Arboreal Galois representation

In arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism group of an infinite, regular, rooted tree.

The study of arboreal Galois representations of goes back to the works of Odoni in 1980s.

is called the absolute Galois group of

This is a profinite group and it is therefore endowed with its natural Krull topology.

be the infinite regular rooted tree of degree

This is an infinite tree where one node is labeled as the root of the tree and every node has exactly

is a bijection of the set of nodes that preserves vertex-edge connectivity.

is a profinite group as well, as it can be seen as the inverse limit of the automorphism groups of the finite sub-trees

copies of the symmetric group of degree

An arboreal Galois representation is a continuous group homomorphism

The most natural source of arboreal Galois representations is the theory of iterations of self-rational functions on the projective line.

Then one can construct an infinite, regular, rooted

is continuous, and therefore is called the arboreal Galois representation attached to

Arboreal representations attached to rational functions can be seen as a wide generalization of Galois representations on Tate modules of abelian varieties.

The simplest non-trivial case is that of monic quadratic polynomials.

In 1992, Stoll proved the following theorem:[2] The following are examples of polynomials that satisfy the conditions of Stoll's Theorem, and that therefore have surjective arboreal representations.

[4] Although in this very general form the conjecture has been shown to be false by Dittmann and Kadets,[5] there are several results when

Benedetto and Juul proved Odoni's conjecture for

are odd,[6] Looper independently proved Odoni's conjecture for

is a rational function of degree 2, the image of

The following conjecture quantifies the previous statement, and it was formulated by Jones in 2013.

[8] Jones' conjecture is considered to be a dynamical analogue of Serre's open image theorem.

One direction of Jones' conjecture is known to be true: if

is a topologically finitely generated closed subgroup of

In the other direction, Juul et al. proved that if the abc conjecture holds for number fields,

is post-critically finite or not eventually stable.

is a quadratic polynomial, conditions (2) and (4) in Jones' conjecture are never satisfied.

[9] In 2020, Andrews and Petsche formulated the following conjecture.

The Chebyshev polynomials the conjecture refers to are a normalized version, conjugate by the Möbius transformation

It has been proven that Andrews and Petsche's conjecture holds true when