Settled Elements in Profinite Groups 3

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DEFINITION 1.1. Let G be a compact subgroup of Aut(T ). A maximal torus is a maximal closed abelian subgroup A of G which acts freely and transitively on ∂T .

Throughout the paper, for an element a ∈ Aut(T ) we denote by A the cyclic subgroup of Aut(T ) generated by a, and we denote by A the closure of A in Homeo(∂T ).

We say that a ∈ Aut(T ) is a minimal element, if A acts transitively on each level Vn, n ≥ 1. We study procyclic groups in detail in Section 4. In particular, we show that if a is minimal, then the closure A in Homeo(∂T ) is a maximal torus. We reserve the letter a for minimal elements in Aut(T ). The next deﬁnition can be given for arbitrary spherically homogeneous tree but we formulate it for a d-ary tree T for simplicity. We remark that for σ ∈ Aut(T ) the restriction σ|Vn, for n ≥ 1, is a n permutation of a set of d elements, and so we can talk about the cycle decomposition of σ|Vn.

DEFINITION 1.2. [6] Let σ ∈ Aut(T ), let n ≥ 1 and let v ∈ Vn be a vertex. We say that v is in a cycle of length k ≥ 0 of σ if σk(v)= v and σj (v) 6= v for every 1 n and m−n m−n w ∈{0, ··· , d − 1} , the vertex vw ∈ Vm is in a cycle of length d k.

The meaning of this deﬁnition is the following. Observe that for a cycle in σ|Vn, its preimage in σ|Vn+1 may be a single cycle, or it may consist of two or more (at most d) disjoint cycles. The preimage of a stable cycle in Vn consists of a single cycle in each Vn+m, m ≥ 1. DEFINITION 1.3. [6] We say that σ ∈ Aut(T ) is settled if |{v ∈ V : v is in a stable cycle}| lim n =1. n→∞ |Vn|

We say that σ ∈ Aut(T ) is strongly settled if there exists n ≥ 1 such that every vertex in Vn is in a stable cycle.

A minimal element a ∈ Aut(T ) is strongly settled, with a|Vn consisting of a single cycle for all n ≥ 1. DEFINITION 1.4. We say that a proﬁnite subgroup G ⊂ Aut(T ) is densely settled if the set of settled elements of G is dense in G.

The simplest example of a densely settled group is Aut(T ) itself, see Section 6.2. Every element in a maximal torus A generated by a minimal element a is strongly settled, so A is densely settled. The question whether an arbitrary proﬁnite subgroup of Aut(T ) is densely settled is considerably more complicated. In Sections 5 and 7, we give a dynamical proof of the following statement. THEOREM 1.5. Let T be a d-ary tree, for d prime, and let a ∈ Aut(T ) be a minimal element. Then the normalizer N(A) of the maximal torus A is densely settled.

A strongly settled element has no fixed points in Vn for n sufficiently large, while in general settled elements may have fixed points at any level Vn, n ≥ 1. Non-abelian profinite groups often have elements which have multiple fixed points in Vn. To approximate such elements by a sequence of settled elements we need settled elements with fixed points at any finite level. In Theorem 1.5 such elements are contained, for instance, in N(A) − A. If a ∈ G is minimal, where G ⊂ Aut(T ) is a profinite group, then the information about settled elements of N(A) contained in G is captured by the Weyl group of A, which we define now. DEFINITION 1.6. Let G ⊂ Aut(T ) be a compact group, and let A ⊂ G be a maximal torus. Then:

(1) The absolute Weyl group of a maximal torus A is the quotient W (A)= N(A)/A. (2) The Weyl group of A in G is the quotient W (A, G) = (N(A) ∩ G)/A. SETTLED ELEMENTS IN PROFINITE GROUPS 3

We now discuss the significance of Weyl groups for the study of the conjecture by Boston and Jones for arboreal representations of Galois groups of number fields. We define the arboreal representations first. Let K be an algebraic number field, and denote by K a separable closure of K. Given a polynomial f(x) of degree d ≥ 2 with coefficients in the ring of integers of K, an arboreal representation of K associated to f(x) is a homomorphism ρf,α : Gal(K/K) → Aut(T ), where T is a d-ary tree, and α ∈ K, see Section 2. A similar construction over the field of rational functions K(t) where t is transcendental gives a subgroup Garith(f) ⊂ Aut(T ), called the profinite arithmetic iterated monodromy group associated to f(x). An arboreal representation ρf,α for a given α ∈ K is related to Garith(f) via a specialization, and Garith(f) is thought of as the image of ρf,α for a generic choice of α. The profinite geometric iterated monodromy group Ggeom(f) is a normal subgroup of Garith(f), which is known to be isomorphic to the closure of the discrete iterated monodromy group (a self-similar group generated by finite automata) of f(x) in geometric group theory [16, 17, 27]. Arboreal representations of Galois groups are relatively little studied, but they have been receiving more attention lately, see [20, 3] for recent surveys. One of the main questions is under what conditions the image of ρf,α has finite index in Aut(T ). This problem is motivated by questions about the density of prime divisors in forward orbits of points under iterations of a polynomial f(x), first asked by Odoni [28]. The methods used to study arboreal representations in the literature are mostly number-theoretical, although techniques from geometric group theory were used by Jones [21], Pink [29] and the second author [25]. Jones [19, 20, 21] also applied theory of stochastic processes to arboreal representations. Some topological dynamical properties of arboreal representations and profinite iterated monodromy groups were studied by the second author [24, 25]. Boston and Jones conjectured in [5, 6, 7] that the set of settled elements is dense in the image of an arboreal representation, relating this property with the factorization of polynomials into irreducible factors. We refer to [5, 6, 7] for details and the arguments behind the conjecture, and also for a discussion of its significance for the development of a theory of arboreal representations similar to that of linear representations of absolute Galois groups of fields. The conjecture is true for arboreal representations which have finite index in Aut(T ) since Aut(T ) is densely settled. The conjecture also holds for arboreal representations isomorphic to the infinite wreath products of certain finite groups, see Remark 6.10. The most difficult and interesting case is when an arboreal representation has infinite index in Aut(T ), in particular, this happens if f(x) is a post-critically finite (PCF) polynomial, see Definition 1.7 and Section 2. Boston and Jones studied arboreal representations associated to quadratic PCF polynomials over Q in [6]. In that case there are three distinct conjugacy classes of polynomials, and it was shown in [6] that the conjecture is true for two of these classes. The method in [6, 7] is based on considering the factorization of iterates f n(x) modulo primes, and the behavior of irreducible factors is conjectured to be approximated by a certain Markov process [7]. The Markov model was refined by Goksel, Xia and Boston [15], and both the model and the methods were further used by Goksel [12, 13, 14] for quadratic polynomials over Q. The conjecture by Boston and Jones still remains unproved for polynomials over Q for which the profinite geometric iterated monodromy group Ggeom(f) is conjugate to the Basilica group. If K is an algebraic number field then there are further conjugacy classes of polynomials, and one expects more diverse behavior. An extensive study of the properties of the profinite arithmetic and geometric iterated monodromy groups for quadratic f(x) over any number field K was done by Pink [29]. Paper [29] gives an explicit representation of Ggeom(f) using its topological generators, and studies the properties of the normalizers of these groups (which contain Garith(f)) in Aut(T ). We study the Weyl groups of maximal tori in profinite arithmetic and geometric iterated monodromy groups associated to a class of PCF quadratic polynomials specified below.

DEFINITION 1.7. Let f(x) be a quadratic polynomial, and let c be the critical point of f(x). The n post-critical orbit is the orbit Pc = {f (c) | n ≥ 1} of the critical point c. A quadratic polynomial f(x) is a post-critically finite (PCF) if Pc is a finite set. 4 MARÍA ISABEL CORTEZ AND OLGA LUKINA

For quadratic polynomials, there are two possibilities for the dynamics on Pc: either the post-critical orbit is strictly periodic and consists of a single cycle of length #Pc, or it is strictly pre-periodic, with a periodic cycle of length strictly smaller than #Pc. For instance, if f(x) ∈ Z[x] is PCF, then either f(x) has a strictly periodic post-critical orbit of length 1 or 2, or f(x) has a strictly pre-periodic post-critical orbit of length 2 with a cycle of length 1 [6]. For the cases when there is a cycle of length 1, it was shown in [6] that the corresponding profinite arithmetic iterated monodromy group is densely settled. When the post-critical orbit is periodic of length 2, then Garith(f) contains a subgroup conjugate to the Basilica group [6, 29]. The question whether this group is densely settled is still open. We study the case when f(x) is a quadratic PCF polynomial over an algebraic number field K with strictly pre-periodic orbit of length r ≥ 2, with periodic cycle of length r − s, for s ≥ 1, r, s ∈ N. Recall from [29] that for r = 2 and s = 1, the polynomial f(x) is conjugate to the quadratic Chebyshev polynomial, and its profinite geometric iterated monodromy group Ggeom(f) is conjugate in Aut(T ) to the closure G of the action of the dihedral group G = ha,b | b2 =1,bab = a−1i. For a set S ⊂ Aut(T ) denote by Min(S) the subset of S which consists of minimal elements. Recall that for a ∈ Min(S) we denote by A the corresponding maximal torus. Note that if S is a closed subgroup of Aut(T ), then A ⊂ S. For a group S, we denote by N(S) the normalizer of S in Aut(T ). THEOREM 1.8. Let f(x) be a quadratic PCF polynomial over an algebraic number field K with strictly pre-periodic post-critical orbit of length r =2. Then for any a ∈ Min(Ggeom(f)) we have the following:

(1) The Weyl group W (A, Ggeom(f)) is ﬁnite of order 2. (2) The Weyl group W (A, N(Ggeom(f))) = W (A), where W (A) is the absolute Weyl group of A. (3) The Weyl group W (A, Garith) has ﬁnite index in W (A). In addition:

(4) The proﬁnite geometric iterated monodromy group Ggeom(f) is not densely settled. (5) The normalizer N(Ggeom(f)) is densely settled. (6) The proﬁnite arithmetic iterated monodromy group Garith(f) is densely settled.

Intuitively, W (A, Ggeom(f)) being ﬁnite says that only a ‘small portion’ of settled elements in N(A) is contained in Ggeom(f). On the other hand, if the Weyl group W (A, Garith(f)) has ﬁnite index in the absolute Weyl group W (A), then this shows that a ‘large portion’ of settled elements of N(A) is contained in Garith(f). Since for the group in Theorem 1.8 we have N(A) = N(Ggeom(f)), this also leads to the statement that Garith(f) is densely settled.

If r ≥ 3 then Ggeom(f) is conjugate to a branch group, see [16, 27] for definition and properties of branch groups. The dynamics of a branch group acting on the boundary of a tree can be very complicated. Such groups contain elements which have fixed points and non-trivial elements which fix every point in a collection of disjoint clopen subsets in ∂T . Although we are not able to prove completely the conjecture of Boston and Jones for profinite iterated monodromy groups associated to f(x) for r ≥ 3, we obtain results about maximal tori and the Weyl groups, which allow for a better understanding of the problem. We start by studying maximal tori in the normalizer N(Ggeom(f)). THEOREM 1.9. Let f(x) be a quadratic PCF polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3 over an algebraic number field K. Then for every coset in N(Ggeom(f))/Ggeom(f) there is a representative w such that:

(1) For every a ∈ Min(Ggeom(f)), the element aw = aw is in Min(wGgeom(f)). (2) For a maximal torus Aw generated by aw, the intersection Aw ∩ Ggeom(f) is an index 2 subgroup of Aw, and Aw ⊂ Ggeom(f) ∪ wGgeom(f). Thus we conclude the following: SETTLED ELEMENTS IN PROFINITE GROUPS 5

(3) The normalizer N(Ggeom(f)) contains an uncountable number of distinct maximal tori, which may intersect along proper subgroups of ﬁnite index. (4) Minimal elements in Ggeom(f), and minimal elements in each coset wGgeom(f) in N(Ggeom(f)) are in bijective correspondence.

Next we are interested in the properties of Weyl groups and maximal tori in profinite arithmetic iterated monodromy groups. For a ∈ Min(Ggeom(f)), the corresponding maximal torus A is contained in Ggeom(f), and the results of [29, Section 3] allow us to conclude that its absolute Weyl group W (A) is contained in N(Ggeom(f)). If a minimal element a is in N(Ggeom(f)) but not in Ggeom(f), then we have less information about the Weyl group of the corresponding maximal torus. We prove the following criterion, which allows us to determine when the absolute Weyl group of a maximal torus topologically generated by a ∈ Min(N(Ggeom(f))) is contained in N(Ggeom(f)). THEOREM 1.10. Let f(x) be a quadratic PCF polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3 over an algebraic number field K. Then there is a well-defined action −1 N(Ggeom(f)) × Min(wGgeom(f)) → Min(wGgeom(f)) : (z,g) 7→ zgz .

If this action is transitive on Min(wGgeom(f)) then for any aw ∈ Min(wGgeom(f)) and the corresponding maximal torus Aw we have the equality of the Weyl groups

W (Aw,N(Ggeom(f))) = W (Aw).

We note that Theorems 1.9 and 1.10 exhibit the differences between the properties of maximal tori in compact connected Lie groups and in Aut(T ). For instance, the Weyl group of a maximal torus in a compact Lie group is always finite, while W (A, Ggeom(f)) may be infinite. Also, in a compact Lie group all maximal tori are conjugate in the group, and every element in the group is contained in a maximal torus. We show that in branch groups which appear as profinite geometric iterated monodromy groups associated to quadratic polynomials, either there are maximal tori which are not conjugate in the normalizer of the branch group or there are elements which are not contained in a maximal torus (possibly both). In normalizers of branch groups, maximal tori need not be conjugate within the normalizer. We discuss these statements in more detail in the proof of Theorem 8.9. We collect the consequences of Theorems 1.9 and 1.10 for the densely settled property of profinite arithmetic iterated monodromy groups associated to quadratic polynomials in the theorem below. THEOREM 1.11. Let f(x) be a quadratic PCF polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3 over a number field K, and let Ggeom(f) and Garith(f) denote the profinite geometric and arithmetic iterated monodromy groups associated to f(x). Then the following hold:

(1) For any w ∈ Garith(f) the sets Min(Ggeom(f)) and Min(wGgeom(f)) are in bijective correspondence. Thus for every w ∈ Garith(f) the coset wGgeom(f) contains strongly settled elements. (2) For any a ∈ Min(Ggeom(f)), the index of the Weyl group W (A, Ggeom(f)) in the absolute Weyl group W (A) is ﬁnite. Thus Ggeom(f) contains settled elements which are not minimal. (3) For any a ∈ Min(Ggeom(f)), the index of the Weyl group W (A, Ggeom(f)) in W (A, Garith(f)) is equal to the index of Ggeom(f) in Garith(f). Thus, for any w ∈ Garith(f) the coset wGgeom(f) contains settled elements.

Theorem 1.11 contains information about settled elements in Garith(f) which arise as elements in normalizers of maximal tori topologically generated by minimal elements. We note that not necessarily all settled elements arise as elements in the normalizers of such maximal tori. Conceivably there may be other settled elements with different cycle structure. The following question seems difficult to answer at the moment, and this question is also the main obstacle to proving the conjecture for r ≥ 3 using maximal tori and Weyl groups. PROBLEM 1.12. Let g ∈ Aut(T ) be such that g is the identity on a proper subset U of ∂T , where U is the finite or infinite union of clopen sets. Does there exists a sequence {gn}, n ≥ 1, which converges to g, and such that each gn is a settled element in the normalizer of a maximal torus? 6 MARÍA ISABEL CORTEZ AND OLGA LUKINA

The proofs of Theorem 1.9, 1.10 and 1.11 rely on the descriptions of profinite iterated monodromy groups in [29, Section 3], and depend on the length and the structure of the post-critical orbit of a polynomial f(x). The proofs do not directly extend to the case when f(x) has a strictly periodic post-critical orbit since the properties of profinite iterated monodromy groups in that case are very different. The following question remains open. PROBLEM 1.13. Let f(x) be a quadratic PCF polynomial with strictly periodic post-critical orbit of length r ≥ 2 over an algebraic number field K. Study the maximal tori and their Weyl groups in the associated groups Ggeom(f), N(Ggeom(f)) and Garith(f).

