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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.04524 |
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| _version_ | 1866914448372924416 |
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| author | Ejder, Özlem Kocak, Dilber |
| author_facet | Ejder, Özlem Kocak, Dilber |
| contents | 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 $f(x)=x^2-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04524 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials Ejder, Özlem Kocak, Dilber Number Theory Group Theory 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 $f(x)=x^2-1$. |
| title | Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2604.04524 |