giving the tiles in the seed norm 0 and then usingeither the skinny path norm or the fat path norm to recursively definenorms for the other tiles. The main theorem is that, with respect to ei-ther of these norms, the growth series for (
X, S ) has polynomial growth.Furthermore, the degrees of the growth rates of hyperbolic expansioncomplexes are dense in the ray [2 , ∞ ). Suppose T is the set of tiles in a tiling of a plane and S is a nonempty,finite subset of T (often a single tile). We give T a metric by defining thedistance d ( s, t ) between two tiles s and t to be the minimum nonnegativeinteger n , such that there is a finite sequence s , s , . . . , s n of tiles such that s = s , s n = t , and for i ∈ { , . . . , n } s i − ∩ s i contains an edge. For eachnonnegative integer n , let a n be the number of tiles whose distance from anelement of S is n . The growth series for ( T, S ) is the power series P ∞ n =0 a n z n .We are interested in the growth series that arise from tilings. Unlessone imposes additional structure on the tiling, the only requirement for thegrowth series is that each a n ≥
0. One can see this by starting with thecollection of circles in the plane with center the origin and radius a positiveinteger. These circle decompose the plane into the union of a disk and acountable family of annuli. If P ∞ n =0 a n z n is a power series with each a n > a tiles and for each n > n and n + 1 into a n tiles. This produces a pair ( T, S ) with growth series P ∞ n =0 a n z n .By contrast, consider a tiling T of the Euclidean or hyperbolic planecoming from the images, under a cocompact group G of isometries of theplane, of a Dirichlet region D for the action of D . Let the seed S bethe single tile D . Then the growth series for ( T, S ) is the growth seriesfor the group G with respect to the geometric generating set Σ = { g ∈ G : g ( D ) ∩ D is an edge of D } . Cannon shows in [Can84] that if G is a Date : September 24, 2018.2000
Mathematics Subject Classification.
Primary 52C20, 52C26; Secondary 05B45,30F45.
Key words and phrases. growth series, expansion complex, finite subdivision rule. cocompact discrete group of isometries of H n , then with respect to a finitegenerating set for G the growth series is rational. In [Ben83], Benson provesthe analogous result for groups of Euclidean isometries. In [CanW92], Can-non and Wagreich consider (1) the case that D is a hyperbolic triangle whoseangles are submultiples of π and G is the associated group. They explicitlycompute the rational growth function f and prove that f (1) = 1 /χ ( G ) andthat all of the poles of f lie in the unit circle except for a pair of positivereciprocal poles. They also consider (2) the case that D is a hyperbolicpolygon with 4 g sides and G is the fundamental group of a closed orientablesurface with genus g ≥
2, and prove that the rational growth function f has the same properties. There is extensive literature on the special prop-erties of the rational grouwth functions of hyperbolic surface groups. Seefor example the papers of Bartholdi-Ceccherini-Silberstein [BarC02], Floyd[Flo92], Floyd-Plotnick [FloP87, FloP88, FloP94], and Parry [Par93].Inspired by the growth functions for surface groups, and motivated bya question from Maria Ramirez Solano about the growth rate for the pen-tagonal subdivision rule, we decided to consider growth series for expansioncomplexes. Like the tilings coming from surface groups, expansion com-plexes are essentially determined by a finite amount of combinatoiral data.But the growth series are very different. In Theorem 1 we prove that theassociated growth functions all have polynomial growth, so they rarely haverational growth.1. Finite subdivision rules and expansion complexes
While a finite subdivison rule is defined dynamically, in essence it is afinite combinatorial procedure for recursively subdividing appropriate 2-complexes. A finite subdivision rule R consists of (1) a finite 2-complex S R , (2) a subdivision R ( S R ) of S R , and (3) a continuous cellular map σ R : R ( S R ) → S R whose restriction to every open cell is a homeomor-phism. We further require that S R is the union of its closed 2-cells andeach closed 2-cell is the image of a polygon (called its tile type) with atleast three edges by a continuous cellular map whose restriction to eachopen cell is a homeomorphism. An R -complex is a 2-complex which is theunion of its closed 2-cells together with a structure map f : X → S R ; werequire that f is a continuous cellular map whose restriction to each opencell is a homeomorphism. The subdivision R ( S R ) of S R pulls back under f to a subdivision R ( X ) of X ; R ( X ) is an R -complex with structure map σ R ◦ f : R ( X ) → S R . Since R ( X ) is an R -complex, one can subdivide it;this is how one can recursively subdivide complexes with a finite subdivisionrule . See [CanFP01] for the basic theory of finite subdivision rules.As a simple example, consider the pentagonal subdivision rule P whichwas first described in [CanFP01]. The subdivision complex S P has a singlevertex, a single edge, and a single face (which is the image of a pentagon).The subdivision of the tile type is shown in Figure 1. ROWTH SERIES FOR EXPANSION COMPLEXES 3
Figure 1.
