Combinatorial equivalence of topological polynomials and group theory

Transcription

1 Combinatorial equivalence of topological polynomials and group theory Volodymyr Nekrashevych (joint work with L. Bartholdi) March 11, 2006, Toronto V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 1 / 37

2 Topological Polynomials Topological polynomials A branched covering f : S 2 S 2 is a local homeomorphism at every point, except for a finite set of critical points C f. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 2 / 37

3 Topological Polynomials Topological polynomials A branched covering f : S 2 S 2 is a local homeomorphism at every point, except for a finite set of critical points C f. A topological polynomial is a branched covering f : S 2 S 2 such that f 1 ( ) = { }, where S 2 is a distinguished point at infinity. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 2 / 37

4 Topological Polynomials A post-critically finite branched covering (a Thurston map) is an orientation-preserving branched covering such that the post-critical set f : S 2 S 2 P f = n 1 f n (C f ) is finite. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 3 / 37

5 Topological Polynomials Two Thurston maps f 1 and f 2 are combinatorially equivalent if they are conjugate up to homotopies: V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 4 / 37

6 Topological Polynomials Two Thurston maps f 1 and f 2 are combinatorially equivalent if they are conjugate up to homotopies: there exist homeomorphisms h 1,h 2 : S 2 S 2 such that h i (P f1 ) = P f2, the diagram S 2 f 1 S 2 h 1 h 2 S 2 f 2 S 2 is commutative and h 1 is isotopic to h 2 rel P f1. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 4 / 37

7 Rabbit and Airplane Rabbit and Airplane V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 5 / 37

8 Twisted Rabbit Rabbit and Airplane Let f r be the rabbit and let T be the Dehn twist V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 6 / 37

9 Twisted Rabbit Rabbit and Airplane Let f r be the rabbit and let T be the Dehn twist V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 6 / 37

10 Twisted Rabbit Rabbit and Airplane Let f r be the rabbit and let T be the Dehn twist V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 6 / 37

11 Rabbit and Airplane Twisted rabbit question of J. H. Hubbard The Thurston s theorem implies that the composition f r T m is combinatorially equivalent either to the rabbit f r or to the anti-rabbit, or to the airplane. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 7 / 37

12 Rabbit and Airplane Twisted rabbit question of J. H. Hubbard The Thurston s theorem implies that the composition f r T m is combinatorially equivalent either to the rabbit f r or to the anti-rabbit, or to the airplane. (There are no obstructions.) V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 7 / 37

13 Rabbit and Airplane Twisted rabbit question of J. H. Hubbard The Thurston s theorem implies that the composition f r T m is combinatorially equivalent either to the rabbit f r or to the anti-rabbit, or to the airplane. (There are no obstructions.) Give an answer as a function of m. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 7 / 37

14 Rabbit and Airplane Twisted rabbit question of J. H. Hubbard The Thurston s theorem implies that the composition f r T m is combinatorially equivalent either to the rabbit f r or to the anti-rabbit, or to the airplane. (There are no obstructions.) Give an answer as a function of m. More generally, give an answer for f r g, where g is any homeomorphism fixing {0,c,c 2 + c} pointwise. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 7 / 37

15 Rabbit and Airplane Theorem If the 4-adic expansion of m has digits 1 or 2, then f r T m is equivalent to the airplane, otherwise it is equivalent to the rabbit for m 0 and to the anti-rabbit for m < 0. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 8 / 37

16 Rabbit and Airplane Theorem If the 4-adic expansion of m has digits 1 or 2, then f r T m is equivalent to the airplane, otherwise it is equivalent to the rabbit for m 0 and to the anti-rabbit for m < 0. Here we use 4-adic expansions without sign. For example, 1 =...333, so that f r T 1 is equivalent to the anti-rabbit. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 8 / 37

17 Iterated monodromy groups Iterated monodromy groups Let f : M 1 M be a d-fold covering map, where M 1 is an open subset of M. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 9 / 37

18 Iterated monodromy groups Iterated monodromy groups Let f : M 1 M be a d-fold covering map, where M 1 is an open subset of M. (In our case M = S 2 \ P f and M 1 = f 1 (M).) V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 9 / 37

19 Iterated monodromy groups Iterated monodromy groups Let f : M 1 M be a d-fold covering map, where M 1 is an open subset of M. (In our case M = S 2 \ P f and M 1 = f 1 (M).) Take a basepoint t M. We get the tree of preimages f n (t) n 0 V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 9 / 37

