Exponential growth of some iterated monodromy groups

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 $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpi\'{n}ski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.


Introduction
The iterated monodromy group (IMG) is a group that is defined in a natural way for certain dynamical systems, like iteration of a rational function on the Riemann sphere C. It was defined by Nekrashevych, see for instance [2], and independently by Kameyama in [24], see also [23].
Recently, the theory of IMGs, and its applications to the study of dynamical systems, has been developed rapidly, see in particular [32,35]. Many important problems in complex dynamics have been solved with the help of IMGs, such as the Hubbard twisted rabbit problem [4]. IMGs are in fact self-similar groups, that is, they act on a certain regular rooted tree in a 'self-similar' fashion, and frequently have some interesting algebraic properties. Furthermore, their connection to dynamics provides new methods for the study of self-similar groups. IMGs have been best understood for postcritically finite polynomials (in dynamics a map is said to be postcritically finite if each of its critical points has finite orbit). In particular, Nekrashevych gave a complete description of the IMGs of postcritically finite polynomials in terms of automata generating them [32,34].
The study of algebraic properties of IMGs plays an important role in self-similar group theory as well as dynamics. For example, the growth properties of IMGs have been investigated in the last decade. 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 (z) = z 2 + i is of intermediate growth, see [10]. Note that the Julia set of P 1 is a dendrite, see Figure 1(a). On the other hand, the IMG of the Basilica map P 2 (z) = z 2 − 1 is of exponential growth; this was proved (for an isomorphic group) in [20]. Note that the closures of the Fatou components of P 2 containing 0 and −1 intersect, see Figure 1(b). Having these two examples in mind, various people have been trying to establish connections between dynamical properties of a map and algebraic properties of its IMG. For instance, it can be shown that any postcritically finite polynomial of 'Basilica type' has IMG of exponential growth, see Theorem 2.5. At the same time, for the airplane polynomial, that is, the unique polynomial of the form P 3 (z) = z 2 + c air , where c air = 0 is real and satisfies P 3 3 (c air ) = c air (c air = −1.75488 . . .), the growth of IMG(P 3 ) is unknown. Note that two distinct bounded Fatou components of P 3 have disjoint closures, see Figure 1(c).
For a postcritically finite rational map f that is not a polynomial, the IMG is much less understood. With the exception of Lattès maps (which have IMG that is virtually Z 2 ) and rational maps whose Julia set 'contains a copy' of the Julia set of a polynomial (that is, they are renormalizable and their IMGs contain an IMG of a polynomial in a natural way), the growth of the IMG of a rational map f has previously been unknown.
The first result we show in this paper is the following.
Theorem 1.1. There exists a postcritically finite rational map f : C → C with Julia set equal to the whole sphere C and IMG of exponential growth.
We point out that this is the first non-Lattès example of a rational map whose Julia set is the whole Riemann sphere where the growth is known.
The key ingredient of this paper is to use geometric tilings of the sphere C associated with a rational (or more generally, branched covering) map f . We use them to describe the iterated monodromy action in a simple combinatorial way (via 'rotation of tiles in flowers'). So these tilings may be viewed as representations of the Schreier graphs of the action of IMG(f ) (on the levels of the dynamical preimage tree). To prove exponential growth of IMG(f ), we find a free semigroup inside it using the properties of the constructed tilings. We also provide another proof, similar to the one in [20] for the Basilica group, which uses computations with the wreath recursions associated with the IMG, see Appendix A.2.
Next, we consider some rational maps whose Julia set is a Sierpiński carpet (that is, homeomorphic to the standard Sierpiński carpet). In particular, this means that distinct Fatou components have disjoint closures.
Theorem 1.2. There exists a postcritically finite rational map with Julia set equal to a Sierpiński carpet and IMG of exponential growth.
Again, this is the first example of such a rational map where the growth of the IMG is known. The methods that we develop here use rather the 'combinatorial information' about the maps than their holomorphic nature. In particular, our methods apply to some obstructed Thurston maps, that is, postcritically finite branched covering maps f : S 2 → S 2 that are not 'Thurston equivalent' to a rational map, see Section 2.1 for the definitions. Since it was shown that P 1 (z) = z 2 + i has IMG of intermediate growth, it was conjectured that polynomials that are 'similar to' P 1 have IMG of intermediate growth. However, the precise meaning of 'similar to' is not clear, and indeed has changed over time. As already noted, the Julia set J of P 1 is a dendrite. Furthermore, every finite (that is, distinct from ∞) point in the postcritical set post(P 1 ) = {i, −i, −1 + i, ∞} is a leaf of J , that is, does not separate J , see Figure 1(a). Finally, P 1 is not renormalizable. Roughly speaking, this means that we cannot extract a simpler polynomial from any iterate P n 1 , see Section 10 for details. The following conjecture is quite natural and has been studied by the community for quite some time.
Conjecture. Let P be a postcritically finite non-renormalizable (quadratic) polynomial, such that its Julia set J P is a dendrite. Then IMG(P ) is of intermediate growth.
This conjecture is supported by quite a few examples of polynomials: P (z) = z 2 + i [10], 2 ) + 1 [16], the quadratic polynomials with the kneading sequences 11(0) ω and 0(011) ω [15]. The hypothesis of non-renormalizability rules out the counterexamples that arise from tuning, a reverse operation to renormalization [27]. The tuning operation allows to construct examples of polynomials with dendrite Julia set whose IMGs are of exponential growth, see [33,Section 5.5]. However, those maps will be renormalizable.
In this paper we show the following result. (1) The Julia set J of P is a dendrite.
(2) Every finite postcritical point of P is a leaf of J .
(3) P is not renormalizable. (4) The IMG of P is of exponential growth. Theorem 1.4 and its proof show that the conjecture stated above is not valid for all polynomials, namely for the ones of sufficiently high degree. However it may still be true for quadratic polynomials. This shows that the question when the IMG of a polynomial is of intermediate growth is even more subtle than previously thought.
The maps in Theorems 1.1-1.4 are explicitly constructed in a combinatorial fashion. The proof of exponential growth relies strongly on this combinatorial description and is essentially the same for all considered maps. In each case it is easy to generalize the construction to obtain infinite families of maps with IMGs of exponential growth.
Based on our examples, we provide a quite general criterion for exponential growth of the IMGs of Thurston maps, see Theorem 6.3. This sufficient condition is not the most general that we can obtain, but rather was formulated to be easily applicable to various examples.
The paper is organized as follows. In Section 2 we review some standard material about Thurston maps, complex dynamics, growth of groups, and IMGs. This section also contains an overview of the main properties of IMGs that are relevant for our paper, in particular, in the context of growth of groups. In Section 3 the map f 1 that serves as the example in Theorem 1.1 is constructed. In Section 4 we review the cell decompositions associated with f 1 and describe the iterated monodromy action in terms of them. In Section 5 we show exponential growth of IMG(f 1 ), that is, prove Theorem 1.1. In Section 6 we extract from the proof of Theorem 1.1 a general criterion for exponential growth of IMGs in our setting. In Section 7 we consider a family of postcritically finite rational maps with Sierpiński carpet Julia set, in particular, we prove Theorem 1.2. In Section 8 we consider an obstructed Thurston map that proves Theorem 1.3. In Section 9 we construct the polynomial P from Theorem 1.4 and prove exponential growth of its IMG. In Section 10 we review renormalization theory and prove that P is not renormalizable. In Appendix A.1 we review some standard material about actions on rooted trees and self-similar groups. In Appendix A.2 we give an alternative proof of exponential growth of IMG(f 1 ), show that it is a regular branch group, and conclude with some further properties.