The subdivision of the tile type for the pentago-nal subdivision rule P Figure 2. P ( t ), P ( t ), and P ( t )Bowers and Stephenson created the pentagonal expansion complex in[BowS97] as part of their analysis of the pentagonal subdivision rule. Fig-ure 2 shows the first three subdivisions of the tile type t , drawn with Stephen-son’s program CirclePack [Ste]. We can identify t with the central pentagonin its first subdivision P ( t ), and for each positive integer n this induces aninclusion of the n th subdivision P n ( t ) in P n +1 ( t ). The direct limit of thesequence of inclusions P n ( t ) → P n +1 ( t ) is the pentagonal expansion com-plex. Bowers and Stephenson put a conformal structure on the pentagonalexpansion complex by making each open pentagon conformally a regularpentagon, using butterfiles to give charts for open edges, and using powermaps to give charts for vertices. They showed that the expansion complexis conformally equivalent to the plane, and the expansion map (which takeseach P n ( t ) to P n +1 ( t )) is conformal.In our papers [CanFP06a, CanFP06b] we gave the general definition ofan expansion complex for a finite subdivision rule and developed some ofthe theory. An expansion complex for a finite subdivision rule R is an R -complex X which is homeomorphic to R such that there is an orientation-preserving homeomorphism ϕ : X → X such that σ R ◦ f = f ◦ ϕ , where f is the structure map for X . If X is an expansion complex and S is asubcomplex of X , then S is a seed of X if S is a closed topological disk, S ⊂ ϕ ( S ), and X = ∪ ∞ n =0 ϕ n ( S ). For the pentagonal expansion complex,one can take the tile type t to be a seed. It is possible for an expansion J. W. CANNON, W. J. FLOYD, AND W. R. PARRY complex not to have a seed. For example, if the subdivision map is theidentity map then no expansion complex can have a seed. But if R is afinite subdivision rule with bounded valence and mesh approaching 0, thenit follows from [CanFP06a, Lemma 2.5] that every expansion complex for R has a seed for some iterate of R . As in [BowS97] one can put a conformalstructure on an expansion complex by taking each open tile to be conformallyregular, using “butterflies” as charts in neighborhoods of open edges, andusing power maps to extend the conformal structure over the vertices. Wecall an expansion complex parabolic if with this conformal structure it isconformally equivalent to the plane, and hyperbolic if with this conformalstructure it is conformally equivalent to the open unit disk.2. Growth series for expansion complexes
Let R be a finite subdivision rule, let X be an expansion complex for R ,and let S be a seed for X . We define the skinny path norm | · | on the tilesof X by setting norm | t | = 0 if t is in S and if t is not in S then | t | is theminimal positive integer n such that there exist tiles t , . . . , t n such that t is in S , t n = t , and t i ∩ t i − = ∅ for 1 ≤ i ≤ n . For a nonnegative integer n , let s n = { tiles t ⊂ X : | t | = n } and let b n = { tiles t ⊂ X : | t | ≤ n } (so s n is the number of tiles in the combinatorial sphere of radius n and b n is the number of tiles in the combinatorial ball of radius n ). The growthseries for ( X, S ) is the power series P ∞ n =0 b n z n . The growth series has exponential growth if lim sup n →∞ n √ b n > subexponential growth iflim sup n →∞ n √ b n = 1. The growth series has polynomial growth of degree d if d = lim sup n →∞ ln( b n )ln( n ) . The growth series has intermediate growth if thegrowth is neither exponential nor polynomial. Theorem 1.
