# A large arboreal Galois representation for a cubic postcritically finite polynomial

@article{Benedetto2016ALA,
title={A large arboreal Galois representation for a cubic postcritically finite polynomial},
author={Robert L. Benedetto and X. W. C. Faber and Benjamin Hutz and Jamie Juul and Yu Yasufuku},
journal={Research in Number Theory},
year={2016},
volume={3},
pages={1-21}
}
• Published 11 December 2016
• Mathematics
• Research in Number Theory
We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath…
A Unique Chief Series in the arboreal Galois Group of Belyi Maps
We give a complete description of the normal subgroups of arboreal Galois groups of Belyi maps. The normal groups form a unique chief series. We also carefully compute the discriminate of the iterate
Dynamical Belyi maps and arboreal Galois groups
• Mathematics
manuscripta mathematica
• 2020
We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of
Arboreal Cantor actions
Given an arboreal representation satisfying certain mild conditions, an equicontinuous action of a discrete group, and the asymptotic discriminant are associated to it.
Galois groups and Cantor actions
In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated
N ov 2 01 9 THE IMAGE SIZE OF ITERATED RATIONAL MAPS OVER FINITE FIELDS
Let φ : P1(Fq) → P1(Fq) be a rational map of degree d > 1 on a fixed finite field. We give asymptotic formulas for the size of image sets φn(P1(Fq)) as a function of n. This is done using properties
Arithmetic Monodromy Groups of Dynamical Belyi maps
We consider a large family of dynamical Belyi maps of arbitrary degree and study the arithmetic monodromy groups attached to the iterates of such maps. Building on the results of Bouw-Ejder-Karemaker
A finiteness property of postcritically finite unicritical polynomials
• Mathematics
• 2020
Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map
The set of stable primes for polynomial sequences with large Galois group
Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$,
A question for iterated Galois groups in arithmetic dynamics
• Mathematics
• 2020
Abstract We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our

## References

SHOWING 1-10 OF 18 REFERENCES
Galois groups of iterates of some unicritical polynomials
• Mathematics
• 2016
We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and we prove surjectivity for some
Dynamical Galois groups of trinomials and Odoni's conjecture
We prove that for every prime p , there exists a degree p polynomial whose arboreal Galois representation is surjective, that is, whose iterates have Galois groups over Q that are as large as
Galois theory of quadratic rational functions
• Mathematics
• 2011
For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular
Profinite iterated monodromy groups arising from quadratic polynomials
We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a
Fixed-point-free elements of iterated monodromy groups
The iterated monodromy group of a post-critically finite complex polynomial of degree d \geq 2 acts naturally on the complete d-ary rooted tree T of preimages of a generic point. This group, as well
Galois representations from pre-image trees: an arboreal survey
Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges
Iterates of Generic Polynomials and Generic Rational Functions
In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is
The density of prime divisors in the arithmetic dynamics of quadratic polynomials
Let f ∈ ℤ[x], and consider the recurrence given by an = f(an − 1), with a0 ∈ ℤ. Denote by P(f, a0) the set of prime divisors of this recurrence, that is, the set of primes dividing at least one
On the Number of Places of Convergence for Newton's Method over Number Fields
• Mathematics
• 2010
Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which
QUADRATIC RECURRENCES WITH A POSITIVE DENSITY OF PRIME DIVISORS
• Mathematics
• 2010
For f(x) ∈ ℤ[x] and a ∈ ℤ, we let fn(x) be the nth iterate of f(x), P(f, a) = {p prime: p|fn(a) for some n}, and D(P(f, a)) denote the natural density of P(f, a) within the set of primes. A