20 Iterated monodromy groups Iterated monodromy groups Let f : M 1 M be a d-fold covering map, where M 1 is an open subset of M. (In our case M = S 2 \ P f and M 1 = f 1 (M).) Take a basepoint t M. We get the tree of preimages f n (t) n 0 on which the fundamental group π 1 (M,t) acts. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 9 / 37

21 Iterated monodromy groups V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 10 / 37

22 Iterated monodromy groups The obtained automorphism group of the rooted tree is called the iterated monodromy group of f. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 10 / 37

23 Iterated monodromy groups Iterated monodromy groups can be computed as groups generated by automata. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 11 / 37

24 Iterated monodromy groups Iterated monodromy groups can be computed as groups generated by automata. If two topological polynomials are combinatorially equivalent, then their iterated monodromy groups coincide. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 11 / 37

25 Iterated monodromy groups Iterated monodromy groups can be computed as groups generated by automata. If two topological polynomials are combinatorially equivalent, then their iterated monodromy groups coincide. This makes it possible to distinguish specific Thurston maps. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 11 / 37

26 Iterated monodromy groups V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 12 / 37

27 Solution We prove that the following pairs of branched coverings are combinatorially equivalent: V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 13 / 37

28 Solution We prove that the following pairs of branched coverings are combinatorially equivalent: f r T 4n f r T n, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 13 / 37

29 Solution We prove that the following pairs of branched coverings are combinatorially equivalent: f r T 4n f r T n, f r T 4n+1 f r T, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 13 / 37

30 Solution We prove that the following pairs of branched coverings are combinatorially equivalent: f r T 4n f r T n, f r T 4n+1 f r T, f r T 4n+2 f r T, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 13 / 37

31 Solution We prove that the following pairs of branched coverings are combinatorially equivalent: f r T 4n f r T n, f r T 4n+1 f r T, f r T 4n+2 f r T, f r T 4n+3 f r T n. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 13 / 37

32 Solution Let G be the mapping class group of the plane with three punctures. It is freely generated by two Dehn twists T and S. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 14 / 37

33 Solution Let G be the mapping class group of the plane with three punctures. It is freely generated by two Dehn twists T and S. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 14 / 37

34 Solution Proposition Let ψ be defined on H = T 2,S,S T < G by ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 15 / 37

35 Solution Proposition Let ψ be defined on H = T 2,S,S T < G by ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1. Then g f r and f r ψ(g) are homotopic. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 15 / 37

36 Solution Proposition Let ψ be defined on H = T 2,S,S T < G by ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1. Then g f r and f r ψ(g) are homotopic. Consider { ψ(g) if g belongs to H, ψ : g Tψ ( gt 1) otherwise. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 15 / 37

37 Solution Proposition Let ψ be defined on H = T 2,S,S T < G by ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1. Then g f r and f r ψ(g) are homotopic. Consider { ψ(g) if g belongs to H, ψ : g Tψ ( gt 1) otherwise. Then for every g G the branched coverings f r g and f r ψ(g) are combinatorially equivalent. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 15 / 37

38 Solution f r T 4n f r T n, f r T 4n+1 f r T, f r T 4n+2 f r T, f r T 4n+3 f r T n. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 16 / 37

39 Solution The map ψ is contracting on G: V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 17 / 37

40 Solution The map ψ is contracting on G: for every g G there exists n such that for all m n ψ m (g) {1,T,T 1,T 2 S,S 1 } V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 17 / 37

41 Solution The map ψ is contracting on G: for every g G there exists n such that for all m n ψ m (g) {1,T,T 1,T 2 S,S 1 } Here 1 and T are fixed by ψ and the last three elements form a cycle under the action of ψ. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 17 / 37

42 Solution The map ψ is contracting on G: for every g G there exists n such that for all m n ψ m (g) {1,T,T 1,T 2 S,S 1 } Here 1 and T are fixed by ψ and the last three elements form a cycle under the action of ψ. This solves the problem for every g G. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 17 / 37

43 Dynamics on the Teichmüller space Dynamics on the Teichmüller space The Teichmüller space T Pf modelled on (S 2,P f ) is the space of homeomorphisms τ : S 2 Ĉ, where τ 1 τ 2 if an automorphism h : Ĉ Ĉ such that h τ 1 is isotopic to τ 2 rel P f. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 18 / 37