Let R be a finite subdivision rule with bounded valence andmesh approaching , let X be a R -expansion complex, and let S be a seed for X . Then the growth series for ( X, S ) with respect to the skinny path normhas polynomial growth.Proof. We recall the skinny path distance from [CanFP06a]. If x, y ∈ X , the skinny path distance d ( x, y ) is the minimum integer n such that there is afinite sequence t , . . . , t n of tiles such that x ∈ t , y ∈ t n , and t i − ∩ t n = ∅ for i ∈ { , . . . , n } . The skinny path distance does not define a distance functionon X since two points in the same tile will have skinny path distance 0, butit does define a pseudometric.Since R has mesh approaching 0, there is a positive integer n suchthat the skinny path distance in X from S to ∂ϕ n ( S ) is at least 2. By[CanFP06a, Lemma 2.7], there is a postive integer n such that if x, y ∈ X and d ( x, y ) ≥ d ( ϕ n ( x ) , ϕ n ( y )) ≥ d ( x, y ). It easily follows thatthere are a positive real number a and a real number b > n the skinny path distance from S to ∂ϕ n ( S ) is greaterthan ab n . Since S is compact and there is an upper bound, d , on the number ROWTH SERIES FOR EXPANSION COMPLEXES 5 of subtiles in the first subdivision of a tile type of R , for any positive integer n the number of tiles in ϕ n ( S ) is at most cd n , where c = S . Suppose k ≥ k ) > ln( a ). Then there is a unique positive integer n such that n − < ln( k ) − ln( a )ln( b ) ≤ n . Then ab n − < k ≤ ab n and soln( b k )ln( k ) < ln( cd n )ln( ab n − ) = ln( c ) + n ln( d )ln( a/b ) + n ln( b )and so lim sup k →∞ ln( b k )ln( k ) ≤ ln( d )ln( b ) and the growth series has polynomialgrowth. (cid:3) One can also consider a growth series for (
X, S ) with repect to the fat pathnorm. As above, let R be a finite subdivision rule, let X be an expansioncomplex for R , and let S be a seed for X . We define the fat path norm | · | on the tiles of X by setting the norm | t | = 0 if t is in S and if t is not in S then | t | is the minimal positive integer n such that there exist tiles t , . . . , t n such that t is in S , t n = t , and t i ∩ t i − contains an edge for 1 ≤ i ≤ n .The other definitions in the first paragraph of this section follow exactly asbefore. Since for every nonnegative integer n the number of tiles of fat pathnorm at most n is at most the number of tiles of skinny path norm at most n , one gets the immediate corollary. Corollary 2.
Let R be a finite subdivision rule with bounded valence andmesh approaching , let X be a R -expansion complex, and let S be a seedfor X . Then the growth series for ( X, S ) with respect to the fat path normhas polynomial growth. A family of examples
In all of the examples we consider in this section, the skinny path normsand the fat path norms are the same, so we won’t name the norm.We start with a simple example of an expansion complex for a finitesubdivision rule R . The subdivisions of the two tile types are shown inFigure 3. The subdivision R ( t ) of the tile type t contains a tile in itsinterior which is labeled t , so the tile type t is the seed of an expansioncomplex X . Let ϕ : X → X be the expansion map. For convenience wedenote the seed by S . Figure 4 shows part of the expansion complex, withthe seed in the center. Then s = 1, s n = 2 n +1 if n >
0, and b n = 2 n +2 − n ≥
0. The growth series has exponential growth, but this doesn’tviolate Theorem 1 since R doesn’t have mesh approaching 0.For each positive integer n , let R n = ϕ n ( S ) \ int( S ). By [CanFP06b,Theorem 5.5] X is hyperbolic if lim n →∞ M ( R n , S ( X )) = ∞ , where S ( X ) isthe shingling of X by tiles.Let n be a positive integer. Define a weight function w on R n as follows.If t is a tile of R n , then for some k ∈ { , . . . , n } , t ∈ ϕ k ( S ) \ int( ϕ k − ( S ));we give t weight 2 n − k . The height curves for R n have height H ( R n , w ) = P n − i =0 i = 2 n −
1, and w is the sum of the weights associated to the height J. W. CANNON, W. J. FLOYD, AND W. R. PARRY curves. Hence by [CanFP94, 2.3.6] w is the optimal weight function for R n for fat flow modulus. For each i ∈ { , . . . , n − } , there are 2 n +1 − i tiles in R n with w -weight 2 i . Hence A ( R n , w ) = P n − i =0 n +1 − i · (2 i ) = 2 n +1 (2 n − M ( R n , S ( X )) = M ( R n , w ) = H ( R n , w ) A ( R n , w ) = (2 n − n +1 . Hence lim n →∞ M ( R n , S ( X )) = < ∞ and X is hyperbolic. t t t t t t t t t −→−→ Figure 3.
