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Settled Elements in Profinite Groups 3 SETTLED ELEMENTS IN PROFINITE GROUPS MAR´IA ISABEL CORTEZ AND OLGA LUKINA Abstract. An automorphism of a rooted spherically homogeneous tree is settled if it satisfies certain conditions on the growth of cycles at finite levels of the tree. In this paper, we consider a conjecture by Boston and Jones that the image of an arboreal representation of the absolute Galois group of a number field in the automorphism group of a tree has a dense subset of settled elements. Inspired by analogous notions in theory of compact Lie groups, we introduce the concepts of a maximal torus and a Weyl group for actions of profinite groups on rooted trees, and we show that the Weyl group contains important information about settled elements. We study maximal tori and their Weyl groups in the images of arboreal representations associated to quadratic polynomials over algebraic number fields, and in branch groups. 1. Introduction In this article we consider actions of profinite groups on rooted trees, and introduce the notion of a maximal torus for such an action in Definition 1.1. Our main results concern the structure of the maximal tori for certain arboreal actions and the properties of the normalizer groups of the maximal tori. To this end we introduce and study the properties of the Weyl group of a maximal torus for such an action. The motivation for this paper is the conjecture by Boston and Jones [5, 6, 7] for arboreal repre- sentations of Galois groups of number fields, which we discuss in detail later. Such representations are profinite groups of automorphisms of spherically homogeneous rooted trees, and the conjecture concerns with the properties of Frobenius elements under such representations, as well as with fac- torizations of polynomials over number fields. For some representations, the conjecture was studied by Boston and Jones [5, 6, 7] and by Goksel [13, 14, 12] using factorizations of polynomials modulo primes. In this paper, we approach the problem from a completely different perspective, using the methods of topological dynamics and geometric group theory, and investigating the properties of elements in profinite groups directly. Our main tools are maximal tori and their Weyl groups, which we claim contain the information about the group which pertains to the conjecture. Our approach allows us to consider a much wider class of arboreal representations than in [5, 6, 7, 13, 14, 12], including those whose images contain branch groups as subgroups. arXiv:2106.00631v1 [math.DS] 1 Jun 2021 We now recall a few definitions in order to state rigorously the problem and our results. We start by discussing maximal tori, Weyl groups, the notion of a settled element, and the interrelations between these concepts. We then recall the conjecture of Boston and Jones for arboreal representations, and use maximal tori and Weyl groups to study arboreal representations. We refer to Section 3.2 for a definition of a rooted spherically homogeneous tree T , and to Section 3.3 for a definition of the group of automorphisms Aut(T ). We call a tree T with vertex set · n≥1 Vn a d-ary tree if the cardinality |V | = dn for each vertex level set V , n ≥ 1. We denote by ∂T the n n S set of all infinite paths in T , which is a Cantor set. Key words and phrases. Settled polynomials, equicontinuous group actions, odometers, profinite groups, arboreal representations, Galois groups, cycle structure, maximal torus, Weyl group, normalizers of profinite groups. 2020 Mathematics Subject Classification. Primary 37P05, 37P15, 37E25, 20E08, 20E22, 20E28, 22B05; Secondary 37F10, 37B05, 11R09, 11R42, 22A05, 20E18. Version date: June 1, 2021. The research of M-I. Cortez was supported by proyecto Fondecyt Regular No. 1190538; the research of O. Lukina is supported by the FWF Project P31950-N35. 1 2 MAR´IA ISABEL CORTEZ AND OLGA LUKINA 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 <j<k. We say that v is in a stable cycle of length k if v is in a cycle of length k and for every m>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 ).
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