44 Dynamics on the Teichmüller space Dynamics on the Teichmüller space The Teichmüller space T Pf modelled on (S 2,P f ) is the space of homeomorphisms τ : S 2 Ĉ, where τ 1 τ 2 if an automorphism h : Ĉ Ĉ such that h τ 1 is isotopic to τ 2 rel P f. T Pf is the universal covering of the moduli space M Pf, i.e., the space of injective maps τ : P f Ĉ modulo post-compositions with Möbius transformations. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 18 / 37

45 Dynamics on the Teichmüller space Dynamics on the Teichmüller space The Teichmüller space T Pf modelled on (S 2,P f ) is the space of homeomorphisms τ : S 2 Ĉ, where τ 1 τ 2 if an automorphism h : Ĉ Ĉ such that h τ 1 is isotopic to τ 2 rel P f. T Pf is the universal covering of the moduli space M Pf, i.e., the space of injective maps τ : P f Ĉ modulo post-compositions with Möbius transformations. The covering map is τ τ Pf. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 18 / 37

46 Dynamics on the Teichmüller space Consider f = f r g for g G. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 19 / 37

47 Dynamics on the Teichmüller space Consider f = f r g for g G. Then P f = {,0,c,c 2 + c } and we may assume that τ( ) =, τ(0) = 0, τ(c) = 1. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 19 / 37

48 Dynamics on the Teichmüller space Consider f = f r g for g G. Then P f = {,0,c,c 2 + c } and we may assume that τ( ) =, τ(0) = 0, τ(c) = 1. Then every τ M Pf is determined by p = τ(c 2 + c) Ĉ \ {,1,0}. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 19 / 37

49 Dynamics on the Teichmüller space Consider f = f r g for g G. Then P f = {,0,c,c 2 + c } and we may assume that τ( ) =, τ(0) = 0, τ(c) = 1. Then every τ M Pf is determined by p = τ(c 2 + c) Ĉ \ {,1,0}. Hence, M Pf = Ĉ \ {,1,0}. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 19 / 37

50 Dynamics on the Teichmüller space For every τ T Pf there exist unique τ T Pf and f τ C(z) such that the diagram S 2 f S 2 τ is commutative. τ Ĉ fτ Ĉ V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 20 / 37

51 Dynamics on the Teichmüller space For every τ T Pf there exist unique τ T Pf and f τ C(z) such that the diagram S 2 f S 2 τ τ Ĉ fτ Ĉ is commutative. The map σ f : T Pf T Pf : τ τ has at most one fixed point. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 20 / 37

52 Dynamics on the Teichmüller space For every τ T Pf there exist unique τ T Pf and f τ C(z) such that the diagram S 2 f S 2 τ τ Ĉ fτ Ĉ is commutative. The map σ f : T Pf T Pf : τ τ has at most one fixed point. If τ is a fixed point, then f is combinatorially equivalent to f τ. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 20 / 37

53 Dynamics on the Teichmüller space For every τ T Pf there exist unique τ T Pf and f τ C(z) such that the diagram S 2 f S 2 τ τ Ĉ fτ Ĉ is commutative. The map σ f : T Pf T Pf : τ τ has at most one fixed point. If τ is a fixed point, then f is combinatorially equivalent to f τ. The fixed point (if exists) is lim n σn f (τ 0). V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 20 / 37

54 Dynamics on the Teichmüller space Let us compute σ f. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 21 / 37

55 Dynamics on the Teichmüller space Let us compute σ f. Denote p 0 = τ (c 2 + c) and p 1 = τ(c 2 + c). V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 21 / 37

56 Dynamics on the Teichmüller space Let us compute σ f. Denote p 0 = τ (c 2 + c) and p 1 = τ(c 2 + c). S 2 τ f S 2 τ Ĉ fτ Ĉ V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 21 / 37

57 Dynamics on the Teichmüller space Let us compute σ f. Denote p 0 = τ (c 2 + c) and p 1 = τ(c 2 + c). S 2 τ f S 2 τ Ĉ fτ Ĉ Then f τ is a quadratic polynomial with critical point 0 such that f τ (0) = 1, f τ (1) = p 1, f τ (p 0 ) = 0. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 21 / 37