The subdivisions of the tile types for R The finite subdivision rule R is similar to R but it has been modi-fied to have mesh approaching 0. This time there are three tile types, andthe subdivisions are shown in Figure 5. Tile type t is a seed for an ex-pansion complex X ; part of this expansion complex is shown in Figure 6.One can show as we did for the previous example that X is hyperbolic.The hyperbolicity of X also follows from the proof of [CanFP06b, Lemma5.1]; this example is simpler than the example being analyzed there butthe approach there fits this example as well. Let S be the seed of X consisting of a single tile labeled t , and for each positive integer n let R n = ϕ n ( S ) \ int( S ). Given n , define a weight function w on R n bygiving a tile t in R n weight 3 n − k if t ⊂ ϕ k ( S ) \ int ϕ k − ( S ). The height H ( R n , w ) = P n − k =0 n − − k k = 3 n − n . The weight function w is a sum ofweight functions corresponding to height curves, so it is an optimal weightfunction. The area A ( R n , w ) = 4 P n − k =0 k n − − k = 4 · n − · (3 n − n ), so M ( R n , S ( X )) = M ( R n , w ) = H ( R n ,w ) A ( R n ,w ) = n − n · n − , lim n →∞ M ( R n , S ( X )) = < ∞ , and X is hyperbolic.The finite subdivision rule R is a special case ( R , ) of a two-parameterfamily of finite subdivision rules R p,q for integers p, q ≥
2. For a given p and q , R p,q has three tile types, t (a quadrilateral), t (a quadrilateral), and ROWTH SERIES FOR EXPANSION COMPLEXES 7
Figure 4.
Part of the expansion complex for R t (a ( q +3)-gon) which is viewed as a quadrilateral with the bottom edgesubdivided into q subedges. The tile type t is subdivided into 5 subtiles, acentral tile of type t surrounded by four tiles of type t . The quadrilateral t is subdivided into pq -subtiles, all of type t , arranged in p rows and q columns. The tile type t is also subdivided into pq subtiles arranged in p rows and q columns, with each column in the first p − t and each column in the last row containing a tile of type t . As for the previous two examples, there is an expansion complex X p,q for R p,q whose seed S is a single tile of type t . As before, we denote theexpansion map by ϕ . For each positive integer n we let R n = ϕ n ( S ) \ int( S ),and we put a weight function on R n as follows: if t is a tile of R n and t ⊂ ϕ k ( S ) \ int( ϕ k − ( S )), then the weight of t is q n − k . It follows as for R and R that w is an optimal weight function for M ( R n , S ( X p,q )). If p = q ,the height of R n with respect to w is H ( R n , w ) = P n − k =0 q k p n − − k = q n − p n q − p ,the area is A ( R n , w ) = 4 P n − k =0 ( pq ) n − − k q k = q n − ( q n − p n ) q − p , and the fat flowmodulus of R n is H ( R n , w ) A ( R n , w ) = ( q n − p n ) ( q − p ) q − p q n − ( q n − p n ) = q n − p n q n − ( q − p ) = 1 − ( p/q ) n − p/q ) . J. W. CANNON, W. J. FLOYD, AND W. R. PARRY t t t t t t t t t t t t t t t t t t t t −→−→−→ Figure 5.