Notation
We denote by N the set of positive integers, while N 0 denotes the set of non-negative integers, that is, Let X be a topological space and U ⊂ X. We denote by U , int(U ), and ∂U the topological closure, the interior, and the boundary of the set U , respectively.
The nth iterate of a map f : S → S on a set S is denoted by f n for n ∈ N. We set f 0 := id S . Suppose a subset U ⊂ S is given. We denote by f −n (U ) the preimage of U under f n , that is, f −n (U ) = {x ∈ S : f n (x) ∈ U }. For simplicity, we denote f −n (y) := f −n ({y}) for y ∈ S. Also we denote the restriction of f to U by f |U .
The identity element of a group is denoted by 1. The notation 'H < G' means that H is a subgroup of G, as usual.
We consider right group actions. So if a group G acts on a set X, then the image of x ∈ X under the action of an element g ∈ G is denoted by x g , and in a product g 1 g 2 the element g 1 acts first, that is, x g1g2 = (x g1 ) g2 . We therefore write [g 1 , g 2 ] = g −1 1 g −1 2 g 1 g 2 and g g2 1 = g −1 2 g 1 g 2 . In a group G, ord(g) denotes the order (or period) of a group element g ∈ G, that is, the smallest positive integer n such that g n = 1. If no such n exists, g is said to have infinite order and we set ord(g) = ∞.

Thurston maps
We provide a brief overview here, but refer the reader to [9,Chapter 2] for details. Suppose f : S 2 → S 2 is a branched covering map of the topological 2-sphere S 2 . That is, f is a continuous and surjective map, that we can write locally around each point p ∈ S 2 as z → z d for some d ∈ N (depending on p) in orientation-preserving homeomorphic coordinates in domain and target. The integer d 1 is uniquely determined by f and p, and called the local degree of the map f at p, denoted by deg(f, p).
A point c ∈ S 2 with deg(f, c) 2 is called a critical point of f . The image of a critical point is called a critical value. The set of all critical points of f is finite and denoted by crit(f ). The union post(f ) = Natural examples are given by postcritically finite rational maps on the Riemann sphere C and postcritically finite polynomial maps on the complex plane C. In this paper we consider Thurston maps defined by a subdivision rule on the sphere S 2 . See for example the construction of the map f 1 in Section 3; in greater generality this may be found in [9,Chapter 12;11].
The ramification function of a Thurston map f : S 2 → S 2 is a function α f : S 2 → N ∪ {∞} such that α f (p) is the lowest common multiple of all local degrees deg(f n , q), where q ∈ f −n (p) and n ∈ N are arbitrary. Thus α f (p) = 1 for all p ∈ S 2 post(f ), and α f (p) = ∞ if and only if p is contained in a cycle containing a critical point (that is f n (p) = p and deg(f n , p) 2 for some n 1). The where we use the convention that 1/∞ = 0. It satisfies A Lattès map is a rational Thurston map with parabolic orbifold that does not have periodic critical points (a point p ∈ S 2 is called periodic if f n (p) = p for an n ∈ N).
Two Thurston maps f : S 2 → S 2 and g : S 2 → S 2 , where S 2 is another topological 2-sphere, are called Thurston equivalent if there are homeomorphisms h 0 , h 1 : S 2 → S 2 that are isotopic rel. post(f ) such that h 0 • f = g • h 1 .

Julia and Fatou sets
The reader is referred to [31] for background in complex dynamics.
Let f : C → C be a rational map. Then the Julia set J f of f is the closure of the set of repelling periodic points. If f is postcritically finite then J f coincides with the set of limit points of the full backwards orbit n 0 f −n (t) of any point t ∈ C post(f ). The Fatou set of If f is a polynomial, then J f coincides with the boundary of the set called the filled Julia set.

Growth of groups
Given a finitely generated group G with a finite symmetric set of generators S one defines the word length of an element g ∈ G with respect to S by and the growth function of G with respect to S by growth G,S (n) := #{g ∈ G : G,S (g) n}.
The group G is said to be of (1) polynomial growth, if growth G,S (n) is bounded above by a polynomial, that is, growth G,S (n) Cn k for some constants C > 0, k ∈ N; (2) exponential growth, if growth G,S is bounded below by an exponential function, that is, growth G,S (n) c exp(αn) for some constants c, α > 0; It can be shown that whether G has polynomial, intermediate, or exponential growth does not depend on the choice of the finite generating set S.
Milnor was the first to ask whether groups of intermediate growth exist [29]. This was answered in the positive by Grigorchuk in [17]. The example of a group of intermediate growth constructed by Grigorchuk in this paper, now called the Grigorchuk group, is a selfsimilar group. This means it acts on a binary rooted tree in a certain 'self-similar' fashion, see Definition A.2 and [32, Chapter 1]. Self-similar groups often exhibit very interesting behavior and have been studied intensely in the last decades. For more information on the theory of growth of groups we refer the reader to the recent survey [18] by Grigorchuk, see also [2], a survey on self-similar groups and their properties. Let d be the topological degree of f . Fix a basepoint t ∈ M. We consider the backward orbit of t, meaning the formal disjoint union T f = ∞ n=0 f −n (t). Then T f has a natural structure of a d-ary rooted tree: the root, that is, the unique point on the level 0, is the basepoint t ∈ f −0 (t) = {t}, and a vertex p ∈ f −n (t) (of the nth level) is connected by an edge to the vertex f (p) ∈ f −(n−1) (t) (of the (n − 1)-th level), for n ∈ N. The set T f viewed as a rooted tree is called the dynamical preimage tree of f at the basepoint t.

Iterated monodromy action and group
The fundamental group π 1 (M, t) acts naturally on the set of preimages f −n (t), n ∈ N 0 : the image p [γ] of a point p ∈ f −n (t) under the action of a loop [γ] ∈ π 1 (M, t) is equal to the endpoint of the unique f n -lift of γ that starts at p. It is easy to see that the action of the fundamental group on the vertices of the dynamical preimage tree T f preserves the tree structure, that is, the fundamental group acts on T f by automorphisms of the rooted tree. Thus, we have defined a group homomorphism from the fundamental group of M = S 2 post(f ) to the automorphism group Aut(T f ) of the d-ary rooted tree T f . Definition 2.2. The iterated monodromy action is the action of π 1 (M, t) on the dynamical preimage tree T f . The quotient of π 1 (M, t) by the kernel of its action on T f is called the iterated monodromy group (IMG) of f and is denoted by IMG(f ). That is, (M, t)).

Selected properties of IMGs
The IMG of a Thurston map f is self-similar [32,Proposition 5.2.2]. Furthermore, if f is a postcritically finite rational map, then the IMG (together with the associated wreath recursion, see Appendix A.1) contains all the 'essential' information about the dynamics of the map f : one can reconstruct from IMG(f ) the action of f on its Julia set J f . In this case the limit space of the IMG is homeomorphic to the Julia set of the map [32,Theorem 6.4.4]. Moreover, one can approximate the Julia set J f by a certain sequence of finite graphs. Definition 2.3. Let G be a group generated by a finite set S and acting on a set X. The labeled Schreier graph Γ(G, S, X) is a labeled directed (multi)graph with the set of vertices X and the set of directed edges X × S, where the edge (x, s) starts at x, ends at x s , and is labeled by s, for each x ∈ X and s ∈ S.
Nekrashevych showed that given a postcritically finite rational map f , the sequence of the Schreier graphs of the action of IMG(f ) on the nth level of the dynamical preimage tree T f converges to the Julia set of f , see [32,Chapters 3 and 6].
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 [2]. For instance, IMG(z 2 + i) is a group of intermediate growth [10] and IMG(z 2 − 1) is an amenable group of exponential growth [6,20]. Below we list the most important results relevant to the growth theory of IMGs that are known at the moment.
Theorem 2.4 [35]. Let f be a postcritically finite rational map. Then IMG(f ) does not contain a free group of rank 2.
Theorem 2.5 [35]. If a postcritically finite polynomial f has two finite Fatou components with intersecting closures, then IMG(f ) contains a free semigroup of rank 2 and is of exponential growth.
For more information on the theory of IMGs we refer the reader to [32, Chapters 5-6; 35].