58 Dynamics on the Teichmüller space Let us compute σ f. Denote p 0 = τ (c 2 + c) and p 1 = τ(c 2 + c). S 2 τ f S 2 τ Ĉ fτ Ĉ Then f τ is a quadratic polynomial with critical point 0 such that f τ (0) = 1, f τ (1) = p 1, f τ (p 0 ) = 0. We get f τ (z) = az and ap = 0, hence a = 1 p 2 0 and p 1 = 1 1 p 2 0, f τ (z) = 1 z2 p0 2. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 21 / 37

59 Dynamics on the Teichmüller space We have proved Proposition The correspondence σ f (τ) τ on T Pf is projected by the universal covering map to the rational function P(z) = 1 1 z 2 on the moduli space M Pf = Ĉ \ {,0,1}. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 22 / 37

60 Dynamics on the Teichmüller space We have proved Proposition The correspondence σ f (τ) τ on T Pf is projected by the universal covering map to the rational function P(z) = 1 1 z 2 on the moduli space M Pf = Ĉ \ {,0,1}. If τ is the fixed point of σ f, then f is equivalent to f τ = 1 z2 p 2, where p is the corresponding fixed point of P in the moduli space Ĉ \ {,0,1}. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 22 / 37

61 Dynamics on the Teichmüller space Fix a basepoint t 0 M Pf corresponding to the rabbit. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 23 / 37

62 Dynamics on the Teichmüller space Fix a basepoint t 0 M Pf corresponding to the rabbit. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 23 / 37

63 Dynamics on the Teichmüller space Fix a basepoint t 0 M Pf corresponding to the rabbit. ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 23 / 37

64 Dynamics on the Teichmüller space Fix a basepoint t 0 M Pf corresponding to the rabbit. ψ ( T 2) ( = S 1 T 1, ψ (S) = T, ψ S T) = 1.. g f r f r ψ(g) V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 23 / 37

65 Dynamics on the Teichmüller space Proposition Let g G be represented by a loop γ π 1 (M Pf,t 0 ). Then is projected onto the end of the path lim n σn f r g (τ 0) γγ 1 γ 2... in M Pf, where γ n continues γ n 1 and is a preimage of γ n 1 under 1 1 z 2. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 24 / 37

66 Dynamics on the Teichmüller space V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 25 / 37

67 2-dimensional iteration A 2-dimensional iteration Let us put together iteration on the moduli space and on the plane in one map on P 2. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 26 / 37

68 2-dimensional iteration A 2-dimensional iteration Let us put together iteration on the moduli space and on the plane in one map on P 2. ( ) ( ) z 1 z2 F = p 2 p 1 1, p 2 V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 26 / 37

69 2-dimensional iteration A 2-dimensional iteration Let us put together iteration on the moduli space and on the plane in one map on P 2. ( ) ( ) z 1 z2 F = p 2 p 1 1, p 2 or F([z : p : u]) = [p 2 z 2 : p 2 u 2 : p 2 ]. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 26 / 37

70 2-dimensional iteration A 2-dimensional iteration Let us put together iteration on the moduli space and on the plane in one map on P 2. ( ) ( ) z 1 z2 F = p 2 p 1 1, p 2 or F([z : p : u]) = [p 2 z 2 : p 2 u 2 : p 2 ]. Its post-critical set is {z = 0} {z = 1} {z = p} {p = 0} {p = 1} and the line at infinity. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 26 / 37

71 2-dimensional iteration The Julia set of F is projected by (z,p) p onto the Julia set of P(p) = 1 1 p 2. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 27 / 37

72 2-dimensional iteration The Julia set of F is projected by (z,p) p onto the Julia set of P(p) = 1 1 p 2. The fibers of the projection are the Julia sets of the iteration where p n+1 = P(p n ). z, f p1 (z), f p2 f p1 (z), f p3 f p2 f p1 (z),..., V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 27 / 37

73 2-dimensional iteration A minimal Cantor set of 3-generated groups The iterated monodromy groups of the backward iterations where p n 1 = P(p n ), fp 3 f p2 f p1 z 2 z1 z0, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 28 / 37

74 2-dimensional iteration A minimal Cantor set of 3-generated groups The iterated monodromy groups of the backward iterations fp 3 f p2 f p1 z 2 z1 z0, where p n 1 = P(p n ), is a Cantor set of 3-generated groups G w with countable dense isomorphism classes. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 28 / 37