Another natural open problem is the following. PROBLEM 1.14. Study the maximal tori and the Weyl groups in proﬁnite groups acting on trees.

We do not expect all statements about maximal tori and Weyl groups which are true for compact Lie groups, to carry over verbatim to our setting. Indeed, we have already described some diﬀerences in Theorem 8.9. However, we can always ask how far the similarities go. We remark that looking for notions and concepts for actions on trees, which are similar to those used in smooth dynamics has already proved fruitful, as in [18]. The rest of the paper is organized as follows: Sections 2 and 3 are devoted to recalling general concepts about arboreal representations of polynomials and topological dynamics respectively. In Section 4 we study the procyclic subgroups of Aut(T ) from a dynamical point of view. In Section 5 we study the normalizer N(A) of a maximal torus A topologically generated by a minimal element a ∈ Aut(T ). In Section 6 we give a dynamical characterization of settled elements of Aut(T ), in terms of their minimal components, and we study the conditions under which the property of being densely settled is preserved. Finally, in Section 7 and 8 we use the previous results to show Theorem 1.5 and Theorems 1.8, 1.9, 1.10, 1.11, respectively. Acknowledgments: The authors thank Steven Hurder for pointing out the similarities between abelian groups of tree automorphisms and their normalizers in our work, and the notions of the maximal torus and the Weyl group in theory of compact Lie groups.

2. Representations of Galois groups

We briefly recall some background on arboreal representations and profinite iterated monodromy groups, see [20] and [29] for more details. Let K be a number field, that is, K is a finite algebraic extension of the rational numbers Q. Let f(x) be a polynomial of degree d ≥ 2 with coefficients in the ring of integers of K. Let t be a transcendental element, then K(t) is the field of rational functions with coefficients in K. We first define the profinite arithmetic and geometric iterated monodromy groups. Consider the solutions of the equation f n(x)= t over K(t). The polynomial f n(x)−t is separable and irreducible over K(t) for all n ≥ 1 [1, Lemma 2.1], which means that all dn roots of f n(x) − t = 0 are distinct, n and f (x) − t does not factor over K(t). It follows that the Galois group Hn of the extension Kn obtained by adjoining to K(t) the roots of f n(x) − t acts transitively on the roots.

The tree T has the vertex set V = · n≥0 Vn, where V0 is identified with t, and Vn with the sets of n solutions of f (x)= t. We join β ∈ Vn+1 and α ∈ Vn by an edge if and only if f(β)= α. For each S n n ≥ 1, the Galois group Hn acts transitively on the roots of f (x)= t by field automorphisms, and so induces a permutation of vertices in Vn. Since the field extensions satisfy Kn ⊂ Kn+1, we have a n+1 group homomorphism λn : Hn+1 → Hn. Taking the inverse limit G n+1 arith(f) = lim{λn : Hn+1 → Hn}, ←− we obtain a profinite group called the arithmetic iterated monodromy group associated to the polynomial f(x). For n ≥ 1, the action of Hn on T preserves the connectedness of paths in T , and so Garith(f) is identified with a subgroup of the automorphism group Aut(T ) of the tree T . SETTLED ELEMENTS IN PROFINITE GROUPS 7

Denote by K a separable closure of K. An arboreal representation of the absolute Galois group Gal(K/K) is obtained by a construction similar to the one described above by choosing α ∈ K −n −n and setting Vn = {f (α)}, with Yn the Galois group of the extension K(f (α)). The image of the representation is the proﬁnite group Y∞ = lim{Yn+1 → Yn}. If Q ⊂ K, then the polynomial ←− f n(x) − α is separable but it need not be irreducible, so the construction requires the assumption that f n(x)−α is irreducible for n ≥ 1. The trees obtained by these two constructions are both d-ary trees, and so their automorphism groups are isomorphic. Therefore, one may think of the groups Yn as obtained via the specialization t = α. As explained in [28], Galois groups of polynomials do not increase under such specializations, and certain groups are preserved. In particular, Y∞ is a subgroup of Garith(f), and one can think of Garith(f) as Y∞ for a generic choice of α, in a loose sense [28, 20]. A similar construction can be made with iterations of the same polynomial replaced by compositions of distinct polynomials f1 ◦···◦ fn such that these compositions are separable and irreducible for n ≥ 1, see [11]. Denote by K the extension of K(t) obtained by adding the roots of f n(x) − t for all n ≥ 1. Let L = K ∩ K be the maximal constant ﬁeld extension of K in K, that is, L contains all elements of K algebraic over K. Then the Galois group Ggeom(f) of the extension K/L(t) is a normal subgroup of Garith(f), and there is an exact sequence [29, 20]

/ / / / (1) 1 Ggeom(f) Garith(f) G(L/K) 1 .

The profinite group Ggeom(f) is called the geometric iterated monodromy group, associated to the polynomial f(x). The geometric monodromy group Ggeom(f) does not change under extensions of L, so one can calculate Ggeom(f) over C(t). The profinite arithmetic iterated monodromy group Garith(f) is contained in the normalizer N = N(Ggeom(f)) of Ggeom(f). A relation with iterated monodromy groups in geometric group theory [27, 16] is given by the following construction. Let P1(C) be the projective line over C (the Riemann sphere), and extend the polynomial map f : C → C to P1(C) by setting f(∞)= ∞. Then ∞ is a critical point of f(x). n Let C be the set of all critical points of f(x), and let PC = n≥1 f (C) be the set of the forward orbits of the points in C, called the post-critical set. Suppose P is finite, then the polynomial f(x) SC is called post-critically finite.

REMARK 2.1. The definition of a post-critically finite quadratic polynomial in Definition 1.7 is slightly different to the one we just gave, but the two definitions are equivalent. Indeed, if f(x) is quadratic, then the set C of critical points of f(x) on the Riemann sphere consists of two points, the point ∞ and the critical point c ∈ C. Then PC = {∞} ∪ Pc, where Pc is the orbit of c under the iterations of f. Thus PC is finite if and only if Pc is finite.

If the polynomial f(x) is post-critically finite, then it defines a partial d-to-1 covering f : M1 →M, 1 −1 where M = P (C)\PC and M1 = f (M) are punctured spheres. An element s ∈ M has d preimages under f, and dn preimages under the n-th iterate f n. Define a tree T so that the n- th level vertex set is {f −n(s)}, for n ≥ 1, and there is an edge between α and β if and only if f(β)= α. The fundamental group π1(M,s) acts on the vertex sets of T via path-lifting.e Let Ker be the subgroup of π1(M,s) consisting of elements which act trivially on every vertex level set. The quotient group IMG(f) = π1(M,s)/Ker, called the discrete iterated monodromye group associated to the partial self-covering f : M1 → M, acts on the space of paths of the tree T . If f(x) is post-critically finite, by [27, Proposition 6.4.2] attributed by Nekrashevych to R. Pink, Ggeom(f) is isomorphic to the closure of the action of IMG(f) in Aut(T ). e The actions of discrete iterated monodromy groups IMG(f) are well-studied, with many results e known and many techniques developed, see for example Nekrashevych [27]. Many of iterated monodromy groups are self-similar and can be generated by finite automata. We do not discuss these notions in details, since we are not going to use them explicitly. 8 MARÍA ISABEL CORTEZ AND OLGA LUKINA

3. Definitions and background in dynamics

3.1. Dynamical systems. In this article a dynamical system is the action by homeomorphisms of a group G on a compact metric space X. The action of an element g ∈ G on x ∈ X is denoted by gx. The orbit of x ∈ X with respect to a subgroup H of G is denoted by Hx and it is deﬁned as Hx = {y ∈ X | y = hx, h ∈ H}. If H = hhi for some h ∈ G then we say that Hx is the h-orbit of x. If G = hgi then the dynamical system given by the action of G on X is denoted by (X,g). The dynamical system given by the action of G on X is minimal is Gx is dense in X for every x ∈ X. A minimal component of the dynamical system is a closed subset Y of X such that Y is invariant under the action of G and such that the dynamical system given by the restriction of the action of G to Y is minimal. In the case that G = hgi we say that g is minimal on Y , and if further Y = X then we just say that g is minimal (see Lemma 4.1 for an alternative characterization of minimality in the case when X is the boundary of a spherically homogeneous rooted tree and g is a tree automorphism). See [2] for more details about topological dynamics.

3.2. Rooted trees. For every n ≥ 1, let mn > 1 be an integer and let Σn = {0, ··· ,mn − 1}. The space Σ = n≥1 Σn is a Cantor set if we endow every Σn with the discrete topology and Σ with the product topology. Then Σ is a metric space with the following metric: let (x ) and (y ) be Q n n≥1 n n≥1 two elements in Σ, then 1 (2) d((x ) , (y ) )= if and only if k = min{n ≥ 1 : x 6= y }. n n≥1 n n≥1 2k−1 n n There is a natural way to identify the elements of Σ with the set of infinite paths of a rooted tree. The spherically homogeneous tree associated to Σ is the infinite graph TΣ = T defined as follows:

• The set of vertices V of the tree is equal to the disjoint union · n≥0 Vn, where V0 contains only one element called the root of the tree (we identify this element with the empty word) n S and Vn = i=1 Σi, for every n ≥ 1. • The set of edges E of the tree is equal to the disjoint union · En, where En contains Q n≥1 exactly one edge e from v ∈ Vn−1 to vl ∈ Vn, for every l ∈ Σn and n ≥ 1. In this case we say that the source s(e) of e is equal to v and the target t(e) ofSe is equal to vl.

The sequence mT = (m1,m2,...) is called the spherical index of T . The set of inﬁnite paths of T is given by

∂T = {(en)n≥1 ∈ En : t(en)= s(en+1), ∀n ≥ 1} =∼ {(xn)n≥1 ∈ Σ}. n Y≥1

Here the element (xn)n≥1 ∈ Σ is identified with (en)n≥1 ∈ ∂T given by t(e1) = x1, s(en) = x1 ··· xn−1 and t(en)= x1 ··· xn, for every n> 1. This identification induces a bijection between Σ and ∂T , which is an isometry with the induced metric on ∂T . Thus paths in ∂T can be thought of as infinite sequences x1x2 ··· , where xn ∈ Σn for n ≥ 1.

When mn = 2 for every n ≥ 1 then we say that T is the binary tree. If mn = d, for some d ≥ 2 and all n ≥ 1, then we say that T is the d-ary rooted tree.

3.3. Group of automorphisms. An automorphism of T is a transformation τ : V → V such that ′ τ(Vn)= Vn and if x ∈ Vn is equal to yt for y ∈ Vn−1 and t ∈ Σn, then there exists t ∈ Σn such that ′ τ(x) = τ(y)t , for every n ≥ 0. In other words, an automorphism of T permutes every Vn, n ≥ 1, preserving the structure of T , that is, two vertices in V are joined by an edge if and only if their images under the automorphism of T are joined by an edge. The collection of all the automorphisms of T is a group denoted by Aut(T ). Observe that an automorphism of T deﬁnes an isometry on ∂T with respect to the induced metric on ∂T . SETTLED ELEMENTS IN PROFINITE GROUPS 9

3.3.1. Automorphisms of the binary tree. For the binary tree T and a word y = y1 ··· yi denote (y) (y) by Ty the connected subtree of T with V0 = V0, the vertex level sets Vn = {y1 ··· yn} ⊂ Vn (y) for 1 ≤ n ≤ i, and Vn = {yxi+1 ··· xn | xj ∈ Σj ,i < j ≤ n} for n>i. Then the boundary ∂Ty consists of all paths in ∂T which start with a finite word y. There is a natural isomorphism (y) κy : Ty → T , which maps yxi+1 ··· xn ∈ Vn , for n ≥ i, to xi+1 ··· xn ∈ Vn−i, and every infinite path yxi+1xi+2 ···∈ ∂Ty to an infinite path xi+1xi+2 ···∈ ∂T . It follows that Aut(Ty) is naturally identified with Aut(T ).

Using these identiﬁcations, we can write u = (u0,u1) with u0,u1 ∈ Aut(T ) for u ∈ Aut(T ), if the restriction u|V1 is the trivial permutation. More generally, let η be the non-trivial permutation of V1 = {0, 1}, and denote also by η the automorphism of T given by ix 7→ η(i)x for i ∈ {0, 1} and m ix ∈ Vn, n ≥ 2. Then for any u ∈ Aut(T ) we can write u = (u0,u1)η , where u0,u1 ∈ Aut(T ) and m ∈{0, 1}. These notation obeys the following simple relations:

−1 −1 −1 (3) (u0,u1)η = η(u1,u0), (u0,u1) = (u0 ,u1 ), (u, v)(z, w) = (uz,vw).

The notation u = (u,b), for u,b ∈ Aut(T ) means that u is deﬁned recursively, namely, u|V1 is trivial, u|T0 = u and u|T1 = b, that is, u acts as b on the subtree T1, and as u on the subtree T0. By the recursive formula we have u|T00 = u and u|T01 = b, and, more generally, u|T0n = u and u|T0n−11 = b for n ≥ 1, where 0n denotes the concatenation of n symbols 0. Thus b and the formula u = (u,b) completely determine u. This way to write automorphism of binary trees is standard in geometric group theory, see for instance [8, 16, 27, 29].

3.3.2. Topology on the group of automorphisms. The group Aut(T ) coincides with the group of isometries of ∂T (or equivalently Σ) with respect to the induced metric on ∂T . We can endow Aut(T ) with the uniform topology inherited from the group of homeomorphisms of ∂T . Since Aut(T ) is a group of isometries, the uniform topology and the topology of pointwise convergence coincide in Aut(T ). We denote by D the uniform distance in Aut(T ), that is,

(4) D(u, w) = sup d(ux, wx), for u, w ∈ Aut(T ). x∈∂T

We show that the uniform topology on Aut(T ) coincides with the proﬁnite group topology. For every n ≥ 1 and u ∈ Aut(T ), we denote by un = u|Vn the restriction of u to Vn. The collection

(5) Nn = {u ∈ Aut(T ): un is trivial} is a normal finite index subgroup of Aut(T ), which is open since Aut(T ) is compact [30, Lemma 2.1.2]. We note that {Nn | n ≥ 1} forms a system of open neighborhoods of the identity in Aut(T ), making Aut(T ) a profinite group [30, Theorem 2.1.3]. On the other hand, Nn coincides with the 1 set of all u ∈ Aut(T ) which are at distance at most 2n from the identity automorphism in Aut(T ). Thus the profinite topology and the uniform topology of Aut(T ) coincide. We note that the action of any subgroup Γ of Aut(T ) on ∂T is automatically effective. That is, for every u ∈ Γ there is x ∈ ∂T such that ux 6= x, or, in other words, the homomorphism Γ → Homeo(∂T ) has trivial kernel. A consequence of this is that if u, w ∈ Aut(T ) and for all x ∈ ∂T we have ux = wx, then u = w. For a subset Γ ⊂ Aut(T ), we denote by Γ the closure of Γ with respect to the uniform topology or the topology of pointwise convergence. Observe that if Γ is a subgroup of Aut(T ) then Γ corresponds to the enveloping semigroup of the dynamical system given by the action of Γ on ∂T (see [2] for details). The following results can be found in [2], we include the proof here for the completeness.