Construction of the map f 1
Here we describe the map f 1 : C → C that will serve as our main example. It is a postcritically finite rational map, such that no critical point is periodic. This means that the Julia set of f 1 is the whole Riemann sphere C (see [31,Corollary 16.5]). Since we are mainly interested in the combinatorial behavior of f 1 , we will construct f 1 in a combinatorial fashion.
We consider two polyhedral surfaces Δ and Δ constructed as follows. Let T be a Euclidean triangle with angles π/2, π/3, π/6. The surface Δ is obtained by gluing two identical copies of T together along their boundaries. The surface Δ is obtained by gluing two Euclidean triangles with angles 2π/3, π/6, π/6 together along their boundaries. The two triangles of Δ, as well as Δ , are called the top and bottom faces. They correspond to the top and bottom triangles in Figure 2. The vertices of each such triangle are labeled −1, 1, ∞; they correspond to the vertices of Δ and Δ . We color the top face of Δ white, and the bottom one black. Each face of Δ can be divided into six triangles T that are similar to T ; we color them black and white as shown in Figure 2. The map g 1 : Δ → Δ is now constructed as follows. Each of the six white triangles T ⊂ Δ is mapped by a similarity to the white face of Δ, that is, each vertex of T is mapped to the one of the same angle. Similarly, each of the six black triangles T ⊂ Δ is mapped to the black face of Δ in the same fashion. To illustrate the mapping behavior, the vertices of Δ are colored red, blue, and green in Figure 2, and each vertex v of a triangle T ⊂ Δ is colored the same as g 1 (v).
It is a standard fact that every polyhedral surface can be equipped with a conformal structure in a natural way, see, for example, [7, Section 3.3]. By the uniformization theorem this means that there are conformal maps ϕ : Δ → C and ϕ : Δ → C. To normalize these maps, we demand that the vertices of Δ and Δ labeled −1, 1, ∞ are mapped to −1, 1, ∞ ∈ C, respectively. The symmetry then implies that the top face of Δ, as well as the top face of Δ , are mapped to the upper half-plane H + ; the other face of Δ, and, respectively, of Δ , is mapped to the lower half-plane H − (otherwise, the map p → ϕ(p) is distinct from ϕ violating its uniqueness † ).
In the case at hand, we can actually construct the maps ϕ and ϕ explicitly. Indeed, ϕ maps the white triangle in Δ to the upper half-plane by the Riemann map normalized so that the vertices labeled −1, 1, ∞ are mapped to the points −1, 1, ∞ ∈ C. Also ϕ maps the black triangle to the lower half-plane by a Riemann map with same normalization. Similarly, ϕ is constructed by mapping the top and bottom face of Δ to the upper and lower half-plane with the same normalization.
The map f 1 : C → C given by It is elementary to check that it is indeed a rational map.
By a slight abuse of notation, let us denote the images of the vertices in Δ labeled by a 0 , a 1 , c 0 , c 1 under ϕ again by a 0 , a 1 , c 0 , c 1 . Note also that by symmetry, the vertex in Δ labeled 0 is mapped to 0 ∈ C by ϕ .
Then the set of critical points of f 1 is crit( Recall that the ramification portrait of a Thurston map f : S 2 → S 2 is a directed graph with the vertex set V = n 0 f n (crit(f )) consisting of the union of all orbits of all critical Put differently, the ramification portrait † Here p denotes the point in Δ obtained from p ∈ Δ by reflection along the edge [−1, 1], see Figure 2. ‡ The map f 1 was originally constructed in [28] (where is was called R 6 ). Its purpose then was to construct a quasisymmetric map from a certain fractal surface to C. This, however, will not be relevant here.
illustrates how critical points are mapped by the map f . For instance, the ramification portrait of f 1 is shown below.
We conclude that the ramification function α f1 of f 1 (see Section 2.1) is given by This means that the orbifold associated with f 1 is hyperbolic.
Remark 3.1. The map f 1 may be given explicitly in the following two forms: It is elementary to check that the two expressions agree. Thus the critical points are in fact 2 (as well as 0 and ∞). Nevertheless, these precise values and the explicit formula (3.2) will be of no importance to us.

Tiles, flowers, and the iterated monodromy group
In this section we describe tilings and the iterated monodromy action associated with a Thurston map f : S 2 → S 2 which has an f -invariant Jordan curve C, such that post(f ) ⊂ C. For simplicity, here we only discuss the case f = f 1 . However, all the definitions and statements can be naturally adapted to the general case.
Note that the extended real line The closures of the components of C R, that is, of the upper and lower half-planes, are called 0-tiles. The one containing the upper halfplane is colored white and denoted by X 0 w ; the one containing the lower half-plane is colored black and denoted by X 0 b . Let n be a non-negative integer. The closure of a component of C f −n 1 ( R) is called an n-tile. Note that for any such n-tile X the set f n 1 (X) is one of the two 0-tiles. We color X by the color of f n 1 (X), that is, black or white. The subdivision of the sphere C in 2-tiles is shown in Figure 3 (where C is identified with Δ as in the previous section).
It follows that each (n + 1)-tile is contained in an n-tile. For example, when comparing Figures 2 and 3 one sees that every 1-tile contains (is subdivided into) six 2-tiles.
Each point in f −n 1 (post(f 1 )) is called an n-vertex. Note that the set of n-vertices contains all critical points of f n 1 . Furthermore, each postcritical point is an n-vertex for any n ∈ N 0 , since f 1 (post(f 1 )) ⊂ post(f 1 ). We say that the n-vertex v is of type a, b, or c if f n 1 (v) = ∞, 1, or −1, respectively. Note that each n-vertex v is also an (n + 1)-vertex, but the type of v as an n-vertex may be different from the type of v as an (n + 1)-vertex.  Let v ∈ C be an n-vertex, and let d v := deg(f n , v). Then v is contained in, or incident to, d v white as well as d v black n-tiles. Furthermore, the colors of n-tiles alternate when going around v. Consider the union of all such n-tiles, This set is called the flower of level n centered at v. We call d v the degree of W n (v). We say that the flower W n (v) is an a-, b-, or c-flower, if the n-vertex v is of type a, b, or c, respectively. Note that the terminology is slightly different from the one in [9], where flowers are always open.
Let us now choose generators of the IMG of f 1 . We first choose an arbitrary basepoint t in the (interior of the) upper half-plane. In particular t / ∈ post(f 1 ). Let a = γ ∞ , b = γ 1 , and c = γ −1 be loops based at t around ∞, 1, and −1, respectively. More precisely, we fix a small simple positively oriented circle around ∞ (Here, 'small circle' means that it is inside a neighborhood U of ∞ such that post(f 1 ) {∞} ⊂ C U ). The loop a connects t to this circle in the upper half-plane, traverses it, and returns to t in the upper half-plane. The loops b and c are defined in an analogous fashion, see Figure 4(a). By abuse of notation we identify the equivalence classes of a, b, and c in IMG(f 1 ) = π 1 ( C post(f 1 ), t)/ Ker(φ f1 ) (see Section 2.4) with a, b, and c, respectively. Since a, b, c generate π 1 ( C post(f 1 ), t), they generate IMG(f 1 ). In fact, any two of the elements a, b, c generate IMG(f 1 ), since acb = 1.
Note that for any white n-tile X the map f n 1 : X → X 0 w is a homeomorphism, see [9, Proposition 5.17(i)]. Thus each white n-tile contains exactly one point from f −n 1 (t). So we may identify the set of white n-tiles with the nth level of the dynamical preimage tree T f1 on which IMG(f 1 ) acts. Furthermore, each white n-tile, and thus any p ∈ f −n 1 (t), is contained in exactly one a-flower, exactly one b-flower, as well as exactly one c-flower of level n. Next we are going to describe how to read off the action of IMG(f 1 ) on tiles from the tiling picture. Consider Note that α f1 (1) = 24 is the lowest common multiple of these degrees by definition of the ramification function. Let X 0 , . . . , X dv−1 be the white n-tiles contained in W n (v) labeled mathematically positively around v. Fix one such white n-tile X j ⊂ W n (v). Let us consider the lift b of the loop b starting at the point t j ∈ f −n 1 (t) ∩ X j . Then its endpoint is the unique point t j+1 ∈ X j+1 ∩ f −n 1 (t) (here the index is taken mod(d v )). We conclude that b acts on white n-tiles by rotations around the centers of b-flowers. This is illustrated in Figure 4(b), where blue arrows represent lifts of b (up to homotopy). The analogous description holds for the generators a and c.
The Schreier graph for IMG(f 1 ) acting on f −1 (t) (the first level of the dynamical preimage tree) is shown in Figure 5. Here, we colored the edges red, blue, and greed instead of labeling them by generators a, b, and c, respectively. In this way, we may think of the sphere tiling generated by the n-tiles as a graphical representation of the Schreier graph of the action of IMG(f 1 ) on the nth level of the dynamical preimage tree T f1 .
Proof. If k = α f1 (1) = 24, then k is a multiple of the degree of each b-flower. Hence b k acts trivially on each white n-tile, n ∈ N 0 . This means that b k = 1 in IMG(f 1 ). Conversely, if k 1 is not a multiple of 24, there is a b-flower W n (v) whose degree d v does not divide k. Then b k does not act trivially on the white n-tiles in W n (v), so b k = 1. Thus ord(b) = α f1 (1) = 24. The argument is completely analogous for the generators a and c.
Let us now give the wreath recursions of the generators a, b, c, see Appendix A.1. To do this we label the white 1-tiles as indicated in Figure 6. Let t j ∈ f −1 1 (t) be the preimage of t forms a forward-invariant tree in C joining the postcritical points of f 1 . This allows us to naturally choose connecting paths from the basepoint t to the points in . . , t 6 } and define a labeling on the dynamical preimage tree T f1 . Namely, we meaning that j stays in the interior of the domain in Figure 6. These choices define the label of every vertex of T f1 uniquely by iterative lifting of the paths 1 , . . . , 6 , see [32,Chapter 5.2] for more details. With respect to this labeling the action of IMG(f 1 ) on the regular rooted tree T f1 becomes self-similar, see Definition A.2. The wreath recursions of a, b, and c are then given by We will however not need the wreath recursions to prove exponential growth of IMG(f 1 ).

