Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

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 $τ$ of the tree is called stable if its length strictly increases at each subsequent level, and $τ$ 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 $f(x)=x^2-1$.

Theorems & Definitions (83)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Example 2.1
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• Definition 3.1