LEMMA 3.1. Let Γ be a subgroup of Aut(T ) such that the action of Γ on ∂T is minimal.

(1) Γx = ∂T , for every x ∈ ∂T . (2) If in addition Γ is abelian, then the action of Γ on ∂T is free. 10 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

Proof. For (1) note that for every y ∈ ∂T there exists a subsequence (γix)i≥0 of the Γ-orbit of x that converges to y. By compactness of Γ there exists a subsequence of (γi)i≥0 that converges to some γ in Γ. We get γx = y. For (2) suppose there exist x ∈ ∂T , u and w in Γ such that ux = wx. If Γ is abelian then Γ is also abelian. Thus for every v ∈ Γ we have uvx = vux = vwx = wvx. Since the action of Γ on ∂T is minimal we have that u and w coincide on a dense subset Γx ⊂ ∂T , which implies that u = w.

3.4. Permutations. Let n ≥ 1 and let σ be a permutation on a set S of cardinality n. We denote by o(σ) the order of the permutation σ. It is known that if σ has s cycles whose lengths are ℓ1, ··· ,ℓs, then ℓ1 + ··· + ℓs = n and o(σ) is equal to the least common multiple of ℓ1, ··· ,ℓs. We say that σ acts transitively on S if σ has only one cycle of length n. In this case we have o(σ) = n, however the converse is not necessarily true as it is shown in this example: suppose that n = 30, s = 9 and the lengths of the cycles of σ are given by 2, 2, 2, 5, 5, 5, 3, 3, 3. The permutation σ does not act transitively on S, but since the least common multiple of 2, 5, 3 is 30, its order is o(σ) = 30. LEMMA 3.2. Let n ≥ 1, let p > 1 be a prime number and let S be a set of cardinality pn. The permutation σ acts transitively on S if and only if o(σ)= pn.

n Proof. Let σ be a permutation on S verifying o(σ)= p . Let ℓ1, ··· ,ℓs be the lengths of the cycles n of σ. Since p is the least common multiple of ℓ1, ··· ,ℓs, every ℓi is equal to a power of p. It follows that n is equal to the maximum of these powers of p, and so there is a cycle of order pn. But this implies that s = 1 and then σ acts transitively on S.

4. Procyclic subgroups of Aut(T ) and maximal tori

In this section, T is a tree with arbitrary spherical index mT = (m1,m2,...), see Section 3.2 for notation.

4.1. Odometers. Let p = (pn)n≥1 be an increasing sequence of positive integers such that pn divides pn+1, for every n ≥ 1. We denote

Zp = lim{Z/pnZ, φn,n ≥ 1}, ←− where φn : Z/pn+1Z → Z/pnZ is the natural projection, for every n ≥ 1. If there exists a natural n number q such that pn = q for each n, then we write Zq instead of Zp.

We consider Zp as a topological group, with the topology induced by the following metric [10]: let x = (xk)k≥1 and y = (yk)k≥1 be two elements in Zp, then 1 (6) d(x, y)= where n = min{k ≥ 1 : x 6= y }. 2n−1 k k It is not diﬃcult to see that this topology is spanned by the collection of clopen sets

[g]n = {(xk)k≥1 ∈ Zp : xn = g}, where g ∈ Z/pnZ and n ≥ 1.

Observe that this topology coincides with the profinite topology on Zp, defined similarly to the profinite topology on Aut(T ) in Section 3.3.2.

We denote by 1 the element in Zp corresponding to (1 + pnZ)n≥1, then there is a homeomorphism

cp(x)= x + 1, for every x ∈ Zp, which is a minimal isometry. The dynamical system (Zp,cp) is known as the odometer or the adding machine.

For a sequence (m1,m2,...) of positive integers, it is not difficult to show that when pn = m1 ··· mn, for every n ≥ 1, then the dynamical system (Zp,cp) is conjugate by means of an isometry to (Σ, αΣ), where Σ = n≥1 Σn with Σn = {0,...,mn − 1}, and αΣ is defined as follows: let x = (xn)n≥1 ∈ Σ. If x = m − 1 for every n ≥ 1 then α (x)=0∞, the infinite sequence of 0’s. Otherwise, let n nQ Σ SETTLED ELEMENTS IN PROFINITE GROUPS 11

k k = min{n ≥ 1 : xn

4.2. Procyclic subgroups of Aut(T ). For every a ∈ Aut(T ) we denote by A the cyclic group generated by a. We denote by A the closure of the group A in Aut(T ). It is known that A is a procyclic group, i.e., it is isomorphic to the inverse limit of ﬁnite cyclic groups (see for example [30]).

4.2.1. Maximal tori. By Section 3.1 a ∈ Aut(T ) is minimal if for every x ∈ ∂T its a-orbit Ax is dense in ∂T . Further, since a is an isometry, a is minimal if and only if there exists x ∈ ∂T such that its a-orbit is dense in ∂T (see for example [2]). In the setting of actions on trees, a is minimal if and only if for every n ≥ 1 and w ∈ Vn the a-orbit of any x intersect the cylinder set [w]= {(xi)i≥1 : x1 ··· xn = w}. Recall the notation an = a|Vn for the restriction of a to Vn, n ≥ 1. The argument above gives the following:

LEMMA 4.1. Let a ∈ Aut(T ). Then a is minimal if and only if an acts transitively on Vn, for every n ≥ 1.

Lemma 4.1 shows that the definitions of minimal elements in Section 3.1 and in the Introduction just after Definition 1.1 are equivalent. Recall from Definition 1.1 that a maximal torus in Aut(T ) is a maximal abelian subgroup of Aut(T ) which acts freely and transitively on ∂T . We now can identify some maximal tori in Aut(T ). LEMMA 4.2. If a is minimal then A is a maximal torus in Aut(T ).

Proof. Since a is minimal and A is abelian, by Lemma 3.1 A acts transitively and freely on ∂T , and we only have to show that A is maximal. Let B ⊃ A be a proﬁnite group which acts freely on ∂T . Let u ∈ B, and let x ∈ ∂T . Since A acts transitively on ∂T then there exists w ∈ A such that w(x) = u(x), which implies that w−1 ◦ u(x) = x, and so w = u since the action of B is free. But this implies that B = A, and A is maximal.

Two elements u and v in Aut(T ) are conjugate if there exists w ∈ Aut(T ) such that u = wvw−1. Conjugation induces an equivalence relation on Aut(T ), whose equivalence classes are called conjugacy classes. It is well known that if σ and τ are conjugate permutations of a ﬁnite set S then their cycle structures are the same. This implies, together with Lemma 4.1, the next result. COROLLARY 4.3. The family of minimal elements in Aut(T ) coincides with a conjugacy class of Aut(T ). Thus every minimal element of Aut(T ) is conjugate to αΣ.

Corollary 4.3 states that maximal tori in Aut(T ) that are topologically generated by a single element are conjugate. This does not imply that that all maximal tori are conjugate, since conceivably there may be maximal tori in Aut(T ) with more than one topological generator.

4.2.2. General procyclic subgroups. The next proposition gives us a more precise description of procyclic groups in Aut(T ), without an assumption that a is minimal. Recall the notation an = a|Vn for a restriction to the vertex level set, and |Vn| for the cardinality of a ﬁnite set Vn, n ≥ 1.

PROPOSITION 4.4. Let a ∈ Aut(T ) and p = (pn)n≥1. The following sentences are equivalent:

(1) pn = o(an), for every n ≥ 1. (2) There exists an isomorphism ψ : A → Zp, which is an isometry and such that the image of a is equal to 1 ∈ Zp. (3) A is isometrically isomorphic to Zp.

In addition, if a is minimal, then o(an)= |Vn|, for every n ≥ 1. 12 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

k Proof. For every n ≥ 1, let pn = o(an). For k ∈ Z we deﬁne ψ(a ) = (k + pnZ)n≥1 ∈ Zp. Observe k j 1 k−j k−j that d(a ,a ) = 2n if and only if an = id and an+1 6= id, which is equivalent to k − j ∈ pnZ k ℓ k ℓ and k − j∈ / pn+1Z. This implies that d(ψ(a ), ψ(a )) = d(a ,a ) for k,ℓ ∈ Z. Then ψ extends to a unique isometry ψ : A → Zp, such that ψ(a) = 1. By construction, ψ is a homomorphism. It is surjective since ψ(A) is dense in Zp. We have shown that (1) implies (2). Sentence (3) is a direct consequence of (2).

Now suppose that A is isometrically isomorphic to Zp. Let r = (rn)n≥1 be such that o(an)= rn, for every n ≥ 1. According to what has been shown above, we have A is isometrically isomorphic to Zr. Thus Zr and Zp are isometrically isomorphic. Let φ : Zp → Zr be an isometric isomorphism. For every n ≥ 1, the map φ induces a bijection between the collection of cylinder sets {[ℓ]n : ℓ ∈ Z/pnZ} and {[j]n : j ∈ Z/rnZ}. This implies that the cardinalities of Z/pnZ and Z/rnZ coincide, which is only possible if pn = rn. This shows that o(an) = pn for every n ≥ 1. If a is minimal then o(an)= pn = |Vn|.

In the next proposition, we study the case when a is minimal.

PROPOSITION 4.5. Let p = (pn)n≥1 be such that pn = |Vn|, for every n ≥ 1. Then a ∈ Aut(T ) is minimal if and only if the dynamical systems (∂T,a) and (Zp,cp) are conjugate by means of an isometry.

Proof. If (∂T,a) and (Zp,cp) are conjugate, then the minimality of cp implies that a is minimal.

Suppose now that a is minimal. Corollary 4.2 and Proposition 4.4 implies that A and Zp are isometrically isomorphic. Let ψ be the isometry from A to Zp given in Proposition 4.4. k k Let x ∈ ∂T . Since the a-orbit of x is dense in ∂T , the function w : Ax → Zp given by w(a x)= ψ(a ), for every k ∈ Z, extends to a bijective isometry w from ∂T to Zp. Moreover, this isometry veriﬁes w ◦ a = cp ◦ w. The map w is in Aut(T ) and it is a conjugacy map between (∂T,a) and (Zp,cp).

We highlight the diﬀerence in the statements of Proposition 4.4 and 4.5 in the example below.

n EXAMPLE 4.6. Let T be a binary tree, so |Vn| = 2 for n ≥ 1. Let a be minimal, and let 2 n−1 n−1 b = a = (a,a). Then each restriction b|Vn consists of two cycles of length 2 , and o(bn)=2 by the deﬁnition in Section 3.4. By Proposition 4.4 we have that B is isometrically isomorphic to the dyadic integers Z2. The dynamical system (∂T,b) is not conjugate to (Z2,c2), while each restriction (∂T0,b0) or (∂T1,b1) is conjugate to (Z2,c2).

From these results and Lemma 3.2 we obtain a characterization of the minimal elements of the group of automorphisms of p-ary rooted trees, when p is a prime number. The key step in this case is the characterization of the permutations acting transitively on Vn by means of their orders. PROPOSITION 4.7. Let p> 1 be a prime number, and suppose that T is the p-ary rooted tree, that is, for the spherical index mT = (m1,m2,...) we have mn = p, n ≥ 1. Then the following sentences are equivalent:

(1) a ∈ Aut(T ) is minimal. (2) a is conjugate to αΣ, the odometer of T . (3) an acts transitively on Vn, for every n ≥ 1. n (4) o(an)= p , for every n ≥ 1. (5) A is isometrically isomorphic to Zp. (6) The dynamical systems (∂T,a) and (Zp,cp) are conjugate by means of an isometry. REMARK 4.8. In the case p = 2 statement (4) of Proposition 4.7 can be replaced by a weaker statement that sgn(an) = −1 for n ≥ 1, where sgn(an) denotes the parity of the permutation an, see [29, Proposition 1.6.2]. SETTLED ELEMENTS IN PROFINITE GROUPS 13

5. The normalizer and the absolute Weyl group of a maximal torus

Let T be a rooted tree with spherical index mT = (m1,m2,...), see Section 3.2. For every n ≥ 1, we set pn = m1 ··· mn. Let a ∈ Aut(T ) be a minimal automorphism. Recall that we denote by A the cyclic group generated by a, and by A the closure of A in Aut(T ). By Lemma 4.2 A is a maximal torus in Aut(T ), and its normalizer is deﬁned by N(A)= {w ∈ Aut(T ): wAw−1 = A}. In this section, we study the structure of the normalizer N(A), and of the absolute Weyl group W (A)= N(A)/A, especially in the case when the spherical index mT is bounded. Recall that mT is bounded if there exists d ≥ 2 such that mn ≤ d for all n ≥ 1.

5.1. Elements of the Weyl group. For a ∈ Aut(T ) minimal with the corresponding maximal torus A, the absolute Weyl group is given by W (A)= N(A)/A. Deﬁne −1 (7) Gζ = {w ∈ Aut(T ): waw = ζ}, for a minimal ζ ∈ A.

Since A is abelian, such a set Gζ is a coset of A in Aut(T ). We prove that the sets Gζ are precisely the cosets of A in N(A). LEMMA 5.1. Let w ∈ Aut(T ) and let a ∈ Aut(T ) be a minimal element. Then w ∈ N(A) if and only if waw−1 ∈ A. Therefore:

(1) N(A)= · {Gζ : ζ ∈ A and ζ is minimal }, (2) The absolute Weyl group W (A) of the maximal torus A is the set of equivalence classes S {Gζ : ζ ∈ A and ζ is minimal }.

Proof. If w ∈ N(A) then waw−1 ∈ A by deﬁnition and by Corollary 4.3 waw−1 is minimal. Let ζ = waw−1 ∈ A. We show that wAw−1 = A. Note that wakw−1 = (waw−1)k = ζk, and so we have wAw−1 = hζi⊂ A, which implies wAw−1 ⊆ A. On the other hand, since ζ is minimal, then by Lemma 4.2 its closure hζi is a maximal torus, which implies that hζi = A. The rest of the lemma follows.

Next we observe that if ζ = a then Ga contains all elements in Aut(T ) that commute with a. Denote by Z(A) the centralizer of A in Aut(T ), that is, the set of all elements in Aut(T ) which commute with every element in A.

LEMMA 5.2. For a ∈ Aut(T ) minimal we have Z(A)= Ga = A.

Proof. Since a is a topological generator of A, we have that g ∈ Aut(T ) commutes with a if and only if g commutes with every element in A, so Z(A)= Ga.

Next, since A is abelian, we have A ⊆ Ga. Let x ∈ ∂T and σ ∈ Ga. Since A is the enveloping semigroup of A, and the action of A on ∂T is minimal, we have Ax = ∂T for the orbit of x in ∂T . Thus there exists γ ∈ A such that σx = γx. This implies that for every n ≥ 0 we have σanx = anσx = anγx = γanx, since σ commutes with a. This shows that σ and γ coincide on the a-orbit of x, and since this orbit is dense, we conclude γ = σ, and so σ ∈ A. REMARK 5.3. Lemma 5.2 implies that if a ∈ Aut(T ) is minimal, then we can deﬁne the Weyl group of the maximal torus A by W (A)= N(A)/Z(A). We note that a similar alternative deﬁnition can be made for the Weyl groups of maximal tori in compact Lie groups.

5.2. Rational elements of the Weyl group and the normalizer of a maximal torus. For a minimal a ∈ Aut(T ), we deﬁne the rational spanning set in N(A). The set is called rational since its elements can be indexed by two integers. We prove that if the spherical index mT of T is bounded, then the rational spanning set is dense in N(A). We give an example of a tree with unbounded spherical index, where the rational spanning set is degenerate and thus it is not dense in N(A). 14 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

5.2.1. Rational cosets in the Weyl group. We consider the set of elements which are coprime to every pn, n ≥ 1, that is,

(8) I(T )= {k ≥ 1 : l.c.d{k,pn} = 1 for every n ≥ 1}. k For a with k ∈ I(T ), we write Gk instead of Gak , where Gak is deﬁned by (7). We have the following. LEMMA 5.4. Let k ≥ 1. Then ak is minimal if and only if k ∈ I(T ).