Exponential growth of IMG(f 1 )
In this section we prove that the IMG of the map f 1 has exponential growth. More precisely, we construct a free semigroup inside IMG(f 1 ). We remind the reader that a more classical, a la Basilica, proof which only uses computations with the wreath recursions (4.1) can be found in Appendix A.2.
Recall that the map f 1 has three postcritical points −1, 1, and ∞, which we call 0-vertices, that divide R into the three 0-edges [−1, 1], [1, ∞], and [−∞, −1]. The reader is advised to consult Figure 2. The 0-edge [1, ∞] will be of special importance to us. Proof. From (3.1) we see that every preimage of ∞ is a critical point of local degree 2, which is not a postcritical point. The statement follows.
Note that Lemma 5.1 is stronger than the statement that ord(a) = 2, since the latter does not rule out a-flowers of degree 1.
The situation is illustrated in Figure 7. (4) Recall that X 0 w and X 0 b are the closures of the upper and lower half-planes, respectively. So we need to show that each flower W n (v), v ∈ (1, ∞), contains as many white n-tiles above the real line as below.
First note that f 1 is a real function, meaning that f 1 (z) = f 1 (z). Thus the n-tiles are symmetric with respect to the real axis. Since deg(f n 1 , v) is even, the number of n-tiles in W n (v) below and above the real line is the same even number. As colors of tiles around v alternate the statement follows.
(5) The statement follows from the above considerations.  Let n 2 be an integer. Consider an n-vertex v ∈ [1, ∞) of type b and the white n-tile X(v) as in Lemma 5.2(5). Let v be the n-vertex of type b to the right of v on [1, ∞] and w be the white n-tile according to Lemma 5.2(5). In the latter we assume that v is different from ∞. Then by the description of the iterated monodromy action on tiles from Section 4 as well as Lemmas 5.1 and 5.2 it follows that ab 4 maps X(v) to X(v ). Put differently, ab 4 'shifts white n-tiles in X 0 w on the 0-edge [1, ∞) to the right'. Since the degree of W n (v ) is 8, it follows that ab 12 and ab 20 act on X(v) in exactly the same way as ab 4 , that is, by shifting X(v) to X(v ) on the right.
Note that by Lemma 5.2(3) there are 2 n−1 n-vertices of type b on [1, ∞). From the above considerations, the elements ab 4 , ab 12 , and ab 20 are of infinite order in IMG(f 1 ). Lemma 5.4. The elements ab 4 , ab 12 , and ab 20 generate a free semigroup in IMG(f 1 ).
Put differently, we consider the words of the form ab k1 ab k2 . . . ab kN , (5.1) where N ∈ N 0 and k j ∈ {4, 12, 20} for j = 1, . . . , N. We will show that if two such words are distinct, then they are distinct as elements of IMG(f 1 ).
Proof. Suppose that n 2 is an integer. Let us consider the critical point c 0 , see Figure 2 and (3.1). Note that f n 1 (c 0 ) = 1 and deg(f n 1 , c 0 ) = 3. Thus, c 0 is an n-vertex of type b and the degree of W n (c 0 ) is 3. Figure 8(a) shows such a flower W n (c 0 ).
Note that the 0-edge [1, ∞] has three f 2 1 -preimages at c 0 . More precisely, there are three analytic closed arcs A 1 , A 2 , A 3 , such that they intersect (pairwise) precisely in c 0 and f 2 1 : A j → [1, ∞] is a homeomorphism for j = 1, 2, 3. Denote by A j the arc A j with the endpoint different from c 0 removed.
Let j ∈ {1, 2, 3}. It follows that on each arc A j the combinatorial picture of level n is the same as on [1, ∞) for level n − 2 (the latter one is described by Lemma 5.2). The only difference is that the degree of W n (c 0 ) is 3, while the degree of W n (1) is 1. In particular, we can associate to each n-vertex v ∈ A j of type b a white n-tile X(v) ⊂ W n (v), so that X(v) is mapped by ab k to X(v ), for k = 4, 12, 20. Here, v is the next n-vertex of type b on A j that follows v, when traversing A j starting from c 0 . Figure 8(b) shows the b-flowers on the arcs A 1 , A 2 , A 3 for sufficiently large n (Figure 8(a) is a close-up of Figure 8(b)). For the convenience of the reader we also show a subgraph of the n-level Schreier graph in Figure 9, which may be viewed as a schematic version of Figure 8  Consider now two distinct words w 1 and w 2 of the form (5.1). By multiplying from the left with the inverse of the common initial word, it is enough to assume that where k 1 , . . . , k N , m 1 , . . . , m N ∈ {4, 12, 20}, and k 1 = m 1 . Fix a sufficiently large n (in fact it will be enough to demand that 2 n−3 > N). Fix one white n-tile X in W n (c 0 ). Let us apply the two words w 1 and w 2 to X. Note that b k1 and b m1 map this tile to distinct n-tiles X 1 and X 2 in W n (c 0 ), since 8 is not a multiple of 3. We conclude that the remaining subwords ab k2 . . . ab kN and ab m2 . . . ab mN of w 1 and w 2 shift these n-tiles along two distinct arcs among A 1 , A 2 , A 3 . Thus w 1 and w 2 map X to distinct n-tiles. Consequently, w 1 and w 2 are distinct elements in IMG(f 1 ).
The previous argument can be applied to show that distinct words of the form (5.1) are distinct in IMG(f 1 ) even if they have different length. However, we note that there is a special case when after cancellation of the common initial part of the two given words we are left with w 1 = 1 and w 2 = 1. In this case w 1 fixes the n-tile X selected above, while w 2 shifts it along one of the arcs A 1 , A 2 , A 3 . Thus w 1 and w 2 are distinct elements in IMG(f 1 ). Hence we have proved that the elements ab 4 , ab 12 , ab 20 generate a free semigroup in IMG(f 1 ).
From the previous lemma it follows immediately that IMG(f 1 ) is of exponential growth. This proves Theorem 1.1.

