Expansion complexes for finite subdivision rules

W. Floyd (joint work with J. Cannon and W. Parry)

Department of Virginia Tech

Spring and Dynamics Conference : March, 2013

Bill Floyd Expansion complexes for finite subdivision rules Definition of a finite subdivision rule R

C-F-P, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196 (electronic).

finite subdivision complex SR

SR is the union of its closed 2-cells. Each 2-cell is modeled on an n-gon (called a tile type) with n ≥ 3.

subdivision R(SR) of SR

subdivision map σR : R(SR) → SR

σR is cellular and takes each open cell homeomorphically onto an open cell. R-complex: a 2-complex X which is the closure of its 2-cells, together with a structure map h: X → SR which takes each open cell homeomorphically onto an open cell One can use a finite subdivision rule to recursively subdivide R-complexes. R(X) is the subdivision of X.

Bill Floyd Expansion complexes for finite subdivision rules Example 1. The pentagonal subdivision rule

The subdivision of the tile type and the first three subdivisions. (The subdivisions are drawn using Stephenson’s CirclePack).

→

Bill Floyd Expansion complexes for finite subdivision rules Example 2. The dodecahedral subdivision rule

The subdivisions of the three tile types. Note that there are two edge types.

Bill Floyd Expansion complexes for finite subdivision rules The second subdivision of the quadrilateral

Bill Floyd Expansion complexes for finite subdivision rules The third subdivision of the quadrilateral

Bill Floyd Expansion complexes for finite subdivision rules Expansion complexes

An expansion R-complex is an R-complex X with structure 2 map f : X → SR such that X is homeomorphic to R and there is an orientation-preserving homeomorphism ϕ: X → X with σR ◦ f = f ◦ ϕ. Expansion complexes arise as direct limits of sequences of subdivisions.

Bill Floyd Expansion complexes for finite subdivision rules The pentagonal expansion complex

P.L. Bowers and K. Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68 (electronic.)

Bill Floyd Expansion complexes for finite subdivision rules An expansion complex for the dodecahedral fsr

Bill Floyd Expansion complexes for finite subdivision rules This subdivision rule on the sphere at infinity

The dodecahedral subdivision rule comes from the at infinity for a Kleinian group. The figure was drawn using Weeks’s program SnapPea.

Bill Floyd Expansion complexes for finite subdivision rules Cannon’s Conjecture

Cannon’s Conjecture: If G is a Gromov-hyperbolic discrete group whose space at infinity is S2, then G acts properly discontinuously, cocompactly, and isometrically on H3. While a primary motivation for this was Thurston’s Hyperbolization Conjecture, even after Perelman’s proof of the Geometrization Conjecture this conjecture is still open. How do you proceed from combinatorial/topological information to analytic information?

Given a sequence of subdivisions of a tiling, how do you understand/control the shapes of tiles? When can you realize the subdivisions so that the subtiles stay almost round?

Bill Floyd Expansion complexes for finite subdivision rules Weight functions, combinatorial moduli

shingling (locally-finite covering by compact, connected sets) T on a surface S, ring (or quadrilateral) R ⊂ S

weight function ρ on T : ρ: T → R≥0 ρ-length of a curve, ρ-height Hρ of R, ρ-area Aρ of R, ρ-circumference Cρ of R 2 2 moduli Mρ = Hρ /Aρ and mρ = Aρ/Cρ 2 2 moduli M(R) = supρ Hρ /Aρ and m(R) = infρ Aρ/Cρ The sup and inf exist, and are unique up to scaling. (This follows from compactness and convexity.)

Bill Floyd Expansion complexes for finite subdivision rules Optimal weight functions - an example

162 324 324 162 562 562 405 405 843 281 281 843 281 405 405 281 561 270 561 844 270 270 844 644 644 644 844 483 483 844 483 483

905 905 905 905

Bill Floyd Expansion complexes for finite subdivision rules Combinatorial Riemann Mapping Theorem

Now consider a sequence of shinglings of S. Axiom 1. Nondegeneration, comparability of asymptotic combinatorial moduli Axiom 2. Existence of local rings with large moduli conformal sequence of shinglings: Axioms 1 and 2, plus mesh locally approaching 0.

Theorem (C): If {Si } is a conformal sequence of shinglings on a topological surface S and R is a ring in S, then R has a metric which makes it a right-circular annulus such that analytic moduli and asymptotic combinatorial moduli on R are uniformly comparable. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155–234.

Bill Floyd Expansion complexes for finite subdivision rules The Cannon-Swenson Theorem

Theorem (C-Swenson): In the setting of Cannon’s conjecture,

it suffices to prove that the sequence {D(n)}n∈N of disks at infinity is conformal. Furthermore, the D(n)’s satisfy a linear recursion. J. W. Cannon, E. L. Swenson, Recognizing constant curvature groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998), 809–849.