Proof. Let n ≥ 1. Suppose that l.c.d{k,pn} = 1. By B´ezout’s identity there exist m,ℓ ∈ Z such that mk =1+ ℓpn. Thus for every 1 ≤ s ≤ pn and every v ∈ Vn we have

k ms s sℓpn s (a ) (v)= a a (v)= a (v) ∈ Vn, k s that is, there exists a power of a which maps v to a (v) for any 1 ≤ s ≤ pn. Since a is transitive k k on Vn, then a is also transitive on Vn. Since this is true for every n ≥ 1, we conclude that a is minimal.

Suppose that l.c.d{k,pn} = ℓ> 1. Then there exist 1 ≤ s < k and 1 ≤ r

5.2.2. Rational subsets in cosets Gk. Recall from Section 3.2 that the boundary ∂T of a tree T with spherical index mT = (m1,m2,...) is identified with the set Σ = n≥1 Σn, where |Σn| = mn for n ≥ 1. Thus each vertex in T corresponds to a finite word, and each infinite path in ∂T corresponds Q to an infinite sequence of digits in Σn, n ≥ 1. Let ξ = (ξn)n≥1 ∈ ∂T be the sequence such that ξn = 0 for every n ≥ 1. We show that within cosets Gk, for k ∈ I(T ), the elements are determined by their action on ξ. As a consequence, they are indexed by elements in ∂T .

As earlier in this section, a is a minimal element in Aut(T ), and Gk for k ∈ I(T ) are deﬁned by (7), for ζ = ak.

LEMMA 5.6. An element σ ∈ Gk is completely determined by its value at ξ.

k m km Proof. Let σ ∈ Gk. The relation σa = a σ implies σa = a σ, for every m ≥ 0. Thus σamξ = amkσξ, which implies that σξ determines completely the values of σ on the set S = {amξ : m ≥ 0}. Since a is minimal, then S is dense in ∂T , which implies that σ is completely determined on ∂T .

LEMMA 5.7. For any t ∈ ∂T there exists σ ∈ Gk such that σξ = t.

Proof. Let τ ∈ Gk. Then by Lemma 5.2 we have Gk = τA. This implies that

Gkξ = τAξ = τ∂T = ∂T.

Thus, there exists σ ∈ Gk such that σξ = t.

The two lemmas above motivate the following deﬁnition.

DEFINITION 5.8. Let k ≥ 1 be an integer in I(T ). For every integer m ≥ 1 we deﬁne σm,k as the element of Gk given by m σm,kξ = a ξ.

Indeed, an element satisfying Deﬁnition 5.8 exists by Lemma 5.7 and it is unique by Lemma 5.6. SETTLED ELEMENTS IN PROFINITE GROUPS 15

DEFINITION 5.9. A rational spanning set in N(A) is the set {σm,k | k ∈ I(T ),m ≥ 1}, where σm,k are given by Deﬁnition 5.8.

m REMARK 5.10. Observe that for k = 1 we have G1 = A, and σm,1 = a for every m ≥ 1, so {σm,1 | m ≥ 1} coincides with the semi-group of positive powers of a.

5.2.3. Dynamics of elements in rational cosets. In Lemma 5.11 below, we show that a sequence of elements in a rational coset Gk, k ∈ I(T ), of the Weyl group of A, which converges at a single point ξ ∈ ∂T , converges at every point in ∂T , thus giving an element in Aut(T ). Thus for elements in Gk convergence at a point implies convergence in Aut(T ).

LEMMA 5.11. Let Gk ∈ W (A) be a coset in the Weyl group of the maximal torus A topologically generated by a minimal element a ∈ Aut(T ), for k ∈ I(T ). Let (σi)i≥0 be a sequence in Gk such that limi→∞ σiξ = σξ, where σ ∈ Gk. Then (σi)i≥0 converges to σ in Aut(T ).

k m mk Proof. The relation σia = a σi implies σia = a σi for every m ≥ 0. Thus m mk σia ξ = a σiξ, which implies m mk m lim σia ξ = a σξ = σa ξ. i→∞ m Thus (σi)i≥0 converges pointwise to σ on the set {a ξ : m ≥ 0}. The pointwise convergence on ∂T follows from the density of {amξ : m ≥ 0} in ∂T . Indeed, ﬁrst observe that for every m ≥ 0 we have m m km km (9) d(σia ξ,σa ξ)= d(a σiξ,a σξ)= d(σiξ,σξ).

mi mi Let t ∈ ∂T and let (a ξ)i≥0 be a sequence converging to t. We write a ξ = ti, so by assumption ti →i t. Then we have

d(σit,σt) ≤ d(σit, σiti)+ d(σiti,σti)+ d(σti,σt)

= d(t,ti)+ d(σiξ,σξ)+ d(σti,σt), where the second line is a consequence of (9) and the fact that σi is an isometry. This implies that limi→∞ σit = σt. Since the topology of pointwise convergence in Aut(T ) coincides with uniform topology and proﬁnite topology, we conclude that (σi)i≥0 converges to σ in Aut(T ).

In the following lemma we show that the set of rational elements in Gk is dense in Gk.

LEMMA 5.12. Let k ≥ 1 be an integer in I(T ). The set {σm,k : m ≥ 1} is dense in Gk.

Proof. For k = 1 it is clear from Remark 5.10.

ni Let k > 1 and let σ ∈ Gk. By minimality of a, there exists a sequence {a ξ : ni+1 >ni > 0} that converges to σξ. This means that there is a unique sequence {σni,k}i≥1 given by Deﬁnition 5.8, so that we have

lim σni,kξ = σξ. i→∞ Then the conclusion follows from Lemma 5.11.

5.2.4. Rational spanning set in N(A). In this section we assume that the spherical index mT of the tree T is bounded. That is, there exists d > 0 such that |mn| ≤ d for every n ≥ 1. Note that this is equivalent to saying that there exist integers d1, ··· , dM ≥ 2 such that k = mn for some n ≥ 1 if and only if k ∈{d1, ··· , dM }. Observe that in this case

I(T )= {p ≥ 1 : l.c.d{p, d1, ··· , dM } =1}.

The case when M = 1 corresponds to the case when T is the d-ary tree, with d = d1 and mn = d for every n ≥ 1. 16 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

THEOREM 5.13. Let T be a tree with bounded spherical index mT and let a ∈ Aut(T ) be minimal. Then the rational spanning set

{σm,k : m ≥ 1} k≥1 l.c.d{k,d1[,··· ,dM }=1 given by Deﬁnition 5.9 is dense in N(A).

Proof. We ﬁrst show that the set of rational cosets in the Weyl group

· Gk k≥1 l.c.d{k,d1[,··· ,dM }=1 is dense in N(A).

Let ζ be a minimal automorphism in A, and let σ ∈ Gζ . Since ζ is in A there exists a sequence ki (a )i≥0 that converges to ζ, with ki ∈ Z for i ≥ 0. In particular this means that, given i, for every kj suﬃciently large j we have ζ|Vi = a |Vi . Assume that i is large enough so that |Vi| is divisible by all kj the integers in {d1, ··· , dM }. Since ζ is transitive at each level, we have that every a is transitive on Vi, which implies that l.c.d{kj , |Vi|} = 1 and then l.c.d{kj, d1, ··· , dM } = 1. Thus we can assume ki l.c.d{ki, d1, ··· , dM } = 1 for every i ≥ 0, which by Lemma 5.4 implies that a is minimal.

For every i ≥ 0, let σi ∈ Gki . Then we have

−1 ki −1 lim σiaσi = lim a = ζ = σaσ . i→∞ i→∞ −1 Consider the sequence (σ σi)i≥1. Since Aut(T ) is compact, this sequence has a convergent subse- −1 quence (σ σis )s≥1 to a limit point λ, that veriﬁes

−1 −1 −1 −1 ki −1 −1 λaλ = lim σ σi aσ σ = lim σ a s σ = σ (σaσ )σ = a. s→∞ s is s→∞ −1 Since λ commutes with a, by Lemma 5.2 we get λ ∈ A. Then σiλ ∈ Gki , for every i ≥ 0. −1 On the other hand, since (σij λ ) converges to σ, we get that σ is in the closure of · k≥1 Gk. k∈I(T ) Then the statement of the theorem follows by Lemma 5.12. S

REMARK 5.14. Theorem 5.13 is need not be true if mn’s are not bounded. For example, suppose that mn = n + 1 for each n ≥ 1. We get that pn = (n + 1)! for every n ≥ 1, and then I(T )= {1}. This implies that · Gk = G1 = A. k∈[I(T ) −1 −1 On the other hand, since a is minimal (a is transitive on each Vn), Corollary 4.3 implies there exists w ∈ Aut(T ) such that waw−1 = a−1. From Lemma 5.1 we get that w ∈ N(A) and, since w does not commute with a, by Lemma 5.2 we deduce that w∈ / A. This shows that G1 = A is not dense in N(A) because w ∈ N(A) − A can not be approximated by elements in A = G1.

6. Stable cycles and settled elements in Aut(T )

In this section we give a topological characterization of settled elements in Aut(T ), for T a d-ary rooted tree. However, most of the deﬁnitions and results are valid for more general trees.

6.1. Minimal components of (∂T,σ). Let σ ∈ Aut(T ), and recall that (∂T,σ) denotes the dynamical system given by the action of the cyclic group hσi on ∂T . We denote by σj [v] the image of the cylinder set [v] under the power σj . Recall from Section 3.2 that we say that u is a word of length k if u is a concatenation of k symbols in {0,...,d − 1}. We denote by {0,...,d − 1}k the set of all words of length k. If v is a vertex in Vn and u is a word of length k, then vu is a vertex in Vn+k. For every n ≥ 1 and a vertex v ∈ Vn we denote by [v]= {x ∈ ∂T : x contains v}. SETTLED ELEMENTS IN PROFINITE GROUPS 17 the cylinder set of paths passing through v. k Recall from Deﬁnition 1.2 that we say that v ∈ Vn is in a cycle of length k ≥ 0 of σ if σ (v)= v and σj (v) 6= v for every 1 n and w ∈{0, ··· , d − 1} , the vertex vw ∈ Vm is in a cycle of length dm−nk. Observe this is equivalent to the following: for every w,u ∈{0, ··· , d − 1}m−n and 0 ≤ i,j

LEMMA 6.1. Let σ ∈ Aut(T ) and n ≥ 1. The vertex v ∈ Vn is in a stable cycle of length k ≥ 0 if k−1 j and only if the union of the cylinder sets X = j=0 σ [v] is a minimal component of (∂T,σ). S Proof. Let k ≥ 0 be such that v ∈ Vn is in a cycle of length k. If the cycle is not stable then there exist m>n, words u, w ∈{0, ··· , d − 1}m−n and integers 1 ≤ i,j

As a consequence of Lemma 6.1 we can give another characterization of settled elements, using the uniform (Bernoulli) measure µ on ∂T , deﬁned for every n ≥ 1 and v ∈ Vn by 1 µ([v]) = . |Vn|

LEMMA 6.3. An automorphism σ ∈ Aut(T ) is settled if and only there exists a family {Ci}i∈I of σ-minimal clopen sets such that

µ ∂T − Ci =0. i I ! [∈

Proof. Let {Ci}i∈I be the collection of minimal components which are clopen (I could be empty). For every n ≥ 1 let Jn be the subset of vertices of Vn which are in a stable cycle (Jn could be empty). We have Ci = [v]. i I n v J [∈ [≥1 [∈ n [v] ⊆ [w], we get Since v∈Jn w∈Jn+1

S S |Jn| µ Ci = lim , n→∞ |Vn| i I ! [∈ which is equal to 1 if and only if σ is settled.

Since σ is an isometry, ∂T is equal to the disjoint union of the minimal components of (∂T,σ). Thus Lemma 6.3 is equivalent to the following. PROPOSITION 6.4. An element σ ∈ Aut(T ) is settled if and only for µ-almost all x in ∂T the closure of its σ-orbit is clopen in ∂T . 18 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

EXAMPLE 6.5. We give an example of a settled element with a fixed point. Let T be a binary tree, and let η be a non-trivial permutation of V1. Recursively define an odometer a = (a, 1)η and an element b = (a,b), see Section 3.3.1 for notation. Then the path in T , consisting of the rightmost vertices at each level is a fixed point. This construction can be modified to obtain an element with a countable number of fixed points by taking c = (b,c) with b as above.

Recall from Deﬁnition 1.3 that a settled element σ ∈ Aut(T ) is strongly settled if there exists n ≥ 1 such that every vertex in Vn is in a stable cycle. As a direct consequence of Lemma 6.1 and the compactness of ∂T we obtain the characterization of the closures of orbits for strongly settled elements. PROPOSITION 6.6. Let σ ∈ Aut(T ). The following are equivalent:

(1) σ is strongly settled. (2) For every x ∈ ∂T the closure of the σ-orbit of x is clopen. (3) σ has ﬁnitely many minimal components.

6.2. Settled elements in Aut(T ). We show next that settled elements are abundant in Aut(T ), that is, they form a dense subset of Aut(T ). We assume that T is a d-ary tree. The proof is constructive and easily generalizes to the case when T has arbitrary spherical index. LEMMA 6.7. For every τ ∈ Aut(T ) and every n ≥ 1, there exists a strongly settled element σ of 1 Aut(T ) such that D(τ, σ) ≤ 2n . In other words, the set of (strongly) settled elements of Aut(T ) is dense in Aut(T ).

Proof. Let τ ∈ Aut(T ) and τn be the restriction of τ to Vn. Let n ≥ 1 and let C1, ··· , Ck be the cycles of τn in Vn. We choose σ ∈ Aut(T ) as follows: the restriction σn of σ to Vn is equal to τn.

For every m>n, the restriction σm of σ to Vm must verify the following: let Dm,1, ··· ,Dm,km be the cycles of σm−1 in Vm−1. For 1 ≤ i ≤ km, let v1, ··· , vℓ be the vertices in Dm,i ordered such that σm−1(vj )= vj+1 and σm−1(vℓ)= v1, for every 1 ≤ j<ℓ. We take σm to be the permutation on Vm whose restriction to {vs : v ∈ Dm,i, 0 ≤ s < d}, for any 1 ≤ i ≤ km, is given by

σm(vj s)= vj+1s for every 1 ≤ j<ℓ, σm(vℓs)= v1(s + 1), for every 0 ≤ s < d − 1 and σm(vj (d − 1)) = vj+1(d − 1) for every 1 ≤ j<ℓ, σm(vℓ(d − 1)) = v10.

Observe that the restriction of σm to {vs : v ∈ Dm,i, 0 ≤ s < d} is transitive, for every 1 ≤ i ≤ km. Since this is true for every m>n, we get that every vertex in Vn lies in a stable cycle of σ, which shows that σ is strongly settled. 1 On the other hand the construction of σ ensures that d(σx, τx) ≤ 2n , for every x ∈ ∂T , which means 1 that σ is at distance at most 2n from τ. REMARK 6.8. We say that x is σ-aperiodic, if the orbit of x under the cyclic group generated by σ is not periodic. We observe that if σ is settled then µ-almost every x ∈ ∂T is σ-aperiodic. If σ is strongly settled then every x ∈ ∂T is σ-aperiodic.