A criterion for exponential growth
Here we analyze the essential ingredients used in Section 5 to prove exponential growth of the IMG of the map f 1 . This will result in a somewhat general sufficient condition for the IMG of a Thurston map to be of exponential growth. However, we point out that this criterion is far from being necessary. Moreover, the imposed conditions on the Thurston map can be further relaxed.
Let g : S 2 → S 2 be a Thurston map (see Definition 2.1). We fix a Jordan curve C ⊂ S 2 with post(g) ⊂ C. As in Section 4, the 0-edges are the closed arcs into which post(g) divides C. For the map f 1 we considered the 0-edge [1, ∞] which was f 1 -invariant. So in general we demand that there is a g-invariant 0-edge E with endpoints p, q ∈ post(g) (a) such that g|E : E → E is not a homeomorphism.
Put differently, g(E) = E and d E := deg(g|E) 2. It follows that the 0-edge E is subdivided into d n E n-edges by n-vertices for each n ∈ N 0 . Further, we assume that Note that from g(E) = E it follows that g({p, q}) ⊂ {p, q}, which means that at least one of the points p, q is a fixed point of g 2 . So the main purpose of condition (b) is to fix the notation, since p and q will play somewhat different roles.
Let v be an n-vertex on E for some n ∈ N 0 , meaning that v ∈ g −n (post(g)) ∩ E. Since E is g-invariant, it follows that either g n (v) = p or g n (v) = q. In the first case we say that the n-vertex v is of type p, otherwise we say that it is of type q. We again note that v is an (n + 1)-vertex as well, and as such may be of different type. Recall that each n-edge e in E is mapped by g n homeomorphically to E. If follows that one endpoint of e is an n-vertex of type p and the other one is of type q. Put differently, the types of the n-vertices on E alternate.
The second essential ingredient for the example f 1 was that for all n-vertices v ∈ (1, ∞) of a fixed type the degrees of the flowers W n (v) were the same. In general we demand that there are constants k p , k q ∈ N, such that for each n ∈ N every n-vertex v ∈ E {p, q} of type p, and every n-vertex w ∈ E {p, q} of type q, satisfies deg(g n , v) = k p and deg(g n , w) = k q . (c) There is an elementary way to check that condition (c) is true. Consider the sets V p := g −1 (p) ∩ (E {p, q}) and V q := g −1 (q) ∩ (E {p, q}). These are the 1-vertices on E {p, q} of type p and q, respectively. We first note that (c) implies that there are constants k p , k q ∈ N with Let v ∈ E {p, q} be an n-vertex of type p, then by (b) it is an (n + 1)-vertex of type p as well. Assuming (c) it follows that p).
The other 0-vertex q satisfies either g(q) = q or g(q) = p. In the first case we obtain from an exactly analogous argument that (c) implies deg(g, q) = 1 in the case g(q) = q. (c3) Furthermore, in this case the three conditions (c1), (c2), (c3) imply (c). Indeed, for any n-vertex v ∈ E {p, q} of type p the sequence v, g(v), . . . , g n (v) = p contains exactly one critical point of g, which is contained in V p , and thus has local degree k p . Hence deg(g n , v) = k p . Similarly, deg(g n , w) = k q for each n-vertex w ∈ E {p, q} of type q.
Let us now consider the second case g(q) = p. Suppose w ∈ V q . Then g 2 (w) = p. From (c) it follows that k p = deg(g 2 , w) = deg(g, w) deg(g, q) = k q deg(g, q). Thus we obtain the condition k p = k q deg(g, q) in the case g(q) = p. (c4) Conversely, in this case the conditions (c1), (c2), (c4) imply that for any n-vertex v ∈ E {p, q} of type p, the sequence v, g(v), . . . , g n (v) = p contains either exactly one point in V p or the point q and exactly one point in V q . In both situations deg(g n , v) = k p . The argument that deg(g n , w) = k q for each n-vertex w ∈ E {p, q} of type q is the same as in the first case.
In practice we often consider the ramification portrait of g restricted to E.
Since v is incident to two 1-edges contained in E, both of which are mapped to E, it follows that v is a critical point of g. Thus the ramification portrait of g does indeed contain all 1-vertices in E, and we may restrict it to the set of these vertices. We note that for For example, the ramification portrait of f 1 restricted to [1, ∞] is Given the restricted ramification portrait of a g-invariant 0-edge, it is immediate to verify that conditions (c1), (c2), (c3) (c4) (as well as (b)) are satisfied. Condition (c) is not yet sufficient to ensure that a suitable word (which was ab 4 for the example f 1 ) acts by 'shifting white n-tiles along E'. Let v ∈ E {p, q} be an n-vertex. Then there are n-edges e, e ⊂ E that intersect in v. Then any n-tile in the flower W n (v) of level n centered at v is either contained in the sector between e and e or in the sector between e and e. We refer to these two sectors as the sectors into which E divides W n (v). In this setting we demand that for each n-vertex v ∈ E {p, q} the two sectors into which E (d) divides W n (v) contain the same number of n-tiles. (g n , v). Then there are 2d v n-edges that contain v. Let e 0 , e 1 , . . . , e 2dv−1 be these n-edges labeled cyclically around v so that e = e 0 ⊂ E. Then (d) is equivalent to the requirement that e = e dv ⊂ E.
Since E is invariant, g n (e 0 ) = g n (e dv ) = E. At the same time g n maps e j to E if and only if j is even. It follows that d v is even. Consequently, conditions (a) and (d) imply that the two sectors into which E divides W n (v) contain an (equal) even number of n-tiles. Thus the numbers k p and k q from condition (c) are even.
Let us color the 0-tiles black and white. As in Section 4, the dynamics of g defines a coloring on the n-tiles respecting the above choice for all n ∈ N. The colors of the n-tiles containing v (that is, the n-tiles in the flower W n (v)) alternate cyclically around v. From the discussion above, it follows that conditions (a) and (d) mean that the numbers of white n-tiles in the two sectors into which E divides W n (v) are equal.
Note that (d) is automatically satisfied if E ⊂ R and g is a real rational function, that is, g( R) ⊂ R.
As for condition (c), there is an equivalent condition that only involves 1-vertices, 1-tiles, and 1-flowers. More precisely, we demand that for each 1-vertex v ∈ E {p, q} the two sectors into which E (d ) divides W 1 (v) contain the same number of 1-tiles.
Lemma 6.1. Let E be a g-invariant 0-edge with the endpoints p and q as in (a). Then condition (d) is equivalent to condition (d ).

