Arboreal Cantor Actions Olga Lukina University of Illinois at Chicago March 15, 2018 1 / 19
Group actions on Cantor sets Let X be a Cantor set. Let G be a countably generated discrete group. The action of G on X is given by a homomorphism Φ : G Homeo(X). Let d be a metric on X. Then the action Φ is equicontinuous if and only if there is an ɛ > 0 and δ(ɛ) > 0 such that for all g G and all x, y X with d(x, y) < δ(ɛ) we have d(φ(g)(x), Φ(g)(y)) < ɛ. Example: Let X = P d be the space of paths in a spherically homogeneous rooted tree T d. Let G be any discrete group, acting on T d by permuting edges at each level so that the paths are preserved. This action is equicontinuous. 2 / 19
Classification of Cantor actions Classification problem Find invariants, which classify actions on Cantor set up to return equivalence. If an action (X, G, Φ) is equicontinuous, then every clopen subset U X contains an adapted set V U such that the restriction of the action of G to V is the action of a subgroup G V which has finite index in G. The actions (X, G, Φ G ) and (X, H, Φ H ) are return equivalent if there exist adapted clopen subsets V, U X, and subgroups G V, H U such that the restricted actions (V, G V, Φ G ) and (U, H U, Φ H ) are conjugate. The classification of Cantor actions up to return equivalence is motivated by the classification problem for the suspensions of Cantor actions, called weak solenoids. 3 / 19
The asymptotic discriminant The asymptotic discriminant of an equicontinuous Cantor action of discrete group G was introduced in S. Hurder and O. Lukina, Wild solenoids, arxiv:1702.03032, to appear in Transactions of the A.M.S. In this talk, we consider the asymptotic discriminant for a special type of actions on the path space of a spherically homogeneous rooted tree, which arise as arboreal representations of absolute Galois groups of number fields in arithmetic dynamics. The talk is based on the results of the paper O. Lukina, Arboreal Cantor actions, arxiv:1801.01440, 2018. 4 / 19
Galois groups of finite extensions of fields Let f(x) be an irreducible polynomial of degree d over a number field K. Let α K, and suppose f(x) = α has d distinct solutions. Then K(f 1 (α))/k is a Galois extension of K. A Galois group Gal(K(f 1 (α))/k) is a subgroup of automorphisms of the field K(f 1 (α)) which fix K. The Galois group Gal(K(f 1 (α))/k) permutes the roots of f(x). Example (textbook): Let p be a prime, and F p be a field of characteristic p. Let f(x) = x d 1, where d is coprime to p. Then the Galois group Gal(K(f 1 (α))/k) is cyclic, generated by the Frobenius automorphism σ : ζ ζ p. 5 / 19
Arboreal representations of Galois groups Identify α with the root of a d-ary tree T d, and identify every solution α 11, α 12,..., α 1d of f(x) = α with a vertex at level 1 in the tree. Gal(K(f 1 (α))/k) is identified with a subgroup of the symmetric group S d. For every α 1i, consider the equation f(x) = α 1i, so f f(x) = f(α 1i ) = α. Suppose there are d 2 distinct roots. Identify the solutions of f(x) = α 1i with the d vertices at level 2 connected with α 1i at level 1. The action of Gal(K(f 2 (α))/k) preserves the structure of the tree, so Gal(K(f 2 (α))/k) [S d ] 2, where [S d ] 2 denotes the two-fold wreath product of symmetric groups S d. 6 / 19
Arboreal representations of Galois groups Continue by induction, assuming that for each i > 0 the polynomial f i (x) has d i distinct roots. In the limit, we get a d-ary infinite tree T d of preimages of α under the iterations of f(x), and the profinite group Gal (f) = lim {Gal(K(f i (α))/k) Gal(K(f (i 1) (α))/k)}, which is a subgroup of the infinite wreath product Aut(T d ) = [S d ]. Thus Gal (f) is a profinite group acting on the Cantor set of paths in the tree T d. Example (Odoni 1985): If K = Q, α = 0, f(x) = x 2 x + 1, then Aut(T ) = [S 2 ]. 7 / 19
Arboreal representations and actions of discrete groups An arboreal representation of the absolute Galois group Gal(K sep /K) into Aut(T d ) is given by the map ρ f,α : Gal (f) Aut(T d ). The study of the arboreal representations of Galois groups was started by Odoni 1985 in order to answer certain questions in number theory. There has been much development in this area of arithmetic dynamics, see Jones 2013 for a survey. We would like to study topological properties of such actions. Problem Study the invariants of the actions of arboreal representations, which classify such actions up to return equivalence. 8 / 19
Group chains for arboreal actions First difficulty: our tools, such as group chains, are developed for actions of discrete groups, while an arboreal representation is a profinite group. Theorem 1 (Lukina 2018) Let f(x) be a polynomial of degree d 2 over a field K, suppose all roots of f i (x) are distinct and f i (x) α is irreducible for all i 0. Let v be a path in P d. Then there exists a countably generated group G 0, a homomorphism Φ : G 0 Homeo(P d ) and a chain {G i } i 0 of subgroups in G 0 such that (1) There is an isomorphism φ : Φ(G 0 ) ρ f,α (Gal (f)), (2) There is a homeomorphism φ : lim {G 0 /G i } P d with φ(eg i ) = v, (3) For all u P d and g Φ(G 0 ) we have φ(g) φ(u) = φ(g(u)). 9 / 19