We note that it is possible to construct examples of elements in Aut(T ) which are not settled and completely aperiodic. EXAMPLE 6.9. Let T be the binary tree, and let h ∈ Aut(T ) be such that the length of every cycle doubles at levels Vn for n odd, and stays the same for n even. Such an element h has no stable cycles, which implies that h is not settled. On the other hand, since the length of cycles in Vn is increasing with n, the h-orbit of every point in ∂T is aperiodic (inﬁnite). We claim that the closure of each such orbit is not open, since it does not contain any cylinder sets. Indeed, let x = (x1, x2,...) ∈ ∂T , then x ∈ [xn] for n ≥ 1. Every cylinder set [xn] is a union of the sets [xn0] and [xn1]. Suppose that n is odd and for deﬁnitiveness, assume that xn+1 = xn0. Then the cycle in Vn+1 which contains xn+1 has the same length as the cycle in Vn containing xn. It follows that SETTLED ELEMENTS IN PROFINITE GROUPS 19 the h-orbit of x never visits the cylinder set [xn1], and so its closure intersects but does not contain the cylinder set [xn]. Since n is arbitrary and [xn] for n even contains cylinder sets [xm], m>n for m odd, this argument shows that the closure of any h-orbit in ∂T is not open.

REMARK 6.10. The proof of Lemma 6.7 implicitly uses the well-known fact that Aut(T ) is ∞ isomorphic to the wreath product [Sd] of an infinite number of copies of the symmetric group on d symbols. To recall the definition of the wreath product, let G and H be finite groups acting on the sets X and Y respectively, and let f : X → H be a function. An element (g,f) ∈ G ⋉ HX of the wreath product acts on the product X × Y in such a way that the action on X is that of g, and the action on copies of {x}× Y is determined by the values of the function f at point in X. In particular, f(x) can be any permutation in H, and for different x1, x2 ∈ X the values of f at these points are independent.

Let T be a d-ary tree with Σn = Σ1, and let H be a finite group acting transitively on Σ1 = V1. Suppose there is g ∈ H such that the action of the cyclic group hgi is transitive on Σ1. Then ∞ the infinite wreath product [H] acts minimally on ∂T = n≥1 Σ, and the proof of Lemma 6.7 generalizes straightforwardly to show that [H]∞ is densely settled. It follows that if the image of Q an arboreal representation ρf,α : Gal(K/K) → Aut(T ), for a polynomial f(x) of degree d and some α ∈ K, has finite index in such an infinite wreath product [H]∞, then it is densely settled. Thus the conjecture of Boston and Jones is true for such representations. In the current literature on arboreal representations, there are many examples of arboreal representations whose image has finite index in the wreath product of groups, as described above. Examples of arboreal representations which have finite index in Aut(T ) can be found, for instance, in the survey [20] and other publications. If the image of an arboreal representation has infinite index in Aut(T ), then it is more difficult to determine if it is densely settled. We now show how to find examples of arboreal representations which have infinite index in Aut(T ) and which are conjugate to the infinite wreath product [H]∞ with H containing a transitive permutation. The images of such arboreal representations are densely settled by the argument above. If a profinite subgroup of Aut(T ) is topologically finitely generated, then it has infinite index in Aut(T ), see for instance [20]. A criterion for when that happens for infinite wreath products may be found in [4]. For example, this criterion is satisfied when the group H has no non-trivial normal subgroup, which is true, for instance, if H = Ad for d ≥ 5, where Ad denotes the alternating group on d symbols. The group Ad contains a transitive cyclic subgroup, and so a transitive permutation, when d > 4 and d is odd, see [23, Table 1]. Paper [9] specifies a class of maps, called the normalized (dynamical) Belyi polynomials, for which the profinite geometric iterated monodromy group ∞ is isomorphic to the infinite wreath product [Ad] where d is the degree of the map. Thus by the argument above, if d> 4 and d is odd, then the profinite geometric iterated monodromy group of a normalized Belyi polynomial is densely settled. In [22, Section 3], the authors give conditions under which the arithmetic iterated monodromy group associated to a rational function r(x) of degree d ≥ 2 is isomorphic to the infinite wreath product [H]∞ for H finite (although H is not specified in these results). The study of permutation groups with a transitive cyclic subgroup is in itself a topic of research in group theory, and a list of examples and the description of the properties of these groups can be found, for instance in [23], see also the references therein. If the wreath product of groups in [23] can be obtained as the Galois groups of field extensions in [22], then we will have another class of arboreal representations, for which the conjecture of Boston and Jones is true.

6.3. Stability of the settled property. In general the property of settledness of elements is not very well-behaved, since the product of two settled elements need not be settled, while the product of two elements which are not settled can be settled, see Example 6.12. However, as we show in this section, the powers of a settled element are settled, and the settledness is preserved under conjugacy. Recall that for h ∈ Aut(T ) we denote by H the cyclic subgroup of Aut(T ) generated by h. For a point x ∈ ∂T the h-orbit of x is the orbit of x under H. 20 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

LEMMA 6.11. An automorphism h ∈ Aut(T ) is (strongly) settled if and only if hk is (strongly) settled, for every k ∈ Z −{0}.

Proof. Suppose that h is (strongly) settled. Let x ∈ ∂T be and let ℓ ∈ Z −{0}. Observe that

ℓ−1 Hx = hhℓihrx, r=0 [ which implies, since the union is ﬁnite, that

ℓ−1 Hx = hhℓihrx. r=0 [ Recall from Section 3.1 that, for g ∈ Aut(T ), a minimal component of the action of hgi is a closed subset Y of ∂T such that Y is invariant under the action, and such that the g-orbit of every y ∈ Y is dense in Y . In a degenerate case, Y may be a ﬁxed point of the action. In our setting, since hℓ is an isometry, for every y ∈ hhℓihrx the points in the hℓ-orbit of y stay at the same distance from the points in the hℓ-orbit hrx, and so if the hℓ-orbit of hrx accumulates at a point z ∈ ∂T , then the hℓ-orbit of y accumulates at z as well. Thus the set hhℓihrx is a minimal component of hℓ, for every 0 ≤ r<ℓ. Then there exists a subset I ⊆{0, ··· ,ℓ − 1} such that

Hx = · hhℓihrx. r I [∈ Note that this union is a disjoint union of sets. Now suppose that the set Hx is clopen. Then every hℓ-minimal component contained in the ﬁnite disjoint union Hx is clopen. This implies that if x ∈ ∂T is in a clopen h-minimal component then x is in a clopen hℓ-minimal component. Thus if h is (strongly) settled then hℓ is (strongly) settled. Let k ∈ Z −{0}. Suppose that hk is (strongly) settled. Since for every x ∈ ∂T the hk-orbit of x is included in the h-orbit of x, then we get that the hk-minimal component containing x is included in the h-minimal component containing x. Thus if the minimal component of x with respect to hk is clopen, the minimal component of x with respect to h is also clopen. Then by Lemma 6.3 we conclude that h is settled.

EXAMPLE 6.12. From Lemma 6.11 we have that if h is (strongly) settled, then h−1 is also (strongly) settled. Since h−1h = id is not settled, we conclude that the product of two (strongly) settled elements is not necessarily (strongly) settled.

On the other hand, the product of two non-settled elements can be (strongly) settled. Let u1 = η be the non-trivial permutation of two elements, and deﬁne u2 = (u1,u2). Then both u1 and u2 have order 2, so they are not settled. However, the product u1u2 = η(u1,u2) is minimal, so it is strongly settled. We discuss the action of the group generated by u1 and u2 in detail in Section 8.1.

The property of being (strongly) settled is preserved under conjugacy in Aut(T ) as shown in the next result. This result is a direct consequence from the fact that the minimal components of a dynamical systems are preserved under conjugacy. PROPOSITION 6.13. Let w,h ∈ Aut(T ). Then h is (strongly) settled if and only if whw−1 is (strongly) settled.

Proof. Let x ∈ ∂T . Since hwhw−1i = wHw−1, we have hwhw−1iwx = wHx. In other words, the whw−1-orbit of wx is equal to the image by w of the h-orbit of x. This implies that the closure of the whw−1-orbit of wx is equal to the image by w of the closure of the h-orbit of x. Thus if the closure of the h-orbit of x is clopen then the closure of the whw−1-orbit of wx is also clopen. Then w{x ∈ ∂T : Hx is clopen }⊆{x ∈ ∂T : hwhw−1ix is clopen}. SETTLED ELEMENTS IN PROFINITE GROUPS 21

Since the uniform measure µ is invariant under the action of Aut(T ) we conclude that if h is (strongly) settled then whw−1 is (strongly) settled. From this we also get that h is (strongly) settled if and only if whw−1 is strongly settled. COROLLARY 6.14. Let Γ ⊆ Aut(T ) and w ∈ Aut(T ). Then Γ is densely settled if and only if wΓw−1 is densely settled.

Recall that we denote by H the closure in Aut(T ) of the cyclic group H generated by h ∈ Aut(T ). PROPOSITION 6.15. The following sentences are equivalent:

(1) h is (strongly) settled. (2) H contains a (strongly) settled element. (3) The set of (strongly) settled elements of H is dense in H.

Proof. It is obvious that (3) implies (2). The implication (1) to (3) is consequence of Lemma 6.11. It is not diﬃcult to see that if ζ ∈ H then the minimal components of ζ are contained in the minimal components of h. Thus, if the h-minimal component of x is not clopen, then neither is the ζ-minimal component of x. From this we deduce that if h is not (strongly) settled then H has no (strongly) settled elements. This shows (2) that implies (1).

REMARK 6.16. Let H1, H2 be profinite groups acting effectively on ∂T , that is, we have that the homomorphisms Hi → Homeo(∂T ) are injective for i =1, 2. Then we can identify H1, H2 with subgroups of Aut(T ). Next, suppose the actions are conjugate by an isometry as dynamical systems, that is, there exists a topological isomorphism Φ : H1 → H2 and an isometry φ : ∂T → ∂T , such that φ(hx)=Φ(h)(φ(x)) for all h ∈ H1 and all x ∈ ∂T . We note that this is equivalent to the subgroups H1, H2 ⊂ Aut(T ) being conjugate in Aut(T ). Indeed, since φ is an isometry, it must restrict to a permutation on each level Vn, and so it must map cylinder sets in ∂T to cylinder sets in ∂T . It follows that φ preserves the structure of the tree T and defines an automorphism wφ ∈ Aut(T ). It follows that Φ : H1 → H2 is given by the conjugation by wφ. We note that the settled property is not preserved under the topological isomorphism of profinite groups. To see an example, suppose T is the binary tree, a is a minimal element in Aut(T ), and h is the non-settled aperiodic automorphism in Aut(T ) introduced in Example 6.9. The order of k the restriction of h to Vn is equal to 2 if n = 2k or n = 2k − 1, for each n ≥ 1. Thus, according to Proposition 4.4, the group H = hhi is isomorphic to the group given by the inverse limit of the following sequence: Z/2Z ← Z/2Z ← Z/22Z ← Z/22Z ← Z/23Z ← Z/23Z ← · · · .

This group is topologically isomorphic to Z2, which is by Proposition 4.4 isomorphic to A = hai. This shows that A and H are topologically isomorphic. However, their actions on ∂T cannot be conjugate since the action of A is minimal, while the action of H is not. By Proposition 6.15, H is not densely settled whereas A is densely settled. In the rest of the article, in all cases when we say that two groups are (topologically) isomorphic they are (topologically) isomorphic and their actions are conjugate by an isometry, as in the ﬁrst paragraph of this remark. The term ‘isomorphism’ in this context is often used to describe the subgroups of Aut(T ) algebraically, for instance, as wreath products or other known algebraic groups, and by construction the isometry in question is often the identity map in Aut(T ).

7. Maximal tori with densely settled normalizers

In this section we assume that d ≥ 2 is a prime number, and T is the d-ary tree, see Section 3.2 for the necessary background on trees. In particular, the spherical index of T is mT = (d, d, . . .), n and for each vertex level set its cardinality is given by the number pn = |Vn| = d . Then the set of numbers coprime with the entries of the spherical index of T deﬁned in (8) becomes I(T )= {k ≥ 1 : k is not divisible by d}. 22 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

Recall that for a ∈ Aut(T ) we denote by A the cyclic subgroup of Aut(T ) generated by a, and we denote by A the closure of A in Aut(T ). If a is minimal, then by Lemma 4.2 A is a maximal torus in Aut(T ). In this section we study the properties of the normalizer of such a maximal torus, and give a dynamical proof of Theorem 1.5. We restate the theorem now for the convenience of the reader. THEOREM 7.1. Let d > 1 be a prime number and let T be the d-ary tree. For every minimal automorphism a ∈ Aut(T ) the normalizer N(A) of the maximal torus A is densely settled.

The proof proceeds by carefully examining elements in the rational spanning set of N(A). Recall that the sets Gk, for k ∈ I(T ), defined in Section 5, are the cosets in N(A) given by −1 k Gk = {w ∈ N(A): waw = a }, k ∈ I(T ). A rational spanning set of N(A) defined in Definition 5.9, consists of the union of countable subsets in Gk, k ∈ I(T ). If the spherical index of T is bounded (for d-ary trees it is constant), then the rational spanning set is dense in N(A). In Remark 5.14 we give an example of a tree with unbounded spherical index, for which the rational spanning set is degenerate, that is, it is not dense in N(A). Our method to prove Theorem 1.5 does not extend to trees with degenerate rational spanning sets. Our method also does not extend to the case when the spherical index of T is bounded but not constant, since in that case we do not have an analog of Lemma 7.1.

7.1. A few technical preliminaries. Let a ∈ Aut(T ) be minimal, then A acts freely and transitively on ∂T , and so there is a bijection between A and ∂T which depends on a choice of a point in ∂T . We make this identiﬁcation explicit.

As it was described in Section 3.2, for every n ≥ 1 the vertices in Vn are labelled by the words in n n j=1{0, ··· , d − 1}. Since a is transitive on Vn, the map φn : Vn → Z/d Z given by Q aj(0n)= j + dnZ, for every j ≥ 0, is a well-deﬁned bijection. The action of a on Z/dnZ reads as a(j + dnZ)= j +1+ dnZ, or, which is equivalent, a(j)= j + 1 mod dn, for every 0 ≤ j < dn.

n Let k ∈ I(T ) and σ ∈ Gk. For n ≥ 1, and v ∈ Vn (or for v ∈ Z/d Z, which is equivalent) the relation σa = akσ together with the identiﬁcation described above implies that (10) σav(0) = σ(v)= σ(0) + kv mod dn.

Applying Equation (10) p> 1 times we get

p−1 p kpv + σ(0) k −1 mod dn if k> 1 (11) σp(v)= kpv + σ(0) ki = k−1 i=0 ( v + σ(0)p if k =1 X A proof of the next lemma can be found at [26]. LEMMA 7.1. Let k> 1 and n ≥ 1 be integers. Consider the following geometric series n−1 kn − 1 r(n) := ki = . k − 1 i=0 X Then for any n ≥ 1 the following holds.

(1) If d is an odd prime and k = sd +1, for some s ≥ 0, then the highest power of d dividing r(n) is equal to the highest power of d dividing n. (2) if d = 2 and k = s4+1, for some s ≥ 0, then the highest power of 2 dividing r(n) is equal to the highest power of 2 dividing n. SETTLED ELEMENTS IN PROFINITE GROUPS 23

We say that an integer k > 1 satisﬁes the conditions of Lemma 7.1 if k = 4s + 1 whenever d = 2 and k = ds + 1 when d is an odd prime, for some s ∈ Z.