Proof. The implication (d) ⇒ (d ) is trivial.
Let us show the other implication. First consider a 1-vertex v ∈ E {p, q}; recall that v then is an n-vertex as well for all n ∈ N.
Since the 0-edge E is g-invariant, g(v) ∈ {p, q}. Assume that g(v) = p. The other case is completely analogous and will not be treated separately. Let E and E be the 0-edges incident to p. Then the 1-edges containing v are alternatingly mapped to E and E by g.  1 , g(v)). Then every 1-edge e j , defined above, contains an n-edge k j,0 incident to v, such that g(k j,0 ) = K 0 . Moreover, between e j and e j+1 there are exactly (m − 1) n-edges containing v. It follows that the two sectors in W n (v) between the two n-edges in E containing v, namely k 0,0 and k dv/2,0 , contain the same number of n-edges, proving the lemma in this case.
Consider now an n-vertex v ∈ E {p, q} that is not a 1-vertex. Then there is a j n − 1 such that w := g j (v) ∈ E {p, q} is a 1-vertex. Since w is not a postcritical point of g, it follows that v is not a critical point of g j . Thus there is a neighborhood of v on which g j is a homeomorphism. Consequently, g j maps n-edges containing v to (n − j)-edges containing w bijectively, and at the same time E to E. The desired statement for v follows now from the corresponding statement for the 1-vertex w already proved above.
Finally, we assume the following condition.
There is a point c ∈ S 2 with g k (c) = p for a k ∈ N (e) such that d c := deg(g k , c) is not a divisor of k p .
As for the map f 1 we define the elements a and b of IMG(g) as being represented by loops around q and p, respectively. More precisely, we fix a basepoint t in the interior of the white 0-tile denoted by X 0 w . Then the loop a is represented by connecting t in X 0 w to a small circle around q. Similarly, the loop b is represented by connecting t in X 0 w to a small circle around p. Assuming conditions (a)-(d), one shows exactly analogous to Corollary 5.3 that a kq/2 b kp/2 is of infinite order in IMG(g). If we also assume condition (e), it follows exactly as in Lemma 5.4 that a kq/2 b kp/2 and a kq/2 b kp/2 b kp generate a free semigroup in IMG(g). This means we have proved the following.
Theorem 6.2. Let g : S 2 → S 2 be a Thurston map, and C ⊂ S 2 be a Jordan curve with post(g) ⊂ C that satisfies conditions (a)-(d). Then IMG(g) contains an element of infinite order. Theorem 6.3. Let g : S 2 → S 2 be a Thurston map, and C ⊂ S 2 be a Jordan curve with post(g) ⊂ C that satisfies conditions (a)-(e). Then IMG(g) is of exponential growth.
Let us provide some remarks on how our conditions may be relaxed.
Remarks 6.4. (1) For any Thurston map g : S 2 → S 2 , IMG(g) is isomorphic to IMG(g n ) for any n ∈ N. Thus to prove exponential growth of IMG(g) it is enough to check that our conditions are satisfied for some iterate g n .
(2) Two Thurston maps f : S 2 → S 2 and g : S 2 → S 2 that are Thurston equivalent have isomorphic IMGs [32,Corollary 6.5.3]. Thus our conditions, in particular (a), need only be satisfied up to isotopy rel. post(g).
(3) For simplicity, let us denote x := a kp/2 b kq/2 . By (b) there exists a unique point t 0 ∈ g −1 (t) ∩ W 1 (p). Conditions (a)-(d) imply that there is a sequence of distinct points t 0 , t 1 , . . . , t n−1 ∈ g −1 (t), such that t j+1 mod (n) = t x j , that is, the lift x j of x starting at t j ends at t j+1 mod (n) for each j = 0, . . . , n − 1. In fact, n = d E = deg(g|E). Let γ be the closed curve in S 2 post(g) obtained by concatenation of the paths x j , j = 0, . . . , n − 1. It follows that the curve γ is homotopic to x rel. post(g). Furthermore deg(g|γ) = n > 1. Loosely speaking, we can say that x acts on the white n-tiles sharing an edge with E by "permuting them cyclically around E". In this way, one can relax conditions (a), (c), and (d) by requiring existence of an element x ∈ IMG(g) with the properties mentioned above. This element x will be automatically of infinite order in IMG(g). However, one will also need to adapt condition (e) in a certain way to conclude the exponential growth of IMG(g). The resulting criterion however is somewhat cumbersome and will not be formulated here.

Sierpiński carpet rational maps
In this section we present a family of postcritically finite rational maps, such that each map in the family has a Sierpiński carpet as its Julia set, and IMG of exponential growth. Here, we call a set S ⊂ C a Sierpiński carpet if and only if it is homeomorphic to the standard Sierpiński carpet. By Whyburn's characterization, this is the case if and only if S is compact, connected, locally connected, has topological dimension 1, and no local cut-points. Equivalently, there exists a sequence {D j } j∈N of Jordan domains in C having pairwise disjoint closures, such that S = C ( j D j ) has empty interior and diam(D j ) → 0 as j → ∞, see [37]. These two characterizations show that a Sierpiński carpet is a universal object and justify the study of rational maps with Sierpiński carpet Julia set. The first example of such a rational map is due to Milnor and Tan Lei, see [30,Appendix].
The rational maps we will consider now have originally been studied by Haïssinsky and Pilgrim in [21] (with one difference that we address at the end of the section). We briefly review the construction, which is similar to the way the map f 1 from Section 3 was defined.
To keep the discussion elementary, we first consider one concrete map from our family. Let Δ and Δ be the two polyhedral surfaces shown in Figure 10. Here, Δ is a pillow obtained by gluing two copies of the unit square [0, 1] 2 , called faces of Δ, together along their boundaries. The polyhedral surface Δ is constructed from 2 × 11 squares of side length 1/3, called 1-squares of Δ (they correspond to 1-tiles in the terminology of Section 4). More precisely we take two copies of and glue them together along their boundaries. We call each copy of Y ⊂ Δ a face of Δ . Clearly Δ and Δ are homeomorphic to S 2 . As polyhedral surfaces, Δ and Δ may both be By symmetry, we may assume that ϕ and ϕ map one face of Δ and, respectively, Δ to the upper half-plane and the other face to the lower half-plane, so that the points marked −1, 0, 1, ∞ in Figure 10 are mapped to −1, 0, 1, ∞ ∈ R ⊂ C, respectively. In fact, ϕ and ϕ may be constructed explicitly by mapping each face of Δ and, respectively, Δ by a Riemann map to the upper or lower half-plane, where the four marked points are mapped as indicated. The map g : Δ → Δ is now given as follows. Each 1-square of Δ is mapped (conformally) by a similarity that scales by the factor 3 to a face of Δ as indicated in Figure 10. We define the map Note that f is holomorphic, hence a rational map. Consider the points in Δ where at least four 1-squares of Δ intersect. Their images under ϕ are exactly the critical points of f . Furthermore, the points −1, 0, 1, ∞ ∈ C, which are the images of the vertices of Δ under ϕ, are exactly the postcritical points of f . So, f is a rational Thurston map with post(f ) = {−1, 0, 1, ∞}. Moreover, the Julia set of f is a Sierpiński carpet, see [21].
We mention in passing that we may think of f as being constructed via a two-tile subdivision rule in the sense of Cannon-Floyd-Parry (see [9,Chapter 12;11]). Alternatively, f is constructed from a Lattès map by 'adding a flap' or 'blowing up an arc' in the sense of [36].
Let C := R, E := [1, ∞] ⊂ C, p := 1, and q := ∞. The ramification portrait of f restricted to E is Again, by a slight abuse of notation, we denote the images of the points labeled by a 0 , b 0 , and c 0 in Figure 10 under ϕ by a 0 , b 0 , and c 0 , respectively.
Note that the critical point c 0 of f satisfies deg(f, c 0 ) = 3, see Figure 10. It is now elementary to check that with the above choices conditions (a)-(e) from Section 6 are satisfied. It follows from Theorem 6.3 that IMG(f ) is of exponential growth. Thus we have proved Theorem 1.2.
We may vary the construction of the map f . In particular, divide each of the two faces of the pillow Δ in n × n squares of side length 1/n, where n 3 is an odd number, and add two diagonally symmetric flaps as in Figure 10 to obtain a polyhedral surface Δ . Then repeat the above construction. Again the resulting rational map f has a Sierpiński carpet as its Julia set, and IMG(f ) is of exponential growth. The maps in [21] are constructed in the same fashion, Figure 11 (colour online). Construction of an obstructed map. but there n 2 is even. This results in a hyperbolic rational map f , meaning that every critical point of f is contained in the Fatou set F f . For each such map f condition (c) is not satisfied. Consequently, Theorem 6.3 does not apply. We do not know if these hyperbolic maps have IMGs of exponential growth. However, the argument in [21], showing that f has a Sierpiński carpet Julia set, is equally valid independently of whether n is even or odd.

