On the growth of iterated monodromy groups

Bux, Kai‐Uwe; Pérez, Rodrigo A.

doi:10.1090/conm/394/07434

Cited by 14 publications

(21 citation statements)

References 3 publications

(3 reference statements)

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“…This conjecture is supported by quite a few examples of polynomials:

P false(z false) = z^{2} + i

P false(z false) = z^{3} false(- \frac{3}{2} + i \frac{\sqrt{3}}{2} false) + 1

, the quadratic polynomials with the kneading sequences

11 {(0)}^{ω}

and

0 {(011)}^{ω}

. The hypothesis of non‐renormalizability rules out the counterexamples that arise from tuning , a reverse operation to renormalization .…”

Section: Introductionmentioning

confidence: 91%

“…Recall that a finitely generated group has either polynomial, intermediate , or exponential growth depending on the volume growth of balls in the Cayley graph of the group, see Section 2.3. It has been known for a while that the IMG of the polynomial

P_{1} false(z false) = z^{2} + i

is of intermediate growth, see . Note that the Julia set of

P_{1}

is a dendrite , see Figure (a).…”

Section: Introductionmentioning

confidence: 99%

“…It was observed that even very simple maps generate IMGs with complicated structure and exotic properties which are hard to find among groups defined by more ‘classical’ methods, see . For instance,

IMG (z^{2} + i)

is a group of intermediate growth and

IMG (z^{2} - 1)

is an amenable group of exponential growth . Below we list the most important results relevant to the growth theory of IMGs that are known at the moment.…”

Section: Introductionmentioning

confidence: 99%

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Exponential growth of some iterated monodromy groups

Hlushchanka

Meyer

2018

Proc. London Math. Soc.

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Abstract. Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth.

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“…This conjecture is supported by quite a few examples of polynomials:

P false(z false) = z^{2} + i

P false(z false) = z^{3} false(- \frac{3}{2} + i \frac{\sqrt{3}}{2} false) + 1

, the quadratic polynomials with the kneading sequences

11 {(0)}^{ω}

and

0 {(011)}^{ω}

. The hypothesis of non‐renormalizability rules out the counterexamples that arise from tuning , a reverse operation to renormalization .…”

Section: Introductionmentioning

confidence: 91%

P_{1} false(z false) = z^{2} + i

is of intermediate growth, see . Note that the Julia set of

P_{1}

is a dendrite , see Figure (a).…”

Section: Introductionmentioning

confidence: 99%

IMG (z^{2} + i)

is a group of intermediate growth and

IMG (z^{2} - 1)

is an amenable group of exponential growth . Below we list the most important results relevant to the growth theory of IMGs that are known at the moment.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exponential growth of some iterated monodromy groups

Hlushchanka

Meyer

2018

Proc. London Math. Soc.

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“…The method relies on building real graphs with certain properties, and gives polynomials with real coefficients. For instance, in Theorem 4.6 we give a polynomial h of degree 6 such that R 2 contains an MG 2 …”

Section: Theorem 11 Let F ∈ C[x] Have Degree D ≥ 2 F •N Be the Nth mentioning

confidence: 99%

Blocks of monodromy groups in complex dynamics

Jones

Peters

2010

Geom Dedicata

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Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a nontrivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.

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“…The structure of nonabelian subgroups of these automorphism groups appears to be quite different from that of linear p-adic groups (see the papers just cited as well as Bux and Perez [7]). The natural action of iterated monodromy groups on rooted trees leads us to the expectation that iterated monodromy representations ρ ϕ,t 0 attached to postcritically finite polynomials ϕ ∈ K[x] have the potential of revealing aspects of arithmetic fundamental groups which are not visible to p-adic representations; see the discussion in Section 7 as well as Boston's preprint [4], where tree representations are suggested as the proper framework for studying finitely ramified tame extensions.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Aitken¹,

Hajir²,

Maire³

2005

Internat. Math. Res. Notices

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Downloaded from856 Wayne Aitken et al.In this work, we discuss a method for studying finitely ramified extensions of number fields via arithmetic dynamical systems on P 1 . At least conjecturally, this method provides a vista on a part of G K,S invisible to p-adic representations. We now sketch the construction, which is quite elementary. Suppose ϕ ∈ K[x] is a polynomial of degree d ≥ 1. 1 For each n ≥ 0, let ϕ •n be the n-fold iterate of ϕ, that is, ϕ •0 (x) = x and ϕ •n+1 (x) = ϕ(ϕ •n (x)) = ϕ •n (ϕ(x)) for n ≥ 0. Let t be a parameter for P 1 /K with function field F = K(t). We are interested in the tower of branched covers of P 1 given by(1.1)as well as extensions of K obtained by adjoining roots of its specializations at arbitraryFix an algebraic closure F of F, and let K be the algebraic closure of K determined by this choice, that is, the subfield of F consisting of elements algebraic over K. For n ≥ 0, let T ϕ,n be the set of roots in F of Φ n (x, t); it has cardinality d n . We denote by T ϕ the d-regular rooted tree whose vertex set is ∪ n≥0 T ϕ,n , and whose edges point from v to w exactly when ϕ(v) = w; its root (at ground level) is t.We choose and fix an end ξ = (ξ 0 , ξ 1 , ξ 2 , . . .) of this tree; in other words, we choose a compatible system of preimages of t under the iterates of ϕ: ϕ(ξ 1 ) = ξ 0 = t and ϕ(ξ n+1 ) = ξ n for n ≥ 1. For each n ≥ 1, we consider the field F n = F(ξ n ) F[x]/(Φ n ) andits Galois closure F n = F(T ϕ,n ) over F. Let O Fn be the integral closure of K[t] in F n . Corresponding to each t 0 ∈ K, we may fix compatible specialization maps σ n,t 0 : O Fn → K with image K n,t 0 , a normal extension field of K, and put ξ n | t 0 = σ n,t 0 (ξ n ) for the corresponding compatible system of roots of Φ n (x, t 0 ). We denote by K n,t 0 the image of the restriction of σ n,t 0 to O Fn . We refer the reader to Section 2.2 for more details, but we should emphasize here that Φ n (x, t 0 ) is not necessarily irreducible over K; hence, although K n,t 0 depends only on ϕ, n, and t 0 , the isomorphism class of K n,t 0 depends a priori on the choice of ξ as well as on the choice of compatible σ n,t 0 . Also, the Galois closure of K n,t 0 /K is contained in, but possibly distinct from, K n,t 0 .Taking the compositum over all n ≥ 1, we obtain the iterated extension F ϕ = ∪ n F n attached to ϕ, with Galois closure F ϕ = ∪ n F n over F. Similarly for each t 0 ∈ K, we obtain a specialized iterated extension K ϕ,t 0 = ∪ n K n,t 0 with Galois closure over K contained in K ϕ,t 0 = ∪ n K n,t 0 . We put M ϕ = Gal(F ϕ /F) for the iterated monodromy group of ϕ and for t 0 ∈ K, we denote by M ϕ,t 0 = Gal(K ϕ,t 0 /K) its specialization at t 0 .The group M ϕ has a natural and faithful action on the tree T ϕ , hence comes equipped 1 This construction actually works for any perfect K as long as the derivative ϕ is not identically zero in K[x].

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On the growth of iterated monodromy groups

Cited by 14 publications

References 3 publications

Exponential growth of some iterated monodromy groups

Exponential growth of some iterated monodromy groups

Blocks of monodromy groups in complex dynamics

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