Bill Floyd Expansion complexes for finite subdivision rules Applications of expansion complexes

The pentagonal subdivision rule is conformal; the dihedral makes showing this much easier. For potential applications, you would like to be able to make use of rotational symmetry. The intuition is that for Cannon’s Conjecture expansion complexes correspond to tangent spaces at infinity, and at fixed points of loxodromic elements you will see rotational (but not dihedral) symmetry.

Theorem: If a one-tile rotationally invariant finite subdivision rule has bounded valence and mesh approaching zero, then it has an invariant (partial) conformal structure.

Theorem: If a finite subdivision rule R has bounded valence, mesh approaching zero, and an invariant (partial) conformal structure, then it is conformal.

Bill Floyd Expansion complexes for finite subdivision rules Example 3, a one-tile rotationally invariant fsr

Bill Floyd Expansion complexes for finite subdivision rules A key observation for rotational invariance

Bill Floyd Expansion complexes for finite subdivision rules The expansion complex

One can put a piecewise conformal structure on the expansion complex X with regular pentagons, and then use power maps to extend over the vertices. (This is inspired by the Bowers-Stephenson construction.) The expansion map agrees with a conformal map on the vertices. One can conjugate to get a new fsr for which this conformal map is the expansion map. The subdivision map is conformal with respect to the induced conformal structure on the subdivsion complex. C-F-P, Expansion complexes for finite subdivision rules I, Conform. Geom. Dyn. 10 63–99 (2006) (electronic) C-F-P, Expansion complexes for finite subdivision rules II, Conform. Geom. Dyn. 10 326–354 (2006) (electronic)

Bill Floyd Expansion complexes for finite subdivision rules Growth series for expansion complexes

Suppose X is an expansion complex and S is a seed for X. You can define a norm on the tiles of X by assigning norm 0 to the tiles of S, assigning norm 1 to the tiles of X \ int(S) that intersect S in at least an edge, and continuing recursively. For n ≥ 0, let sn be the number of tiles of norm n and let bn be the number of tiles of norm at most n. Maria Ramirez-Solano asked last fall whether, for the s pentagonal expansion complex, lim n = 0? n→∞ bn More generally, what kind of growth can an expansion complex have? And how nice is the growth series ∞ X n snx ? n=0

Bill Floyd Expansion complexes for finite subdivision rules Growth series for expansion complexes

Suppose R is a finite subdivision rule with bounded valence and mesh approaching zero. Let X be an expansion complex for R, let S be a seed of X, and let p be the center (fixed point of the expansion map). There exist a > 0 and b > 1 such that for any n ∈ N,

dist(p, ∂Rn(S)) ≥ abn.

There exist c > 0 and d > 1 such that for any n ∈ N,

#(Rn(S)) ≤ cd n.

#(B(abn)) ≤ cd n. n lim (cd n)1/(ab ) = 1, so the growth of the series is n→∞ subexponential.

Bill Floyd Expansion complexes for finite subdivision rules Example 4

Bill Floyd Expansion complexes for finite subdivision rules Example 4

∞ X n The growth series snx = n=0 4(1 + x + 3x2 + 3x3 + 9x4 + 9x5 + 9x6 + 9x7 + 27x8 + ... ). ln(3)/ ln(2) sn ≤ n , with equality if n is a power of 2.

Bill Floyd Expansion complexes for finite subdivision rules The type problem

How do you tell if an expansion complex is parabolic or hyperbolic? (Here the type can be defined in terms of circle packings or in terms of conformal structures on the complex.) Bowers and Stephenson showed that the pentagonal expansion complex is parabolic by putting a conformal structure on it for which the expansion map is conformal. A conformal fsr cannot have a hyperbolic expansion complex, but in general the type problem is delicate, as shown by the following example.

Bill Floyd Expansion complexes for finite subdivision rules Example 5

Bill Floyd Expansion complexes for finite subdivision rules A hyperbolic expansion complex

Bill Floyd Expansion complexes for finite subdivision rules Part of a parabolic expansion complex

Bill Floyd Expansion complexes for finite subdivision rules laminations

This fsr has bounded valence and mesh approaching zero, and is irreducible. Any compact subcomplex of one of those two expansion complexes is isomorphic to a subcomplex of the other. This example can be viewed in the context of Riemann surface laminations. In a survey paper, Ghys gave an example of Kenyon’s of a Riemann surface lamination that has a hyperbolic leaf and parabolic leaves, and Blanc generalized this greatly in his thesis. E. Ghys, Laminations par surfaces de riemann, Panoramas & Synthèses 8 (2000), 49–95. E. Blanc, Propriétés génériques de laminations, Ph.D. thesis, Université Lyon-1, 2001.

Bill Floyd Expansion complexes for finite subdivision rules