75 2-dimensional iteration A minimal Cantor set of 3-generated groups The iterated monodromy groups of the backward iterations fp 3 f p2 f p1 z 2 z1 z0, where p n 1 = P(p n ), is a Cantor set of 3-generated groups G w with countable dense isomorphism classes. For any finite set of relations between the generators of G w1 there exists a generating set of G w2 with the same relations. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 28 / 37

76 z 2 + i Twisting z 2 + i Let a and b be the Dehn twists V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 29 / 37

77 z 2 + i The corresponding map on the moduli space is P(p) = ( 1 2 ) 2 p V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 30 / 37

78 z 2 + i The corresponding map on the moduli space is P(p) = ( 1 2 ) 2 p and the map on P 2 is F ( z p ) = ( ) 2 1 2z p ( ) 2, 1 2 p or F[z : p : u] = [(p 2z) 2 : (p 2u) 2 : p 2 ]. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 30 / 37

79 z 2 + i The corresponding map on the moduli space is P(p) = ( 1 2 ) 2 p and the map on P 2 is F ( z p ) = ( ) 2 1 2z p ( ) 2, 1 2 p or F[z : p : u] = [(p 2z) 2 : (p 2u) 2 : p 2 ]. This map was studied by J. E. Fornæss and N. Sibony (1992). V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 30 / 37

80 z 2 + i Solution Consider the group Z 2 C 4 of affine transformations of C where k Z and z 0 Z[i]. z i k z + z 0, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 31 / 37

81 z 2 + i Solution Consider the group Z 2 C 4 of affine transformations of C z i k z + z 0, where k Z and z 0 Z[i]. It is the fundamental group of the orbifold (4,4,2), which is associated with (1 2/p) 2. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 31 / 37

82 z 2 + i Solution Consider the group Z 2 C 4 of affine transformations of C z i k z + z 0, where k Z and z 0 Z[i]. It is the fundamental group of the orbifold (4,4,2), which is associated with (1 2/p) 2. We have a natural homomorphism φ : G Z 2 C 4 a z + 1, b iz V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 31 / 37

83 z 2 + i Let g G be arbitrary. Then f i g is either equivalent to z 2 + i, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 32 / 37

84 z 2 + i Let g G be arbitrary. Then f i g is either equivalent to z 2 + i, or to z 2 i V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 32 / 37

85 z 2 + i Let g G be arbitrary. Then f i g is either equivalent to z 2 + i, or to z 2 i or is obstructed, V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 32 / 37

86 z 2 + i Let g G be arbitrary. Then f i g is either equivalent to z 2 + i, or to z 2 i or is obstructed, i.e., is not equivalent to any rational function. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 32 / 37

87 z 2 + i Let g G be arbitrary. Then f i g is either equivalent to z 2 + i, or to z 2 i or is obstructed, i.e., is not equivalent to any rational function. It depends only on φ(g), which of these cases takes place. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 32 / 37

88 z 2 + i The answer V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 33 / 37

89 z 2 + i Obstructed polynomials V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 34 / 37

90 z 2 + i The iterated monodromy group of obstructed polynomials f i g is a Grigorchuk group (very similar to the example constructed as a solution of a problem posed by John Milnor in 1968). V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 35 / 37

91 z 2 + i The iterated monodromy group of obstructed polynomials f i g is a Grigorchuk group (very similar to the example constructed as a solution of a problem posed by John Milnor in 1968). A. Erschler proved that the growth of this group has bounds ( ) ( ) n n exp ln 2+ɛ v(n) exp (n) ln 1 ɛ (n) for all ɛ > 0. V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 35 / 37

92 z 2 + i The iterated monodromy group of obstructed polynomials f i g is a Grigorchuk group (very similar to the example constructed as a solution of a problem posed by John Milnor in 1968). A. Erschler proved that the growth of this group has bounds ( ) ( ) n n exp ln 2+ɛ v(n) exp (n) ln 1 ɛ (n) for all ɛ > 0. IMG ( z 2 + i ) also has intermediate growth (Kai-Uwe Bux and Rodrigo Pérez, 2004) V. Nekrashevych (Texas A&M) Topological Polynomials March 11, 2006, Toronto 35 / 37