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Galois theory of quadratic rational functions. (English) Zbl 1316.11104
Let $$K$$ be a number field, $$\bar{K}$$ its algebraic closure and $$G_{K} :=\mathrm{Gal}(\bar{K}/K)$$ its absolute Galois group. Let $$\phi\in K(x)$$ be a rational function of degree $$2$$. For each $$\alpha\in K$$ we consider the tree $$T_{\alpha}$$ whose vertex set is the disjoint union $${\textstyle\bigsqcup_{n\geq1}\phi^{-n}}(\alpha)$$ of the iterated preimages with edges between $$\phi ^{-n}(\alpha)$$ and $$\phi^{-n+1}(\alpha)$$ defined by the action of $$\phi$$. The elements of $$G_{K}$$ commute with $$\phi$$ and so we have a homomorphism $$\rho:G_{K}\rightarrow \mathrm{Aut}(T_{\alpha})$$ called the arboreal Galois representation attached to $$(\phi,\alpha)$$. The object of the paper is to study the image $$G_{\infty}$$ of $$\rho$$ for particular functions $$\phi$$.
If $$\phi$$ commutes with some $$f\in \mathrm{PGL}_{2}(K)$$ and $$f(\alpha)=\alpha$$, then the Galois action on $$T_{\alpha}$$ commutes with the action of $$f$$. Define $$A_{\phi}:=\left\{ f\in \mathrm{PGL}_{2}(\bar{K})\, |\, \phi\circ f=f\circ\phi\right\}$$ and let $$A_{\phi,\alpha}$$ be the stabilizer of $$\alpha$$ in $$A_{\phi}$$. Let $$C_{\infty}\leq G_{\infty}$$ be the centralizer of action of $$A_{\phi,\alpha}$$ on $$\mathrm{Aut}(T_{\alpha})$$. The authors are interested in the following conjecture. (Conjecture 1.1): If $$\phi$$ is not post-critically finite (that is, the orbit of at least one critical point under $$\phi$$ is infinite), then $$\left| C_{\infty}:G_{\infty}\right| <\infty$$. In an earlier paper [J. Lond. Math. Soc., II. Ser. 78, No. 2, 523–544 (2008; Zbl 1193.37144)] the first author proved that this conjecture holds for two special polynomials. In the present paper it is proved in some other cases. For example, suppose that $$K=\mathbb{Q}$$, $$\phi(x)=k(x^{2}+1)/x$$ and $$\alpha=0$$. Then it is shown that there exists an effectively computable set $$\Sigma$$ of primes of natural density $$0$$ in $$\mathbb{Z}$$ such that $$G_{\infty}\cong C_{\infty}$$ provided the valuations $$v_{p}(k)$$ are $$0$$ for all $$p\in\Sigma$$. More concretely the authors show that if $$k\in\mathbb{Z}$$ then Conjecture 1.1 holds whenever $$\left| k\right| <10000$$.

##### MSC:
 11R32 Galois theory 37P15 Dynamical systems over global ground fields
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