7.2. Cycle structure of elements in N(A). Recall that for integers m ≥ 0 and k ≥ 1, the element m σm,k ∈ Gk of the rational spanning set is the unique automorphism in Gk verifying σm,kξ = a ξ, n where ξ is an inﬁnite sequence of 0’s. In particular, using (10) we get in Z/d Z that σm,k(0) = m. We now determine what points in ∂T are in stable cycles of σm,k, depending on k and m. LEMMA 7.2. Let k> 1 be an integer that satisﬁes the conditions of Lemma 7.1. Let m ≥ 1 be an integer which is not divisible by d. Then σm,k is minimal and so strongly settled.

Proof. Let n ≥ 1. Since m is not divisible by d, according to (11) and Lemma 7.1, for 0 ∈ Vn we have that if p> 0 is the smallest positive integer such that kp − 1 σp (0) = m = 0 mod dn then p = dn. m,k k − 1 n n This implies that the orbit of 0 in Vn by σm,k is in a cycle of length d . Since |Vn| = d , σm,k is transitive in Vn. Since this is true for any n ≥ 1 we conclude that σm,k is minimal.

m REMARK 7.3. Recall that σm,1ξ = a ξ by deﬁnition. Thus by Lemma 5.4 we have σm,1 is minimal if and only if m is not divisible by d. LEMMA 7.4. Let k > 1 be an integer that satisﬁes the conditions of Lemma 7.1. Let m ≥ 1 be an integer which is divisible by d. Let j ≥ 1 be the biggest integer such that dj divides m. Then for n−j every n > j the vertex 0 is in a stable cycle of σm,k of length d .

p n Proof. Let p ≥ 1 be such that σm,k(0) = 0 mod d , that is, 0 is in a cycle of length p in Vn. Then n kp−1 (11) implies that d divides mr(p) = m k−1 . By the choice of j and Lemma 7.1, we get that the smallest p such that dn−j divides r(p) is dn−j , and dn−j is the highest power of d which divides n−j n−j r(d ). This implies that 0 is in a cycle of length d of σ in Vn, which lifts to an open path in Vn+1. Since n > j is arbitrary, we deduce that 0 is in a stable cycle of σm,k in Vn, for every n > j.

In Lemma 7.4 for a given σm,k with m ≥ 1 we studied the stability of the cycle containing 0 for the n restriction of σm,k to Vn =∼ Z/d Z. Lemma 7.5 below investigates a similar question for an arbitrary vertex in Vn. LEMMA 7.5. Let k> 1 be an integer that satisﬁes the conditions of Lemma 7.1. Let m ≥ 1 be an integer which is divisible by d and let j ≥ 1 be the biggest integer such that dj divides m. Then the automorphism σm,k is settled, and for every n ≥ 1 the lengths of cycles of the restriction σm,k|Vn are powers of d.

Proof. For simplicity we will write σ instead of σm,k. j Let n > j and v ∈ Vn. Let q ≥ 1 be a number not divisible by d such that m = d q. By Lemma 7.4 we can assume that v 6= 0. Let p ≥ 1 be such that σp(v)= v mod dn. Then dn divides kp − 1 (12) σp(v) − v = (m + (k − 1)v) = (m + (k − 1)v)r(p), k − 1 and, since k satisﬁes Lemma 7.1, dn−1 divides (k − 1) dj−1q + v r(p) = (dj−1q + sv)r(p). d Let i ≥ 0 be the biggest integer such that di divides sv. In the arguments below, we use the fact that sv/di is not divisible by d, and for any ℓ

1 j−1 n−(i+1) Case 1: If i < j − 1 then di (d q + sv) is not divisible by d, which implies that d must divide r(p). From Lemma 7.1 the smallest p for which that happens is p = dn−(i+1). This implies n−(i+1) that v is in a cycle of length d of σ in Vn. Since this is true for any n > j, we conclude that v is in a stable cycle of Vn. 1 j−1 n−j Case 2: If i > j − 1 then dj−1 (d q + sv) is not divisible by d, which implies that d must divide n−j n−j r(p). From Lemma 7.1 we get p = d , which implies that v is in a cycle of length d of σ in Vn. Since this is true for any n > j, we conclude that v is in a stable cycle of Vn. Case 3: If i = j − 1 then sv = dj−1r, for some positive integer r not divisible by d. Let t ≥ 0 be the biggest integer such that dt divides the number (q + r). If t j, we conclude that v is in a stable cycle of Vn. Now let t ≥ n − j, and write q + r = dtr′ for an integer r′ > 0 non divisible by d. Then (dj−1q + sv)r(p)= dj−1(q + r)r(p)= dj+t−1r′r(p). Since dj+t−1 ≥ dj+n−j−1 = dn−1, dn−1 divides this expression for any p, in particular for p = 1. This implies that v is a ﬁxed point of σ in Vn.

We now estimate the number of fixed points of σ in Vn. For that let ℓ ≥ 1 be an integer such that dj−1q + sv = dn−1ℓ. Solving for the second term, since v < dn we obtain the estimate (k − 1) (k − 1) 0 < −dj−1q + ℓdn−1 = sv = v < dn, d d which implies that q l< (k − 1) + . dn−j q Since q is fixed we can assume that n is sufficiently large so that dn−j < 1 and then ℓ ∈{1,...,k−1}. This implies that in Vn for n large there are at most k − 1 vertices which are not in a stable cycle.

Since there are at most k − 1 vertices in Vn which are not in a stable cycle of σ, σ cannot have more than k − 1 ﬁxed points. We conclude that σ is settled.

Thus Lemmas 7.2, 7.4 and 7.5 yield the following statement. PROPOSITION 7.6. Let k > 1 be an integer that satisﬁes the conditions of Lemma 7.1. For every m ≥ 1 the automorphism σm,k is settled, and for every n ≥ 1 the lengths of cycles of the restriction σm,k|Vn are powers of d.

7.3. Proof of the densely settled property of N(A). Proposition 7.6 shows that if k = sd + 1, for some s ≥ 0, then σm,k is settled. We now show that σm,k is settled for any k ≥ 1 such that l.c.d(d, k) = 1. Since by Theorem 5.13 the rational spanning set {σk,m | m ≥ 1, l.c.d.(d, k)=1} is dense in N(A) this ﬁnishes the proof of Theorem 7.1. PROPOSITION 7.7. For every integer k ≥ 1 such that k is not divisible by d, and every m ≥ 1, the automorphism σm,k is settled.

m Proof. If k = 1, then σm,1 = a by Remark 5.10. Then by Lemma 6.11 σm,1 is strongly settled.

For simplicity, we will write σ instead of σm,k. Let k > 1 be an integer which is not divisible by d. We set n = 4 if d = 2 and n = d if d is an odd prime. Let p = ϕ(n), where ϕ is the Euler function. Then kp = ns +1 for some s ≥ 1. Observe that σa = akσ implies p σpa = ak σp. p r rk This implies that σ is in Gkp . In addition, using that σa = a σ for every r ≥ 1, we get p−1 σpξ = σp−1(σξ)= σp−1amξ = σp−2amkσξ = σp−2am(k+1)ξ = am(1+k+···+k )ξ, SETTLED ELEMENTS IN PROFINITE GROUPS 25

p p from which we get that σ = σm(1+k+···+kp−1),kp . From Proposition 7.6 it follows that σ is settled. Then by Lemma 6.11, we conclude that σ is settled.

7.4. Application: minimal automorphisms of N(A). In this section we give a characterization of the minimal elements in N(A). LEMMA 7.8. Let k ≥ 1 be an integer which is not divisible by d. Let m ≥ 1.

(1) If d =2 then the following sentences are equivalent: (a) σm,k is transitive on V2. (b) m is odd and k =4s +1, for some s ≥ 0. (c) σm,k is minimal. (2) If d is an odd prime then the following sentences are equivalent: (a) σm,k is transitive on V1. (b) m is not divisible by d and k = ds +1, for some s ≥ 0. (c) σm,k is minimal.

Thus σm,k is minimal if and only if m is not divisible by d and k satisﬁes the conditions of Lemma 7.1.

Proof. Both in (1) and (2), Lemma 7.2 implies the equivalence between (c) and (b). The implication (c) to (a) follows from Lemma 4.1. So it is enough to show the implication (a) to (b), in both cases.

For the case d = 2, observe that if σm,k is transitive on V2, then it is transitive on V1. For any prime d, observe that if m ≥ 1 is divisible by d, then formula (11) yields σm,k(0) = m = 0 mod d. This implies that σm,k has a ﬁxed point at 0 in V1. So if σm,k is transitive on V1, then m cannot be divisible by d.

Case d =2. Suppose that k =4s +3, for some s ≥ 0. Then by (11) we get (4s + 3)2 − 1 (4s +3 − 1)(4s +3+1) σ2 (0) = m = m =4m(s + 1), m,k 4s +3 − 1 4s +3 − 1 2 2 2 which implies that σm,k(0)=0 mod 2 . Thus σm,k has a ﬁxed point at 0 in V2, and since |V2| = 4, then σm,k is not transitive on V2. Thus if σm,k is transitive on V2 then k =4s + 1 and m is odd.

Case d an odd prime. Suppose that k = sd + r for some 2 ≤ r < d and s ≥ 0. Then d−1 d−2 (13) σm,k (0) = m(1 + (sd + r)+ ··· + (sd + r) ). Observe that for every 1 ≤ ℓ ≤ d − 2 ℓ ℓ ℓ ℓ (sd + r)ℓ = (sd)irℓ−i = rℓ + (sd)irℓ−i = rℓ + dN , i i ℓ i=0 i=1 X X d−1 where Nℓ is an integer. Recall that for r 6= 0 mod d we have r = 1 mod d (see for example [?]). Substituting the formula above and this into (13) we get rd−1 − 1 q σd−1(0) = m(1 + r + ··· rd−2 + dN)= m + dN = m d + dN , m,k r − 1 r − 1 d−2 d−1 d−1 where N = i=1 Ni and q is the integer such that r − 1= dq. Since σm,k (0) is an integer, q is divisible by r − 1 and P d−1 σm,k (0) = 0 mod d. d−1 Thus σm,k has a ﬁxed point at 0 in V1, and so σm,k cannot be transitive on V1. Thus if σm,k is transitive on V1 then m is not divisible by d and k = ds +1, for some s ≥ 0.

For d ≥ 2 prime consider the set Md of all minimal elements in the rational spanning set, namely, by Lemma 7.8

Md = {σm,k : m> 0 is not divisible by d and k = 1 mod d} for d ≥ 3. 26 MAR´IA ISABEL CORTEZ AND OLGA LUKINA and M2 = {σm,k : m> 0 is odd and k =1 mod4}.

The next proposition characterizes the minimal elements of N(A) as those in the closure of Md. PROPOSITION 7.9. Let σ ∈ N(A). The following sentences are equivalent:

(1) σ is minimal. (2) There exists w ∈ Aut(T ) such that σ = waw−1. (3) σ ∈ Md.

Proof. The equivalence between (1) and (2) was shown in Corollary 4.3.

To show the implication (3) to (2), suppose that σ ∈ Md. Then there exists a sequence (σmi,ki )i≥0 in Md that converges to σ. Since every σmi,ki is minimal, there exists wi ∈ Aut(T ) such that −1 σmi,ki = wiawi . By compactness of Aut(T ) there exists a subsequence (wij )j≥0 of (wi)i≥0 that converges to some w ∈ Aut(T ). This implies that (σ ) converges to waw−1, which shows mij ,kij j≥0 that σ = waw−1. We conclude that σ is minimal. Suppose that σ ∈ N(A) is minimal. Since the rational spanning set is dense in N(A) there exists a sequence (σmi,ki )i≥0 converging to σ. We need to show that this sequence can be chosen to be in

Md. Observe that there exists i0 ≥ 0 such that for every i ≥ i0 the restrictions σ|V2 and σmi,ki |V2 coincide. Since σ is transitive on V2 (because it is minimal), σmi,ki is transitive in V2 (and then in

V1) too. Then Lemma 7.8 implies that for every i ≥ i0 the automorphism σmi,ki is in Md, which shows that σ ∈ Md.

8. Iterated monodromy groups associated to PCF polynomials

Let f(x) be a quadratic polynomial with strictly pre-periodic post-critical orbit of length r ≥ 2 with periodic cycle of length r − s, for 1 ≤ s < r. Consider the associated proﬁnite arithmetic and geometric iterated monodromy groups Garith(f) and Ggeom(f), as deﬁned in Section 2.

In this section we study the densely settled property for Garith(f) and Ggeom(f) by considering the maximal tori in these groups, and the Weyl groups of these maximal tori. We recall necessary concepts and notation ﬁrst.

Let T be a binary tree, and recall that for f(x) as above by [29, Proposition 1.7.15] Ggeom(f) is conjugate in Aut(T ) to the closure of the group generated by the elements

(14) u1 = η,ui = (ui−1, 1) if 2 ≤ i ≤ r and i 6= s +1,us+1 = (us,ur). The recursive notation used in formula (14) was explained in Section 3.3.1. We denote the countable group generated by (14) by G, and its closure in Aut(T ) by G. By Corollary 6.14 Ggeom(f) is densely settled if and only if G is densely settled. As earlier in the paper, given an element a ∈ Aut(T ) we denote by A the cyclic group generated by a, and by A the closure of A in Aut(T ). We denote by N(A) the normalizer of A in Aut(T ).

We claim that G contains maximal tori. Indeed, the product a0 = u1u2 ··· ur is minimal by [29, Proposition 3.1.3], and so the closure A0 is a maximal torus in G. Now let a ∈ N(G) be any minimal element. By Proposition 4.4 the maximal torus A is isometrically isomorphic to the dyadic integers Z2, and by Lemma 5.1 its absolute Weyl group satisﬁes −1 W (A)= {Gζ | ζ ∈ A is minimal }, where Gζ = {w ∈ N(A) | waw = ζ}.

ki i Every ζ ∈ A can be written down as a sequence ζ = (a )i≥1, where ki is odd, and ki+1 mod 2 = ki. × It is also known that the normalizer N(A) can be identiﬁed with the semi-direct product Z2 ⋉ Z2 × (see [29, Proposition 1.6.3] or [30, Theorem 4.4.7]), where Z2 is the multiplicative group in Z2, and ∼ × × then W (A) = Z2 . An element in Z2 is represented by a sequence (ki)i≥1, where each ki is coprime × to 2, and so ki is odd. Thus every equivalence class Gζ is represented by a sequence (ki)i≥1 ∈ Z2 . SETTLED ELEMENTS IN PROFINITE GROUPS 27

Our approach to investigating the densely settled property of the profinite arithmetic and geometric iterated monodromy groups is based on the study of the Weyl groups of maximal tori in G. Recall from Definition 1.6 that the Weyl group of a maximal torus A in G is the quotient W (A, G) = (N(A) ∩ G)/A. Similarly, one can define the Weyl group of A in the normalizer N(G) by W (A, N(G)) = (N(A) ∩ N(G))/A. By Theorem 1.5 N(A) is densely settled, and by [29, Proposition 3.9.5] we have that N(A) ⊂ N(G), but apriori we do not know whether N(A) is contained in G. If a ∈ N(G)−G is minimal, apriori we do not know if its normalizer N(A) is contained in N(G). Thus the indices of the Weyl groups W (A, G) and W (A,N(G)) in the absolute Weyl group W (A) of the maximal torus A contain information about settled elements in G and N(G). Below, we prove Theorems 1.8, 1.9, 1.10 and 1.11. We consider four cases for different values of r and s, since the groups G and the normalizer N(G) have different descriptions in these cases.

8.1. The case s =1 and r =2. We prove Theorem 1.8, which says that if r = 2 and s = 1, then Ggeom(f) is not densely settled, while the normalizer N(Ggeom(f)) is densely settled. It will follow that Garith(f) is densely settled.

For r = 2 and s = 1, the group G = hu1,u2i = hη, (u1,u2)i is the inﬁnite dihedral group, and a = u1u2 is an element of inﬁnite order, see for instance [25] for a detailed proof. Moreover, a is transitive on each Vn, n ≥ 1, and so it is minimal.

