Rational maps, subdivision rules, and Kleinian groups

Transcription

1 Rational maps, subdivision rules, and Kleinian groups J. Cannon 1 W. Floyd 2 W. Parry 3 1 Department of Mathematics Brigham Young University 2 Department of Mathematics Virginia Tech 3 Department of Mathematics Eastern Michigan University Dynamical Systems seminar, 2 May 2008

2 Cannon s Conjecture Conjecture: If G is a Gromov-hyperbolic discrete group whose space at infinity is S 2, then G acts properly discontinuously, cocompactly, and isometrically on H 3. Suppose G is a group and Γ is a locally finite Cayley graph. G is Gromov-hyperbolic if Γ has thin triangles. Points in the space at infinity are equivalence classes of geodesic rays; R S if sup{d(r(t), S(t)) : t 0} <. How do you proceed from combinatorial/topological information to analytic information?

3 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 H ρ of R, ρ-area A ρ of R, ρ-circumference C ρ of R moduli M ρ = H 2 ρ /A ρ and m ρ = A ρ /C 2 ρ moduli M(R) = sup ρ H 2 ρ /A ρ and m(r) = inf ρ A ρ /C 2 ρ The sup and inf exist, and are unique up to scaling. (This follows from compactness and convexity.)

4 Optimal weight functions - an example fat flows skinny cuts

5 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 {S i } 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),

6 Cannon-Swenson G a Gromov-hyperbolic group, Γ a locally finite Cayley graph, base vertex O space at infinity Γ : points are equivalence classes of geodesic rays based at O half-space H(R, n) = {x Γ: d(x, R([n, )) d(x, R([0, n])} disk at infinity D(R, n) = {[S] Γ : lim t d(s(t), Γ \ H(R, n)) = } cover D(n) = {D(R, n): R is a geodesic ray based at O}

7 Cannon-Swenson Theorem (C-Swenson): In the setting of Cannon s conjecture, it suffices to prove that the sequence {D(n)} n N 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), The disks at infinity give a basis for the topology of Γ. The CRMT implies there is a quasiconformal structure on Γ. It is quasiconformally equivalent to an analytic structure. The group action is uniformly quasiconformal so by Sullivan/Tukia it is conjugate to a conformal action. The linear recursion follows from finite cone types. Theorem (C): If G is a cocompact, discrete group of isometries of hyperbolic space, then G has a linear recursion. J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984),

8 Definition of a finite subdivision rule R C-F-P, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), (electronic). subdivision complex S R S R 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(S R ) of S R subdivision map σ R : R(S R ) S R σ R 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 S R 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.

9 Example 1. Pentagonal subdivision rule The subdivision of the tile type and the first three subdivisions. (The subdivisions are drawn using Stephenson s CirclePack).

10 Example 2. The barycentric subdivision rule 0! 0! e 1 e 1! 1 e 1 d! e 3 e 1 e 2 e 2 e 3!! a b e 2 c! e 3 e 2 e 2 e 2! e e 1 e 3 e 3 e 3 e 1! "! #!! 1 e 2 e 3 e 1! "!

11 Example 3. The dodecahedral subdivision rule The subdivisions of the three tile types. Note that there are two edge types.

12 Example 3. The dodecahedral subdivision rule The first two subdivisions of the pentagonal tile type.

13 This subdivision rule on the sphere at infinity The dodecahedral subdivision rules comes from the recursion at infinity for a Kleinian group.

14 Critically finite branched maps An orientation-preserving branched map f : S 2 S 2 is critically finite if P f, the set of post-critical points, is finite. Two such maps f and g are equivalent if there is a homeomorphism h : S 2 S 2 such that h(p f ) = P g, (h f ) Pf = (g h) Pf, and h f is isotopic, rel P f, to g h. If a finite subdivision rule R is orientation-preserving and has subdivision complex a 2-sphere, then the subdivision map σ is critically finite. (The post-critical points are vertices of R.)

15 Realizing subdivision maps by rational maps When can the subdivision map of a finite subdivision rule be realized by a rational map? Theorem (C-F-Kenyon-P): Suppose R is an orientation-preserving finite subdivision rule which has bounded valence, mesh approaching zero, and subdivision complex a 2-sphere. If R is conformal (in the sense of Cannon), then it is equivalent to a rational map.