A family of obstructed maps
Here, the example from the previous section is slightly modified to obtain an infinite family of obstructed Thurston maps with IMGs of exponential growth.
The construction is very similar to the one in Section 7. The only difference is that instead of 'adding two flaps' to a Lattès map we add only one; see Figure 11 and compare with Figure 10. As before, we obtain a map g : Δ → Δ and define homeomorphisms ϕ : Δ → C and ϕ : Δ → C that are normalized as in the previous section. That is, the top and bottom faces of Δ (and Δ ) are mapped to the upper and lower half-planes in C, respectively, so that the points labeled by −1, 0, 1, ∞ in Δ (and Δ ) are mapped to −1, 0, 1, ∞ ∈ C, respectively. We obtain a Thurston map f := ϕ • g • (ϕ ) −1 : C → C such that IMG(f ) is of exponential growth (using Theorem 6.3 in the exact same form as in the previous section).
However, we want to point out a crucial difference from the situation in Section 7. Namely, here each face of Δ is not symmetric with respect to the (diagonal) geodesic joining the vertices labeled by 0 and ∞ (in the respective face). This means we cannot choose the map ϕ : Δ → C to be conformal with our normalization. Consequently, the constructed map f is not rational.
With a slight abuse of notation consider now an arbitrary Thurston map f : S 2 → S 2 . Thurston gave a criterion when the map f is Thurston equivalent to a rational map. We present it only in the case when f has four postcritical points and a hyperbolic orbifold. We refer to [14] for the general statement, as well as the terminology. Let γ ⊂ S 2 post(f ) be a Jordan curve that is non-peripheral, meaning that each component of S 2 γ contains at least two postcritical points (since # post(f ) = 4 this means that each component contains exactly two postcritical points). Let γ 1 , . . . , γ k be the components of f −1 (γ) that are nonperipheral. The curve γ is called invariant if one (or, equivalently in our case, each) of the curves γ j , j = 1, . . . , k, is homotopic to γ rel. post(f ). Denote by d j the degree of the restriction f |γ j : γ j → γ for j = 1, . . . , k. Assume that γ is invariant and define Then the curve γ is called a Thurston obstruction if λ f (γ) 1. Thurston's theorem now says that f is Thurston equivalent to a rational map if and only if f has no Thurston obstruction. Otherwise f is called an obstructed Thurston map. It follows that the map f : C → C constructed in this section has a Thurston obstruction γ as shown in Figure 11. Here, we choose to draw γ in Δ and its preimages γ 1 , . . . , γ 4 in Δ for simplicity. In reality, we consider the images of these curves under ϕ and ϕ in C, respectively. Then λ f (γ) = 1 (note that the component γ 4 of f −1 (γ) is peripheral, thus it does not contribute to the sum (8.1)). Consequently, γ is a Thurston obstruction. Thus f is not Thurston equivalent to a rational map by Thurston's theorem, meaning it is obstructed. We have proved Theorem 1.3.

A non-renormalizable polynomial P with IMG of exponential growth
Here, we present an example of a postcritically finite, non-renormalizable polynomial P : C → C with dendrite Julia set and IMG of exponential growth. This polynomial serves as an example in Theorem 1.4. In fact, P is given by From these two expressions we immediately see that P has the (finite) critical points ± √ 3i of local degree 3, and 0 of local degree 4. Furthermore, they are mapped as follows Clearly P is postcritically finite and has no (finite) periodic critical points. It follows that the Julia set J of P , shown in Figure 12(b), is a dendrite (that is, a compact, connected, locally connected set with empty interior which does not separate the plane), see [8, § 11.2]. Consequently, P satisfies property (1) of Theorem 1.4.
Since J is a dendrite, the Hubbard tree of P may be defined as the smallest continuum in J containing all (finite) postcritical points (see [12]).  Figure 12(a). Here, we color the points in P −1 (1) black and the points in P −1 (−1) white, and indicate how they are mapped for convenience (even though this information is already contained in the coloring). We also label the critical and postcritical points.
Let us recall that a point x of a dendrite X is called a leaf of X if X {x} is connected. Clearly, the points 1 and −1 are leaves of the Hubbard tree H and the pre-Hubbard tree H . It follows that each set P −n (H), n ∈ N 0 , is a planar tree and its set of leaves contains the points 1, −1. Consequently, the postcritical points 1 and −1 are leaves of the dendrite Julia set J . Thus P satisfies property (2) of Theorem 1.4.
The polynomial P is in fact a Shabat (or Belyȋ) polynomial. This means that P has exactly 2 finite critical values. The diagram in Figure 12(a) is the dessin d'enfant of P . For a general Shabat polynomial, its dessin d'enfant is obtained as the preimage of an arc connecting its that is, two half-planes for each edge e of the tree T . Then, one constructs an abstract Riemann surface Δ by gluing these half-planes together as indicated by the structure of T . There is a natural holomorphic map g : Δ → C that maps the half-planes in Δ alternatingly to the upper and lower half-planes in C. Using the uniformization theorem one obtains a conformal map ϕ : Δ → C. The map g • ϕ −1 is then the desired map, which can be shown to be a Shabat polynomial. Put differently, the construction of a Shabat polynomial is very similar to the construction of the rational maps in Sections 3 and 7. The half-planes H + e and H − e correspond to the white and black 1-tiles from which Δ was constructed there. Finally, we note that a general dessin d'enfant is not necessarily planar or a tree. The concept was introduced by Grothendieck in [19] as a way to describe algebraic curves.

P is not renormalizable
Here, we show that the polynomial P from the previous section is not renormalizable. That is, we verify that P satisfies property (3) of Theorem 1.4. This means P is a counterexample to the conjecture stated in Section 1. However the conjecture may still be valid for quadratic polynomials. We first recall some relevant definitions.
A continuous map f : U → V is said to be proper if f −1 (K) is compact in U for every compact set K ⊂ V . Assume now in addition that U and V are Jordan domains in C and that f is holomorphic. In this case f extends continuously to ∂U and this extension (still denoted by f ) satisfies f (∂U ) = ∂V . The map f : ∂U → ∂V is topologically conjugate to z d : S 1 → S 1 for a d ∈ N (here we view the unit circle S 1 as the boundary of the unit disk D). Furthermore, every point in V has exactly d preimages in U when counted with multiplicity (see [31,). The number d is then called the degree of the proper map f : U → V . Polynomial-like maps were introduced by Douady and Hubbard in [13]. The above definition differs slightly from the typical one found in the literature, as we allow the case d = 1. The filled Julia set of a polynomial-like map (f, U, V ), denoted by K(f |U ), is the set of points in U that never leave U under iteration of f , that is, It is a compact subset of U .
Definition 10.2. Let P : C → C be a polynomial of degree d 2 with connected Julia set. Let n ∈ N, then P n : U → V is called a renormalization of P if (1) (P n , U, V ) is a polynomial-like map; (2) K(P n |U ) is connected; (3) the degree δ of (P n , U, V ) satisfies 2 δ < d n .
If such a renormalization P n : U → V exists for some n ∈ N, then we call the polynomial P renormalizable.
Condition (3) in the above definition excludes trivial polynomial-like restrictions, that is, P itself or maps of degree 1. Renormalization has been mostly considered in the case of quadratic polynomials, see, for example, [27]. The higher degree case has been considered in [22], see also [26]. Note that in the non-quadratic case there are several distinct definitions of renormalization. We have chosen the most general one. Proof. Let P be a polynomial as in the statement. Assume that P is renormalizable, that is, there exists a polynomial-like map (P n , U, V ) as in Definition 10.2 for some n ∈ N. Denote by δ its degree. Let p and q be the two finite postcritical points of P and K(P n |U ) be the filled Julia set of (P n , U, V ).
Clearly, if V contains no postcritical point of P , then the polynomial-like map (P n , U, V ) has degree δ = 1 by the Riemann-Hurwitz formula. Thus (P n , U, V ) is not a renormalization of P .
Without loss of generality, we may assume from now on that p ∈ V . Two cases remain, namely q ∈ V and q / ∈ V . First suppose that q ∈ V . This means that Thus the closed Jordan domain A ∞ := C V contains a single postcritical point of P n , namely ∞, in its interior. Note that every component of P −n (A ∞ ) contains a preimage of ∞, hence there is only one such component. Using the remark before the proof, we conclude that P −n (A ∞ ) = C P −n (V ) is a Jordan domain. It follows that P −n (V ) consists of a single component, namely U . This in turn implies that the degree of the polynomial-like map (P n , U, V ) is δ = deg(P n ) = deg(P ) n . This contradicts condition (3) in Definition 10.2, meaning that (P n , U, V ) is not a renormalization of P . Now suppose that q / ∈ V , that is, V contains exactly one postcritical point, namely p. By the remark after Theorem 10.3, U contains exactly one point in P −n (p), which we denote by c. Then c is a critical point of P n ; for otherwise the degree of (P n , U, V ) is δ = 1, contradicting condition (3) in Definition 10.2. Since P has no periodic critical points, c = p. Thus the set (P n |U ) −1 (c) = (P n |U ) −2 (p) contains exactly δ distinct points c 1 , . . . , c δ . It follows that (P n |U ) −1 (U ) = (P n |U ) −2 (V ) has exactly δ components U 1 , . . . , U δ (where c j ∈ U j for j = 1, . . . , δ). Furthermore, U j ⊂ U for each j, since U ⊂ V and P n : U → V is proper. Iterating this process, we see that each U j contains δ components of (P n |U ) −2 (U ), and so on. It follows that U j ∩ K(P n |U ) = ∅ for all j = 1, . . . , δ. Hence K(P n |U ) is disconnected, contradicting condition (2) in Definition 10.2. Thus (P n , U, V ) is not a renormalization of P . Theorem 10.3 implies that the polynomial P from Section 9 is not renormalizable. Thus P satisfies property (3) of Theorem 1.4. This finishes the proof of Theorem 1.4.