Set b = u1. Since u1 and u2 have order 2, then −1 −1 bab = u1u1u2u1 = (u1u2) = a , and one obtains another standard presentation of the dihedral group, namely (15) G = ha,b | bab = a−1,b2 =1i, were 1 denotes the identity element in G. The proof of the ﬁrst part of Proposition 8.1 below is based on the fact that the only settled elements in G are powers of a, and all other elements are periodic of period 2. PROPOSITION 8.1. The proﬁnite group G is not densely settled, while the normalizer N = N(G) is densely settled.

Proof. Every element in G can be presented as a word bmak, for some k ∈ Z and m ∈ {0, 1}. If m = 1, then (bak)2 = (bak)(bak)= b2a−kak =1, so each such element has order 2. Therefore, the only settled elements in G are the powers of a. By Corollary 4.2 the maximal torus A, topologically generated by a, is isomorphic to the group of dyadic integers Z2. Since every element in Z2 is of infinite order, G is strictly larger than A. Let g ∈ G − A, and consider a sequence (bakn ) ⊂ G converging to g. Then for every n ≥ 1 there exists ki in ≥ 1 such that g|Vn = ba |Vn, for all i ≥ in. It follows that for any point in ∂T , the projection of its orbit under the action of the cyclic group generated by g onto Vn has length 2, and so g is periodic with period 2. Thus any element in G − A is periodic of period 2. Since A is closed, every sequence of elements in G converging to g ∈ G − A is eventually in G − A, which contains only periodic elements which are not settled. Thus G is not densely settled. On the other hand, if w ∈ N(G) then waw−1 is minimal (Corollary 4.3), which implies that waw−1 has infinite order and then waw−1 ∈ A. It follows from Lemma 5.1 that w ∈ N(A), which implies that N(G) ⊆ N(A). Then N(A)= N(G), and Theorem 7.1 implies that N(G) is densely settled. 28 MARÍA ISABEL CORTEZ AND OLGA LUKINA

Proof. (of Theorem 1.8) Let K be a number ﬁeld, and let f(x) be a quadratic polynomial with post-critical orbit of length r = 2, which is strictly pre-periodic with s = 1. Then the proﬁnite geometric iterated monodromy group Ggeom(f) is conjugate to the group G in Proposition 8.1 and so it is not densely settled. The normalizer N(Ggeom(f)) is conjugate to N(G) and so it is densely settled. Moreover, for any a ∈ Ggeom(f) minimal we have for the Weyl groups

W (A)= N(Ggeom(f)) = W (A,N(Ggeom(f))), and W (A, Ggeom(f)) = Ggeom(f)/A = {−1, 1}.

By [29, Corollary 3.10.6(g)] Garith(f)/Ggeom(f) has ﬁnite index in × ∼ G G Z2 \{−1, 1} = N( geom(f))/ geom(f).

Thus W (A, Garith(f)) has ﬁnite index in W (A). In particular, Garith(f) is a ﬁnite index closed subgroup of N(Ggeom(f)). It follows that Garith(f) is also an open subgroup, and so it is densely settled since N(Ggeom(f)) is densely settled.

8.2. Normalizer of G for r ≥ 3. We recall from [29] the description of the normalizer of G in Aut(T ) for diﬀerent values of r ≥ 3 and s ≥ 1. × Denote by Cn the additive cyclic group on n symbols, and by Cn the multiplicative group of Cn. × ∼ × Recall that Z2 = C4 × Z2, with equivalence classes [−1] and [5] generating the respective factors × [30, Theorem 4.4.7]. Composing the projection onto the ﬁrst factor with the isomorphism from C4 to C2, we obtain, as in [29, Section 3.9], the homomorphism k − 1 (16) θ : Z× → C : k 7→ mod 2. 1 2 2 2 2 × × Next, notice that the projection of the set of squares {k | k ∈ Z2 } to C8 is trivial, and the × projection of the same set onto C16 is equal to the order 2 subgroup {1, 9}. Composing squaring × with the projection onto C16 and the isomorphism of the image subgroup to C2, one obtains the homomorphism k2 − 1 (17) θ : Z× → C : k 7→ mod 2. 2 2 2 8 LEMMA 8.2. We have the following:

(1) k = (ki) ∈ ker(θ1) if and only if ki =4ℓi +1 for all ℓi ≥ 0, i ≥ 2. (2) k = (ki) ∈ ker(θ2) if and only if either ki =8ℓi +1 or ki =8ℓi +7, for all ℓi ≥ 0, i ≥ 4.

We now consider diﬀerent cases of r ≥ 3 and s ≥ 1. We recursively deﬁne the following elements: • (a) Case r ≥ 4 and s ≥ 2: Set

(18) w1 = (us,us), wi+1 = (wi, wi) for all i ≥ 1.

Since us has order 2, every wi has order 2. Also note that wi|Vk = id for 1 ≤ k k − s then the restriction wi|Vk is trivial. • (b) Case r ≥ 3 and s =1: Set 2 (19) w1 = (1, (usur) ), wi+1 = (wi, wi) for all i ≥ 1. 2 Since by [29, Proposition 3.1.9] usur has order 4 then (usur) has order 2, and so every wi has order 2. Also note that wi|Vk = id for 1 ≤ k k − 1, then the restriction wi|Vk is trivial. • (c) Case r =3 and s =2: Set 2 (20) w0 = u3(w0, w0), w1 = (1, (usur) ), wi+1 = (wi, wi) for all i ≥ 1. 2 Since by [29, Proposition 3.1.9] usur has order 4, then (usur) has order 2, and so for i ≥ 1 wi has order 2. Also note that wi|Vk = id for 1 ≤ k

In the cases (a) and (b) deﬁne the map

t1 t2 (21) ϕ : C2 → N(G) : (t1,t2,...) 7→ w1 w2 ··· , i Y≥1 which is continuous but not a homomorphism. In the case (c) set for i ≥ 0

t 1, if t =0, wi = wi if t =1, and then deﬁne the map (which we denote by the same symbol as the map in (21))

t0 t1 t2 (22) ϕ : C2 → N(G) : (t0,t1,t2,...) 7→ w0 w1 w2 ··· , i Y≥0 which is continuous but not a homomorphism. In the cases (b) and (c) the image of ϕ is in fact contained in a speciﬁc subgroup of N(G), but this is not important for our arguments. THEOREM 8.1. (1) [29, Lemma 3.6.9, Theorem 3.6.12]: In the case (a), where r ≥ 4 and s ≥ 2, we have [wi, wj ] ∈ G for all i, j ≥ 1, and so there is an induced isomorphism ∞ ϕ : i=1 C2 → N(G)/G. (2) [29, Lemma 3.7.14, Theorem 3.7.18]: In the case (b), where r ≥ 3 and s = 1, we have Q ∞ [wi, wj ] ∈ G for all i, j ≥ 1, and so there is an induced isomorphism ϕ : i=1 C2 → N(G)/G. (3) [29, Lemma 3.8.23, Theorem 3.8.27]: In the case (c), where r ≥ 3 and s = 1, we have 2 Q ∞ [wi, wj ], w0 ∈ G for all i, j ≥ 0, and there is an induced isomorphism ϕ : i=0 C2 → N(G)/G. Q

t1 t2 Thus every coset in N/G is represented by w1 w2 ··· where ti ∈ {0, 1} for i ≥ 1 in the cases (a) t0 t1 t2 and (b). In the case (c) every coset N/G is represented by w0 w1 w2 ··· where ti ∈{0, 1} for i ≥ 0.

8.3. Weyl groups of some maximal tori. In this section we study the Weyl groups of maximal tori contained in G. We recall again our notation. For a ∈ Aut(T ) we denote by A the cyclic group generated by a, and by A its closure in Aut(T ). If a ∈ Aut(T ) is minimal, then A is a maximal torus by Lemma 4.2. We denote by N(A) the normalizer of A in Aut(T ). We recall that the absolute Weyl group of A is the group W (A)= N(A)/A, and the Weyl group of A in G is the group W (A, G) = (N(A) ∩ G)/A. ∼ × ∼ × Choose a minimal element a ∈ G and ﬁx an isomorphism N(A) = Z2 ⋉ Z2, then W (A) = Z2 . • (a) Case r ≥ 4 and s ≥ 2. Composing the induced map ϕ in Theorem 8.1 with the map

× (θ1,θ1,...): Z2 → C2, i Y≥1 where θ1 is deﬁned by (16), by [29, Proposition 3.9.14] we have a map

× θ1(k) θ1(k) (23) ϕ ◦ (θ1,θ1,...): Z2 → N/G : k 7→ ϕ(k)G = (w1 w2 ··· )G. ∼ × By [29, Proposition 3.9.14] the image of the absolute Weyl group W (A) = Z2 coincides with the × image of (23), where the coset Gζ ∈ W (A) defined by (7) is identified with k = (ki) ∈ Z2 such that ki ζ = (c ). Thus the image of W (A) in i≥1 C2 consists of two points, the infinite sequence of 0’s and the infinite sequence of 1’s, and the normalizer N(A) is contained in the union of two cosets of Q G in N(G). This gives the following. LEMMA 8.3. If r ≥ 4 and s ≥ 2, and let a ∈ G be a minimal element. Then we have:

(1) N(A) ∩ G = {σm,4ℓ+1 : m ≥ 1,ℓ ≥ 0}. (2) W (A, G) has index 2 in the absolute Weyl group W (A). (3) If b ∈ N(A) is minimal, then b ∈ N(A) ∩ G. 30 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

Proof. Since N(A) and G are proﬁnite groups, they are closed. Then their intersection is closed, and the ﬁrst statement follows from Lemma 8.2(1) and the fact that the rational spanning set {σm,k | m ≥ 1, k ≥ 0} is dense in N(A). The second statement follows from the fact that the image of (24) consists of two points. The last statement follows by Lemma 7.8(1) and Proposition 7.9.

• (b) Case r ≥ 3 and s = 1. Composing the induced map ϕ in Theorem 8.1 with the map × (θ2,θ2,...): Z2 → C2, i Y≥1 where θ2 is deﬁned by (17), by [29, Proposition 3.9.14] we obtain the map

× θ2(k) θ2(k) (24) ϕ ◦ (θ2,θ2,...): Z2 → N/G : k 7→ ϕ(k)G = (w1 w2 ··· )G. × Again, using the identification of W (A) with Z2 , we obtain that the image of W (A) in i≥1 C2 consists of two points, the infinite sequence of 0’s, and the infinite sequence of 1’s, and N(A) is Q contained in the union of two cosets of G in N(G). More precisely, we have the following. LEMMA 8.4. If r ≥ 3 and s =1, and let c ∈ G be a minimal element. Then we have:

(1) N(A) ∩ G = {σm,8ℓ+1, σm,8ℓ+7 : m ≥ 1,ℓ ≥ 0}. (2) W (A, G) has index 2 in the absolute Weyl group W (A). (3) Both cosets of N(A) ∩ G in N(A) contain minimal elements of N(A).

Proof. Since N(A) and G are proﬁnite groups, they are closed. Then their intersection is closed, and the ﬁrst statement follows from Lemma 8.2(2) and the fact that the rational spanning set {σm,k | m ≥ 1, k ≥ 0} is dense in N(A). The second statement follows from the fact that the image of (24) consists of two points. The last statement follows by Lemma 7.8(1) and Proposition 7.9.

• (c) Case r =3 and s = 2. Composing the induced map ϕ in Theorem 8.1 with the map × (θ1,θ2,θ2,...): Z2 → C2, i Y≥0 where θ1 is deﬁned by (16) and θ2 is deﬁned by (17), by [29, Proposition 3.9.14] we obtain the map

× θ1(k) θ2(k) θ2(k) (25) ϕ ◦ (θ1,θ2,θ2,...): Z2 → N/G : k 7→ ϕ(k)G = (w0 w1 w2 ··· )G. ∞ ∞ In this case the image of W (A) in i≥0 C2 consists of four points, the inﬁnite sequences 0 , 01 , 10∞ and 1∞, where x∞ denotes the concatenation of an inﬁnite number of symbols x. Then N(A) Q is contained in the union of four cosets of G in N(G). LEMMA 8.5. If r ≥ 3 and s =1, and let a ∈ G be a minimal element. Then we have:

(1) N(A) ∩ G = {σm,8ℓ+1 : m ≥ 1,ℓ ≥ 0}. (2) W (A, G) is a normal index 4 subgroup of the absolute Weyl group W (A). (3) Cosets of N(A) ∩ G in N(A) represented by ϕ(0∞) and ϕ(01∞) contain minimal elements of N(A).

Proof. The kernel (25) is the intersection of normal subgroups ker θ1 ∩ ker θ2, and so it is a subgroup of index 4 in W (A) which is normal since W (A) is abelian. Then arguments similar to the ones in Lemma 8.4 ﬁnish the proof.

Lemmas 8.3, 8.4 and 8.5 show whether, for a given a ∈ G, the maximal tori contained in the normalizer N(A) are also entirely contained in G. In particular, Lemma 8.3 states that for r ≥ 4 and s ≥ 2 all maximal tori contained in N(A) are entirely contained in G. Lemmas 8.4 and 8.5 state that when r ≥ 3 and s = 1, or when r = 3 and r = 2, then some minimal elements in N(A) are not in G. We will see in Section 8.4 that such elements topologically generate maximal tori, which are not contained in G but nevertheless have a non-empty intersection with G. The normalizers of such maximal tori are studied in Section 8.5. SETTLED ELEMENTS IN PROFINITE GROUPS 31

8.4. Maximal tori in the normalizer N(G). As earlier in the paper, for a ∈ Aut(T ) we denote by A the cyclic group generated by a, and by A its closure in Aut(T ). If a ∈ Aut(T ) is minimal, then A is a maximal torus by Lemma 4.2.

THEOREM 8.6. Let G be topologically generated by the elements u1,...,ur given by (14) and let r ≥ 3. Then for every coset in N(G)/G there exists a representative w such that the following holds:

(1) For every minimal a ∈ A, the element aw = aw ∈ wG is minimal. (2) For the maximal torus Aw topologically generated by aw, the intersection Aw ∩ G is an index 2 subgroup of Aw, and Aw ⊂ G ∪ wG.

Thus we conclude the following:

(3) The normalizer N(G) contains an uncountable number of distinct maximal tori. (4) Minimal elements in G and in each coset wG ∈ N(G)/G are in bijective correspondence.

Proof. Following [29, Section 1.5], for n ≥ 1 deﬁne a homomorphism

sgnn : Aut(T ) → {±1} which assigns to each g ∈ Aut(T ) the parity of the permutation g|Vn. Recall that by [29, Proposition 1.6.2] a is minimal if and only if sgnn(a) = −1 for all n ≥ 1. Note that if g = (h1,h2), then sgnn(g) = sgnn(h1)sgnn(h2) since h1 and h2 induce permutations of disjoint sets of vertices on Vn and sgnn is a homomorphism. We start by proving statement (1).

• (a) Case s ≥ 2 and r ≥ 4. Let t ∈ i≥1 C2, and let w = ϕ(t) ∈ N(G), where ϕ is given by (21). By (18) we have Q t1 t2 t1 t2 t1 t2 w = w1 w2 ··· = (us w1 ··· ,us w1 ··· ), and in particular

2 t1 t2 sgnn(w|Vn) = sgnn(us w1 ··· )=1.

Let a ∈ G be minimal, then for aw = aw we have for n ≥ 1

sgnn(aw|Vn) = sgnn(a|Vn)sgnn(w|Vn) = sgnn(a|Vn)= −1, and so aw ∈ wG is a minimal element. Throughout the paper, we use the same notation for the left and the right cosets of G in N(G), since they are equal. • (b) Case s =1 and r ≥ 3. First note that for n ≥ 1 we have

2 2 sgnn(w1) = sgnn((1, (usur) )) = (sgnn(usur)) =1.

