Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

TL;DR

This paper proves that for certain quadratic polynomials over fields, the associated arboreal Galois groups are densely settled, extending previous conjectures and applying to notable maps like the Basilica map.

Contribution

It establishes that arithmetic iterated monodromy groups of postcritically finite quadratic polynomials are densely settled, confirming a conjecture for a broad class of polynomials.

Findings

Densely settled groups are shown for quadratic polynomials with periodic postcritical orbits.

Results apply to the Basilica map, a well-known quadratic polynomial.

In number fields, infinitely many cases have densely settled arboreal Galois representations.

Abstract

Let $f(x) \in K(x)$ be a quadratic polynomial where $K$ is a field of characteristic not equal to $2$. The associated arboreal Galois representation of the absolute Galois group of $K$ acts on a regular rooted binary tree. Boston and Jones conjectured that, for $f \in \mathbb{Z}[x]$, the image of this representation contains a dense set of settled elements. Roughly speaking, a cycle of an automorphism $\tau$ of the tree is called stable if its length strictly increases at each subsequent level, and $\tau$ is called settled if the proportion of vertices contained in stable cycles goes to $1$ as the level goes to infinity. In this article, we prove that the arithmetic iterated monodromy groups of postcritically finite quadratic polynomials in $K[x]$ with periodic postcritical orbits are densely settled. In the number field case, by a result of Benedetto--Ghioca--Juul--Tucker \cite{BGJT2025s}, it follows that for infinitely many $a \in K$, the associated arboreal Galois representations are densely settled. In particular, our results apply to the arithmetic IMG of the Basilica map $f(x)=x^2-1$.