The subdivisions of the tile types for R If p = q , then H ( R n , w ) = n · p n − , A ( R n , w ) = 4 n · p n − , and M ( R n , w ) = n . If p < q , then lim sup n →∞ M ( R n , S ( X p,q )) = q q − p ) < ∞ and X p,q ishyperbolic.We next look in more detail at the growth series for X p,q . Suppose p, q ≥ X p,q as an expansion complex for R p,q with seed S a single tileof type t . For looking at the finer detail of the growth series, it is moreconvenient to look at the growth series g ( z ) = P ∞ n =0 s n z n for spheres insteadof the growth series f ( z ) = P ∞ n =0 b n z n for balls. As we saw above, g ( z ) = 1+4 z +4 qz +4 qz + · · · +4 qz p +1 +4 q z p +2 + . . . q z p + p +1 +4 q z p + p +2 + . . . , where each coefficient 4 q k appears p k consecutive times. For example, when p = 2 and q = 3 (the example R ), g ( z ) = 1+4 z +12 z +12 z +36 z + · · · +36 z +108 z + · · · +108 z +324 z + . . . . Since the sequence { s n } has no upper bound on the number of consecutiveterms which are constant, g ( z ) cannot be rational or even D -finite. However, g ( z ) does satisfy a functional equation. Note that q · g ( z p ) = q + 4 qz p + 4 q ( z p + z p + · · · + z p + p ) + 4 q z p +2 p + . . . , ROWTH SERIES FOR EXPANSION COMPLEXES 9
Figure 6.
Part of the expansion complex for R so q · [ g ( z p ) −
1] (1 + z + · · · + z p − ) = z p − ( g ( z ) − − z )and g satisfies the functional equation q · ( g ( z p ) −
1) ( z p − z −
1) = z p − ( g ( z ) − − z ) . We now consider the growth series P ∞ n =0 b n z n for X p,q with respect to theseed S consisting of a single tile labeled t . For convenience we assume that p, q ≥
2. Let n be a nonnegative integer. Then there is a nonnegative integer k such that p k − p − ≤ n < p k +1 − p − . Let m = n − p k − p − . Then 0 ≤ m ≤ p k − n = p k − p − m and b n = 1 + 4 ( pq ) k − pq − m (4 q k ) . When m = 0, ln( b n )ln( n ) = ln( pq + 4( pq ) k − − ln( pq − p k − − ln( p − and in general ln( b n )ln( n ) < ln( pq + 4( pq ) k +1 − − ln( pq − p k +1 − − ln( p − . It follows that the growth series has polynomial growth of degreelim sup n →∞ ln( b n )ln( n ) = ln( pq )ln( p ) = 1 + ln( q )ln( p ) . Since X p,q is hyperbolic whenever q > p , the degrees of the polynomialgrowth rates of hyperbolic expansion complexes with respect to the fat pathnorm are dense in [2 , ∞ ), and the degrees of the polynomial growth rates ofhyperbolic expansion complexes with respect to the skinny path norm aredense in [2 , ∞ ).In his Ph.D. thesis [Woo06], Wood notes that hyperbolic complexes canhave spherical growth rates of degree 1 + ǫ for ǫ arbitrarily small. References [Ben83] M. Benson,
Growth series of finite extensions of Z n are rational , Invent. math. (1983), 251–269.[BarC02] L. Bartholdi and T. G. Ceccherini-Silberstein, Salem numbers and growth seriesof some hyerbolic graphs , Geom, Dedicata (2002), 107–114. arXiv:math/9910067[BowS97] P. L. Bowers and K. Stephenson, A “regular” pentagonal tiling of the plane ,Conform. Geom. Dyn. (1997), 58–68 (electronic).[Can84] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolicgroups , Geom. Dedicata (1984), 123–148.[CanFP94] J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finiteRiemann mapping theorem , The mathematical legacy of Wilhelm Magnus: groups,geometry and special functions (Brooklyn, NY, 1992), Amer. Math. Soc., Providence,RI, 1994, pp. 133–212.[CanFP01] J. W. Cannon, W. J. Floyd, and W. R. Parry,
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CirclePack
ROWTH SERIES FOR EXPANSION COMPLEXES 11 [Woo06] W. E. Wood,
Combinatorial type problems for triangulation graphs , Ph.D. thesis,Florida State University, 2006.
Department of Mathematics, Brigham Young University, Provo, UT 84602,U.S.A.
E-mail address : [email protected] Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, U.S.A.
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