Group chains in arboreal representations Comments on the proof of Theorem 1: 1. G 0 exists because ρ f,α (Gal (f)) is the inverse limit of a sequence of finite groups indexed by natural numbers; 2. The group G i is the isotropy group of the action of G 0 on the vertex of v at level i; 3. The closure Φ(G 0 ) is identified with the enveloping (Ellis) group of the action of G 0 on P d. As a consequence of Theorem 1, we can now compute the asymptotic discriminant for arboreal representations. 10 / 19
Actions on subtrees of T d Given a path v = (v i ) i 0 T d, the set of paths through the vertex v i at level i is a clopen subset U i of P d. The restricted action on U i is given by Φ i : G i Homeo(U i ). The isotropy group of the action of Φ(G i ) at v D v,i = {g Φ(G i ) g (v j ) j i = (v j ) j i } is called the discriminant group of the action (U i, G i, Φ i ). The discriminant group is a compact profinite group, so it is either finite or a Cantor group. 11 / 19
Discriminant groups of restricted actions Theorem 2 (Dyer, Hurder and Lukina 2017) Let (X, G, Φ) be a Cantor group action with group chain {G i } i 0, let x X be a point. Then for any j > i 0 there is a well-defined surjective homomorphism of discriminant groups. Λ i,j : D j,x D i,x The action is called stable, if there exists i 0, such that for all j > i i 0 the homomorphism Λ i,j is an isomorphism. If such an i 0 does not exist, then the action is called wild. 12 / 19
The asymptotic discriminant of a Cantor group action Let {G i } i 0 be a group chain associated to an action (G 0, X, Φ), x X. Asymptotic discriminant (Hurder and Lukina 2017) The asymptotic discriminant of the action of G 0 is the equivalence class of the chain of surjective group homomorphisms D 0,x D 1,x D 2,x with respect to the equivalence relation, defined in HL 2017. The asymptotic discriminant is an invariant of Cantor actions. Theorem 3 (Hurder and Lukina 2017) The asymptotic discriminant of a Cantor group action (X, G, Φ) is invariant under the return equivalence of actions. 13 / 19
Examples of stable and wild actions 1. All equicontinuous actions of abelian groups on Cantor sets are stable, with trivial discriminant group D 0. 2. Dyer, Hurder and Lukina 2017: every finite group can be realized as the stable discriminant group of an action of a torsion-free finite index subgroup of SL(n, Z) on a Cantor set. 3. Dyer, Hurder and Lukina 2017: every separable profinite group can be realized as the stable discriminant group of an action of a torsion-free finite index subgroup of SL(n, Z) on a Cantor set. 4. Hurder and Lukina 2017: There exists uncountably many wild actions of torsion-free finite index subgroups of SL(n, Z) with distinct asymptotic discriminants. 14 / 19
Strongly quasi-analytic actions In the realm of arboreal representations, we have the following examples. Theorem 4 (Lukina 2018) Suppose the image of an arboreal representation ρ f,α : Gal (f) Aut(T d ) is a subgroup of finite index in Aut(T d ). Then the action of the dense subgroup G 0 on P d is wild. Remarks: 1. The proof of Theorem 4 is geometric, it uses the absence of the strong quasi-analytic property of wild actions. The proof does not require an explicit description of Galois groups. 2. The discrete group G 0 acting on the path space P d in Theorem 4 is infinitely (countably) generated. 15 / 19
Arboreal representations of p-adic numbers Theorem 5 (Lukina 2018) Let p and d be distinct odd primes, let K = Q p be the field of p-adic numbers, and let f(x) = (x + p) d p. Then the arboreal representation ρ f,0 is stable. Remarks: 1. Theorem 5 uses explicit descriptions of Galois groups for computations, and then applies Theorem 1 to obtain a group chain {G i } i 0. 2. The group acting on T d in Theorem 5 is the Baumslag-Solitar group BS(p, 1) = {τ, σ στσ 1 = τ p }. 3. Theorem 5 also holds if K is a finite unramified extension of Q p, for the same polynomial f(x). 16 / 19
Work in progress Theorem 4 gives examples of arboreal representations with wild asymptotic discriminant. Groups, acting on the tree T d in Theorem 4, are infinitely generated. Problem Find an example of an arboreal representation, where the discrete group, acting on the tree T d, is finitely generated, and the asymptotic discriminant of the action is wild. The answer to this question is the work in progress in O. Lukina, Non-Hausdorff elements in arboreal representations, in preparation, 2018. 17 / 19
Open problems One of the open questions in arithmetic dynamics is to find examples of polynomials, especially of degree d 3, whose arboreal representations have finite index in Aut(T d ). This question has some implications in Number Theory. By Theorem 4, all such representations have wild asymptotic discriminant and the acting group is infinitely generated. At the same time, there are examples of group actions of finitely generated groups which are wild. Open problem Relate the wildness of the asymptotic discriminant of an arboreal representation with questions in Number Theory. 18 / 19
References R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. (3), 51 1985, 385-414. R. Jones, Galois representations from pre-image trees: an arboreal survey, in Actes de la Conférence Théorie des Nombres et Applications, 2013, 107-136. J. Dyer, S. Hurder and O. Lukina, Molino s theory for matchbox manifolds, Pacific J. Math., 289(1) 2017, 91-151. S. Hurder and O. Lukina, Wild solenoids, arxiv:1702.03032, 2017, to appear in Transactions of the A.M.S. O. Lukina, Arboreal Cantor actions, arxiv: 1801.01440. O. Lukina, Non-Hausdorff elements in arboreal representations, in preparation, 2018. Thank you for your attention! 19 / 19