Appendix
In the literature, one usually studies properties of a self-similar group G via its wreath recursion. In particular, see the proof of exponential growth of the Basilica group, that is, IMG(z 2 − 1), in [20,Proposition 4] or the study of IMGs of postcritically finite quadratic polynomials in [5]. The goal of this appendix is to show how these techniques apply in our setting, that is, when G is the IMG of a Thurston map.

A.1. Actions on rooted trees and self-similar groups
For the convenience of the reader we review the definition of self-similar groups and some closely related notions, more details can be found in [32, Chapters 1 and 2].
Choose an alphabet X of d letters. The set of all words in X of length n ∈ N is denoted by X n . The empty word is the only word of length 0 and denoted by ∅. Consequently we set X 0 = {∅}. Let X * = n∈N0 X n be the set of all finite words in the alphabet X. Then X * has a natural structure of a d-ary rooted tree: we define the root to be ∅ and connect every word v ∈ X n to all words of the form vx ∈ X n+1 for an arbitrary letter x ∈ X and each n ∈ N 0 . The set X * viewed as a rooted tree is called the tree of words in the alphabet X and is denoted by T . It is evident that the nth level of the tree T is given by the words in X n , n ∈ N 0 .
Let Aut(T ) be the automorphism group of the tree T , that is, the group of all bijective maps g : T → T that preserve the adjacency of the vertices of T . We consider the right action of Aut(T ) on the tree T . So, the image of a vertex v under the action of an element g ∈ Aut(T ) is denoted by v g , and in the product g 1 g 2 the element g 1 acts first.
Definition A.1. Let G be a subgroup of Aut(T ). The nth level stabilizer is the subgroup Stab G (n) of those elements of G that fix pointwise all vertices of the nth level X n of the tree T . That is, The nth level stabilizer is a normal subgroup in G of finite index for all n ∈ N 0 .
Let v ∈ X * be an arbitrary vertex of the tree of words T . Denote by T v the subtree of T rooted at v such that the vertex set of T v is {vx : x ∈ X * }. Clearly, T v is isomorphic to the whole tree of words T via the shift ι v : T v → T defined by vu → u for u ∈ X * .
For every g ∈ Aut(T ) and v ∈ X * , we define an automorphism g| v : T → T , called the restriction of g to the subtree T v , by v . To simplify the notation assume that X = {1, . . . , d}. Denote by Σ(X) the symmetric group of permutations of the set X. Then every element g ∈ Aut(T ) can be written in the following form, called the wreath recursion of g, g = ⟪g| 1 , . . . , g| d ⟫σ g , where ⟪g| 1 , . . . , g| d ⟫ ∈ Stab Aut(T ) (1) ∼ = Aut(T ) X , that is, an element of the direct product Aut(T ) × · · · × Aut(T ) with d = #X factors, and σ g ∈ Σ(X) is the permutation equal to the action of g on X 1 . Formally, there is a canonical isomorphism ψ : Aut(T ) → Aut(T ) X Σ(X), (A.1) where the semidirect product is taken with respect to the natural action of Σ(X) on the factors of Aut(T ) X (an element σ ∈ Σ(X) acts on Aut(T ) X by the permutation of the factors coming from its action on X, that is, (g 1 , . . . , g d ) σ = (g 1 σ , . . . , g d σ )). In other words, the automorphism group Aut(T ) is isomorphic to the permutational wreath product Aut(T ) Σ(X). That is, the wreath recursions of arbitrary elements g, h ∈ Aut(T ) are multiplied according to the rule ⟪g 1 , . . . , g d ⟫σ · ⟪h 1 , . . . , h d ⟫τ = ⟪g 1 h 1 σ , . . . , g d h d σ ⟫στ.
Definition A.2. A group G < Aut(T ) acting faithfully on the tree T = X * is said to be self-similar if for every g ∈ G and every x ∈ X there exist h ∈ G and y ∈ X such that (xv) g = y(v) h for all v ∈ X * . Put differently, G is called self-similar if each restriction g| v belongs to G for all g ∈ G and v ∈ X * .
It is clear from the definition that a group G < Aut(T ) is self-similar if and only if g| x ∈ G for all x ∈ X 1 and all generators g of G. Furthermore, every self-similar group G has an associated wreath recursion, that is, a homomorphism ψ : G → G Σ(X) given by the restriction of the canonical isomorphism (A.1) to G. Definition A.3. Let G < Aut(T ) be a self-similar group. G is said to be recurrent (or self-replicating) if its action is transitive on the first level X 1 of the tree T = X * and for some (and thus for all) x ∈ X the homomorphism ψ x : G x → G given by g → g| x is onto, where G x = {g ∈ G : x g = x} is the stabilizer of x in G.
Note that if a self-similar group is recurrent, then it is transitive on every level of the tree T (the group is then called level-transitive).
Definition A.4. A self-similar group G < Aut(T ) is called regular branch if there exists a finite index subgroup H of G such that H X < ψ(H), where ψ : G → G Σ(X) is the wreath recursion associated with G. In such a case we say that G is regular branch over H.
A.2. Further properties of IMG(f 1 ) Here we revisit the IMG of the map f 1 from Section 3. Let a, b, c be the generators of IMG(f 1 ) as in Section 4. That is, IMG(f 1 ) acts on the dynamical preimage tree T f1 that is identified with {1, . . . , 6} * and the wreath recursions of the generators are given by (4.1). We use this to prove additional properties of IMG(f 1 ). The discussion is kept rather short, to avoid excessive details.
We start with giving alternative proofs of Corollary 5.3 and Lemma 5.4 (which implies exponential growth of IMG(f 1 )).
Proof. To save space we will only prove that x := ab 4 and y := ab 12 generate a free semigroup in IMG(f 1 ) (the proof is analogous if we introduce the third generator z := ab 20 ).
The first two identities imply that x and y are elements of infinite order in IMG(f 1 ). Indeed, the order of x cannot be an odd number, because x acts as the permutation (13)(25)(46) on the first level of the dynamical preimage tree T f1 . If ord(x) = 2n, n ∈ N, then (A.2) imply that x n = 1, contradiction. The same argument applies to y. Now let w be an arbitrary word in x and y. Here and throughout the proof, |w| denotes the length of the word w with respect to the alphabet {x, y}. If |w| is even, w can be written uniquely as a product of x 2 , y 2 , xy, and yx. From (A.2) it follows that ψ 1 (w) is a word in {x, y} and |ψ 1 (w)| = |w|/2, where ψ 1 denotes the projection onto the first coordinate in the canonical isomorphism (A.1), that is, ψ 1 (w) = w| 1 . Moreover, ψ 1 (w) ends with the same letter (x or y) as w.
Consider now any two distinct words w 1 and w 2 in the alphabet {x, y}. We are going to show that they represent different group elements in IMG(f 1 ). The proof is by induction on |w 1 |+|w 2 |.
The base cases |w 1 | + |w 2 | ∈ {1, 2} can easily be verified. Now let |w 1 | + |w 2 | 3. By multiplying from the right with the inverse of the common ending word, it is enough to assume that w 1 and w 2 end with different letters. We can further assume that the parity of |w 1 | and |w 2 | is the same (for otherwise, w 1 w −1 2 does not fix the first level). If |w i | is odd set w i = xw i , otherwise set w i = w i , for i = 1, 2. Then ψ 1 (w 1 ) and ψ 1 (w 2 ) are two distinct words in {x, y}. Indeed, they end with different letters since w 1 and w 2 do. Furthermore, |ψ 1 (w 1 )| + |ψ 1 (w 2 )| |w 1 | + 1 2 + |w 2 | + 1 2 < |w 1 | + |w 2 |.
To simplify the notation we denote IMG(f 1 ) by G from now on.