16 The pentagonal subdivision rule 0 The pentagonal subdivision rule is closely associated with a fsr which is realizable by a rational map. Here are subdivisions drawn by CirclePack and by preimages under the rational map (unfolding by z z 5 ). 0 e 3 e 2 e 1 a b e 2 e 3 e 1 c d e 3 e 2 e 1 e 2 e 3 #" e3 1 e 1 e e 1!" 1 e 2 e 1!"

17 The barycentric subdivision rule The barycentric subdivision rule can be realized by the rational map f (z) = 4(z2 z+1) 3 27z 2 (z 1) 2. Here is the Julia set.

18 The barycentric subdivision rule R { } is the 1-skeleton. Here are f 1 (R { }) and f 2 (R { }).

19 Realizing rational maps by fsr s Theorem (C-F-P, Bonk-Meyer): If f is a critically finite rational map without periodic critical points, then every sufficiently large iterate of f is equivalent to the subdivision rule of a fsr. Our proof: Pick a simple closed curve containing the post-critical points. For a sufficiently large iterate, that curve can be approximated by a curve in its preimage. Now use the expansion complex machinery. Do you need to pass to an iterate of the map? What about critically finite maps with periodic critical points? This corresponds to fsr s with unbounded valence.

20 Definitions for the subdivision space Let G be a Gromov-hyperbolic group, and let Γ be a locally-finite Cayley graph for G with base vertex O. cone to infinity of x Γ C(x) = {y Γ: d(o, y) = d(o, x) + d(x, y)} Γ shadow of x Γ S(x) = {[R]: x R} Γ vertex set of x Γ (and n N) V (x, n) = {g G : g = n and x S(g)} pretile of X {g G : g = n} t(x) = {x Γ : V (x, n) = X} star neighborhood of X {g G : g = n} T (X) = {t(y ): Y X}

21 The pentagonal reflection group As an example, consider the group generated by reflections in the edges of a right-angled pentagon. a e 1 b d c

22 Some pretiles for the pentagonal reflection group a b c d e a a,e e a a a a a b b b b b c c c c c d d d d d e e e e e d,e d c c,d b,c b a,b

23 The dodecahedral reflection group Now consider the group generated by reflections in the faces of a right-angled dodecahedron.

24 Some pretiles for the dodecahedral reflection group

25 O(Γ) Two vertex sets X 1 and X 2 are equivalent if there is a group element g such that g(x 1 ) = X 2 and left-multiplication by g preserves the subdivision lattices of the pretiles t(x 1 ) and t(x 2 ). Let X 1,..., X k be representatives of the equivalence classes of vertex sets. O(Γ) = T (X 1 ) T (X 2 ) T (X K )/ There is a subdivision map S : O(Γ) O(Γ). Theorem: If G is torson free and G = S 2, then O(Γ) is a compact non-hausdorff 2-manifold.

26 Part of O(Γ) for the pentagonal reflection group de de b ab a,bc e d,e d e,cd c,ab b b b,ae ae ae b,ae d,ae d,ae ae d c,d c c,ab c,ab a,e e,ab a b,ae b,c b,ae c d,ae ab c,ab ab e a,de ae a,b c c,de d e,ab e,ab ab a,bc c,de c,de e,ab d,ae de d,e de a,de a,de d,bc a,de b,cd e bc d,bc bc a,b a d,bc bc b,cd a a,bc bc d c,de cd b,c cd a,bc b b,cd b,cd cd e,cd e,cd a,e a d,bc cd e,cd e c,d c

27 Conformal structures on O(Γ) Theorem: If G is a Kleinian group, then there is a conformal structure on O(Γ) with respect to which S is conformal. Theorem: Let G be a torsion-free Gromov-hyperbolic group with G = S 2. Suppose there is a conformal structure on O(Γ) such that S is conformal. Then G is a Kleinian group. One shouldn t need to assume that G is torsion free, but we haven t checked all of the details if G has torsion. Instead of assuming that S is conformal, it should suffice to assume that there is a homeomorphism h : O(Γ) O(Γ) such that h is isotopic to the identity and h S is conformal. This has the feel of a statement that G is Kleinian if the induced map from S on the Teichmüller space of O(Γ) has a fixed point. But what is the Teichmüller space of O(Γ)?