Galois theory of quadratic rational functions with periodic critical points
Özlem Ejder
TL;DR
The work analyzes arboreal Galois representations attached to quadratic rational maps with post-critical finite dynamics, focusing on maps where the two critical points fall into a single periodic orbit. By a detailed study of geometric and arithmetic iterated monodromy groups and their self-similar structure, the authors prove that many such maps yield arboreal Galois groups with no settled elements, providing a negative answer to density questions about settled elements and settled pairs. They establish a criterion linking the presence of settled elements to odometers within self-similar groups, and use this to show non-dense settledness in the PCF single-periodic-orbit setting; they also determine the field of constants associated with these maps, proving containment in cyclotomic extensions generated by all $2^n$-th roots of unity. In particular, for $f(x)=\frac{1}{(x-1)^2}$, they verify that the field of constants is exactly $k(\zeta_{2^n}:n\ge1)$, thereby confirming a conjecture of Ejder, Kara, and Ozman and contributing to a negative answer to Jones and Levy's settled-pairs question. Overall, the paper links dynamical Galois theory, iterated monodromy groups, and cyclotomic constants to illuminate the structure of arboreal representations for PCF quadratic rational maps.
Abstract
Given a number field $k$, and a quadratic rational function $f(x) \in k(x)$, the associated arboreal representation of the absolute Galois group of $k$ is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for $f \in \mathbb{Z}[x]$ contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the $n\text{th}$ level of the tree remains small as $n$ goes to infinity. In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of $f$, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs. Furthermore, we study the iterated extension $k(f^{-\infty}(t))$ generated by adjoining to $k(t)$ all roots of $f^n(x) = t$ for $n \geq 1$ for a parameter $t$. We call the intersection of $k(f^{-\infty}(t))$ with $\bar{k}$, the field of constants associated with $f$. When one of the two critical points of $f$ is the image of the other, we show that the field of constants is contained in the cyclotomic extension of $k$ generated by all $2$-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function $\frac{1}{(x-1)^2}$.