@article{cedfa6cc7be44cceadafdf5e61e6679a,

title = "Finite index theorems for iterated Galois groups of unicritical polynomials",

abstract = "Let K be the function field of a smooth irreducible curve defined over Q¯. Let f ∈ K[x] be of the form f(x) = xq + c, where q = pr, r ≥ 1, is a power of the prime number p, and let β ∈ {\=K}. For all n ∈ ℕ ∪ {∞}, the Galois groups Gn(β) = Gal(K(f-n(β))/K(β)) embed into [Cq]n, the n-fold wreath product of the cyclic group Cq. We show that if f is not isotrivial, then [[Cq]∞ : G∞(β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if f1(x) = xq + c1 and f2(x) = xq + c2 are two such distinct polynomials, then the fields ∪∞n=1 K(f-n1 (β)) and ∪∞n=1 K(f-n2 (β)) are disjoint over a finite extension of K.",

keywords = "Arboreal Galois representations, Arithmetic dynamics, Iterated Galois groups",

author = "Andrew Bridy and Doyle, {John R.} and Dragos Ghioca and Hsia, {Liang Chung} and Tucker, {Thomas J.}",

note = "Funding Information: Received by the editors July 18, 2019, and, in revised form, February 3, 2020, and June 30, 2020. 2010 Mathematics Subject Classification. Primary 37P15; Secondary 11G50, 11R32, 14G25, 37P05, 37P30. Key words and phrases. Arithmetic dynamics, arboreal Galois representations, iterated Galois groups. The third author was partially supported by an NSERC Discovery grant. The fourth author was partially supported by MOST Grant 106-2115-M-003-014-MY2. Funding Information: The third author was partially supported by an NSERC Discovery grant. The fourth author was partially supported by MOST Grant 106-2115-M-003-014-MY2. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society.",

year = "2021",

month = jan,

doi = "10.1090/tran/8242",

language = "English",

volume = "374",

pages = "733--752",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "1",

}