For i ≥ 2 and n ≥ 1 we have for wi deﬁned by (19) 2 sgnn(wi+1) = sgnn((wi, wi)) = (sgnn(wi)) =1,

t1 t2 so clearly for any ﬁnite product sgnn(wi1 ··· wik )=1 for n ≥ 1. Next, let ϕ(t)= w = w1 w2 ··· be an inﬁnite product. Since for i>n − 1 the restriction wi|Vn is a trivial permutation, then for n ≥ 1

t1 tn−1 w|Vn = w1 ··· wn−1 |Vn, and sgnnw = 1 for all n ≥ 1. Now let a be a minimal element in G. Then

sgnn(aw) = sgnn(a)sgnn(w) = sgnn(a)= −1 for n ≥ 1 and so aw is minimal. • (c) Case s = 2 and r = 3. The choice of a suitable w is more complicated in this case. Since wi for i ≥ 1 are deﬁned by the same formulas as in Case (b), by the same argument as in (b) one t1 t2 obtains that sgnn(wi) = 1 for all i ≥ 1 and n ≥ 1, and so sgnn(w1 w2 ··· ) = 1 for any n ≥ 1 and any combination of ti, i ≥ 1. However, it is easy to see that sgnn(w0)= −1 at least for some n ≥ 0. 32 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

If for t ∈ i≥0 C2 we have t0 = 0, then we set w = ϕ(t) with ϕ deﬁned by (22) and the argument showing that aw is minimal if a is minimal proceeds as in Case (b). So assume that t0 = 1 and so Qt1 t2 ϕ(t)= w0w1 w2 ··· . Then for n ≥ 1 we have t1 t2 sgnn(ϕ(t)) = sgnn(w0)sgnn(w1 w2 ··· ) = sgnn(w0) = sgnn(u3(w0, w0)) = sgnn(u3).

Now set w = u3ϕ(t). Since w and ϕ(t) diﬀer by u3 ∈ G, they represent the same coset. For n ≥ 1 2 sgnn(w) = sgnn(u3)sgnn(ϕ(t)) = (sgnn(u3)) =1. Now let a be a minimal element in G, then aw ∈ ϕ(t)G, and aw is minimal since for n ≥ 1

sgnn(aw) = sgnn(a)sgnn(w) = sgnn(a)= −1. This ﬁnishes the proof of statement (1). We now prove statement (2). In our arguments we repeatedly use the fact that the left and the right cosets of G in N(G) are equal, so given g ∈ G and w ∈ N(G), there exists g′ ∈ G such that gw = wg′. The proof proceeds similarly for the cases (a) and (b). The case (c) has some diﬀerences and we prove it separately. In the proofs, w is a representative of a coset in N(G) chosen as in the proof of statement (1) above.

Cases (a) and (b). Consider the cyclic subgroup Aw generated by aw. Let m ≥ 1 be an integer. Note that for i>n − s the restriction wi|Vn is a trivial permutation (recall that s = 1 in the case (b)). Then m t1 tn−s m t1 tn−s m (aw) |Vn = (aw1 ··· wn−s ) |Vn = (w1 ··· wn−s gn) |Vn −1 −1 for some gn ∈ G. Since for any wi, wj the commutator [wi, wj ]= wiwj wi wj ∈ G by [29, Lemma 3.6.9, Lemma 3.7.14], and the left and the right cosets of G in N(G) are equal, then

We obtain that ′ 2m ′ (aw) = lim gn, n→∞ ′ and since G is a closed subgroup of Aut(T ), then (aw)2m ∈ G. On the other hand, ′ 2m −1 t1 tn−s ′ (aw) = lim w ··· wn s gn, n→∞ 1 − where the sequence on the right converges to a point in the coset wG. Here we use that the cosets of G are closed since G is closed and group multiplication in Aut(T ) is a homeomorphism.

We have shown that Aw ⊂ G ∪ wG, and the intersection Aw ∩ G is an index two subgroup of Aw, which implies statement (2). Next note that the maximal tori intersecting distinct cosets in N(G)/G are distinct. Since i≥1 C2 is uncountable and this space is in bijection with N(G)/G, there is an uncountable number of distinct maximal tori in N(G), and statement (3) follows as well. Finally Q note that aw = a′w for a,a′ ∈ G implies that a = a′, which proves statement (4). ∞ Case (c). Let t ∈ i=0 C2. If t0 = 0, then the proof proceeds precisely as in the case (b), so suppose that t = 0. Then ϕ(t)= w wt1 wt2 ··· . Let w = u ϕ(t), and consider the cyclic subgroup 0 Q 0 1 2 3 generated by cw. Since for i ≥ 1 wi|Vn is trivial if i>n − 2, and u3 ∈ G, we compute that

m t1 tn−2 m m mt1 mtn−2 ′ (cw) |Vn = (cu3w0w1 ··· wn−2 ) |Vn = w0 w1 ··· wn−2 gn|Vn ′ 2 for some gn ∈ G. Since w0 ∈ G, we use the equality of left and right cosets of G to obtain

m t0 mt1 mtn−2 ′′ (cw) |Vn = w0 w1 ··· wn−2 gn|Vn ′′ ′ for t0 = m mod 2 and some gn ∈ G. Then for m ≥ 1 ′ ′ 2m ′′ 2m −1 t1 tn−2 ′′ t1 tn−2 (cw) |Vn = gn ∈ G, (cw) |Vn = w0w1 ··· wn−s gn ∈ w0w1 ··· wn−s G. SETTLED ELEMENTS IN PROFINITE GROUPS 33

The rest of the proof proceeds as in cases (a) and (b).

Proof. (of Theorem 1.9) Let f(x) a quadratic polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3. By [29, Proposition 1.7.15] the proﬁnite geometric iterated monodromy group Ggeom(f) associated to f(x) is conjugate to the group generated by (14), and it follows that its normalizer is conjugate to N(G). Moreover, the conjugation maps the cosets of N(G) to the cosets of Ggeom(f) in its normalizer. Then the proof follows from Theorem 8.6.

8.5. Weyl groups of maximal tori in N(G). Theorem 8.6 shows that the normalizer N(G) contains more maximal tori than given by Lemmas 8.3, 8.4 and 8.5, especially in the case r ≥ 4 and s ≥ 2. If a maximal torus A is contained in G, we know that its normalizer N(A) is contained in N(G). In this section we formulate a criterion which ensures that the normalizer of a maximal torus, topologically generated by a minimal element and not entirely contained in G, is in N(G). For wG ∈ N(G)/G denote by Min(wG) the set of all minimal elements in wG.

PROPOSITION 8.7. Let G be topologically generated by the elements u1,...,ur given by (14), and let r ≥ 3. Then there is a well-deﬁned action (26) N(G) × Min(wG) → Min(wG) : (z,aw) 7→ z(aw)z−1.

−1 That is, if z ∈ N(G) satisﬁes za1z = a2 for a1,a2 ∈ Min(G), then we have in wG −1 (27) z(a1w)z = (a2 [z, w]) w ∈ Min(wG).

Proof. By [29, Theorem 3.9.4] all minimal elements in G are conjugate under N(G), so the action (26) is well-defined and transitive on Min(G). We now show that it is well-defined in other cosets in N(G). Note that minimal elements in G and wG are in bijective correspondence by Theorem 1.9, so for any g ∈ Min(wG) and any representative w satisfying Theorem 1.9 we can find a ∈ Min(G) such that g = aw. Since aw is minimal, its conjugate z(aw)z−1 is minimal, and we only must prove that this element is in the same coset as aw.

Recall that for the elements wi, deﬁned by (18), (19) or (20), their commutators are in G by [29]. We note that a similar statement holds for any two elements in N(G). Indeed, for any w, w given by (18), (19) or (20), consider the elements hw and gw with h,g ∈ G. Then, using the fact that the ′ right cosets are equal to the left cosets we have for some g ∈ G b [hw,gw] = (hw)(gw)(hw)−1(gbw)−1 = g′[w, w] ∈ G.

−1 Now suppose za1z = a2 for ab1,a2 ∈ Min(bG) and z ∈bN(G). Multiplyingb za1 = a2z on the left by w and using the commutator relation, we obtain

za1w = a2zw = (a2[z, w])wz. −1 Further multiplying on the right by z gives (27). Since a2[z, w] ∈ G, then (a2[z, w])w ∈ wG.

We now give a criterion for when the Weyl group of a maximal torus generated by an element in Min(N(G)) is contained in N(G).

PROPOSITION 8.8. Let G be topologically generated by the elements u1,...,ur given by (14), and let r ≥ 3. Let aw ∈ Min(wG) ⊂ N(G) and let Aw be the corresponding maximal torus. If the action N(G) × Min(wG) → Min(wG) deﬁned by (26) is transitive, then we have the equality of the Weyl groups W (Aw,N(G)) = W (Aw).

Proof. The inclusion W (Aw,N(G)) ⊂ W (Aw) holds by deﬁnition. We only have to show the reverse inclusion. 34 MAR´IA ISABEL CORTEZ AND OLGA LUKINA

−1 k × Let aw = aw ∈ Min(wG), and let g ∈ N(Aw). Then g(aw)g = ((aw) ) ∈ Aw for some k ∈ Z2 . By Theorem 8.6 we have ((aw)k)= mw for some m ∈ Min(G). If the action (26) is transitive, then there exists h ∈ N(G) such that h(aw)h−1 = mw = g(aw)g−1.

−1 −1 This implies that g h commutes with aw. Then by Lemma 5.2 g h ∈ Aw. Thus g is a product of elements in N(G) and must be in N(G). It follows that N(Aw)= N(Aw,N(G)).

We ﬁnally collect the results proved in this section together to obtain a proof of Theorem 1.10.

Proof. (of Theorem 1.10) Let f(x) be a quadratic polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3. By [29, Proposition 1.7.15] the proﬁnite iterated monodromy group Ggeom(f) associated to f(x) is conjugate to the group generated by (14), and it follows that the normalizer N(Ggeom(f)) of Ggeom(f) in Aut(T ) is conjugate to N(G). Then the proof follows from Propositions 8.7 and 8.8.

We note that Theorems 1.9 and 1.10 highlight the differences between the properties of maximal tori and Weyl groups in subgroups of Aut(T ) and in compact connected Lie groups. For instance, it is known that the Weyl group of a maximal torus in a compact Lie group is always finite, all maximal tori are conjugate in the group, and every element in the group is contained in a maximal torus. In the theorem below, we show that similar statements need not hold for subgroups of Aut(T ). THEOREM 8.9. Let f(x) a quadratic polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3. Let Ggeom(f) be the associated profinite geometric iterated monodromy group, and let N(Ggeom(f)) be its normalizer in Aut(T ). Then the following holds.

(1) For a maximal torus A ⊂ Ggeom(f), the Weyl group W (A, Ggeom(f)) is inﬁnite. (2) For N(Ggeom(f)) and Ggeom(f), at least one of the following alternatives is true: either there are maximal tori in Ggeom(f) which are not conjugate in N(Ggeom), or there are elements in Ggeom(f) which are not contained in a maximal torus. (3) There are maximal tori in N(Ggeom(f)) which are not conjugate in N(Ggeom(f)).

G × Proof. For a maximal torus A ⊂ geom(f), its absolute Weyl group W (A) is isomorphic to Z2 , which is an infinite group. By Lemmas 8.3, 8.4 and 8.5 for r ≥ 3 W (A, Ggeom(f) has finite index in W (A) and so it must be infinite. This shows (1).

To see (2), note that if all maximal tori in Ggeom(f) are conjugate, then they are necessarily generated by minimal elements. Since Ggeom(f) contains elements of order 2 (for instance, elements conjugate to the generators u1,...,ur in (14)), it then cannot be the union of maximal tori. On the other hand, if every element of Ggeom(f) is contained in a maximal torus, then there must be maximal tori which are not the closures of cyclic groups generated by minimal elements. Such tori cannot be conjugate to the torus A topologically generated by a minimal element a, since all elements in A are of inﬁnite order.

To see (3) note that Proposition 8.7 implies that a conjugate of any minimal a ∈ N(Ggeom(f)) by an element of N(Ggeom(f)) is contained in the same coset of Ggeom(f) in N(Ggeom(f)) as a. By Theorem 8.6 there is an uncountable number of cosets in N(Ggeom(f))/Ggeom(f) and each contains minimal elements. Minimal elements in diﬀerent cosets are conjugate in Aut(T ) but the conjugating element cannot be in N(Ggeom(f)).

8.6. The profinite arithmetic iterated monodromy group. We now discuss the consequences of Theorem 1.10 for the question whether the profinite arithmetic iterated monodromy group Garith(f) associated to the quadratic polynomial f(x) with strictly pre-periodic post-critical orbit is densely settled. We first recall some information about Garith(f) from [29, Section 3.10]. SETTLED ELEMENTS IN PROFINITE GROUPS 35

Denote by K a separable closure of K, and by G(K/K) the absolute Galois group of K. By [29, Section 1.7] there is a surjective homomorphism

ρ : G(K/K) → Garith(f)/Ggeom(f) ⊂ N(Garith(f))/Ggeom(f).

G Denote by hbi1 ,...,bir i the generators of geom(f), such that bik is conjugate to uk, for uk given by

(14). Then b∞ = bi1 bi2 ··· bir is conjugate to a = u1 ··· ur and so it is minimal. Then by [29, Lemma 3.10.4] for any τ ∈ G(K/K) we have ρ(τ) = wGgeom(f) for some w ∈ N(Ggeom(f)) satisfying −1 c(τ) G × G wb∞w = b , where c : (K/K) → Z2 is the cyclotomic character. It follows that arith(f) is contained precisely in the image of the map (23) if r ≥ 4 and s ≥ 2, the map (24) if r ≥ 3 and s = 1, and the map (25) if r = 3 and s = 2. Summarizing this, the results of [29, Corollary 3.10.6], Theorem 1.9 and 1.10, we obtain the statement of Theorem 1.11. Recall that every element in a maximal torus is strongly settled, and settled elements are dense in the normalizer of a maximal torus. The following is a restatement of Theorem 1.11 which follows directly from the arguments above. THEOREM 8.10. Let f(x) be a quadratic polynomial with strictly pre-periodic post-critical orbit of length r ≥ 3 over a number ﬁeld K, and let Ggeom(f) and Garith(f) denote the proﬁnite geometric and arithmetic iterated monodromy groups associated to f(x). Then the following holds.

(1) For any w ∈ Garith(f) the set Min(Ggeom(f)) and the set Min(wGgeom(f)) of minimal elements are in bijective correspondence, and so wGgeom(f) contains strongly settled elements. (2) For any a ∈ Min(Ggeom(f)), the index of the Weyl group W (A, Ggeom(f)) in the absolute Weyl group W (A) is ﬁnite. In particular, Ggeom(f) contains settled elements which are not minimal. (3) For any a ∈ Min(Ggeom(f)) and the corresponding maximal torus A, the index of the Weyl group W (A, Ggeom(f)) in W (A, Garith(f)) is equal to the index of Ggeom(f) in Garith(f). In particular, for any w ∈ Garith(f) − Ggeom(f) the coset wGgeom(f) contains settled elements which are not minimal.

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Email address: [email protected]

Facultad de Matematicas,´ Pontificia Universidad Catolica´ de Chile, Edificio Rolando Chuaqui, Vicuna˜ Mackenna 4860, Stgo-Chile

Email address: [email protected]

Faculty of Mathematics, University of Vienna, Oskar-Morgensternplatz 1, 1090 Vienna, Austria