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Finite index theorems for iterated Galois groups of cubic polynomials. (English) Zbl 07051737
Summary: Let $$K$$ be a number field or a function field. Let $$f\in K(x)$$ be a rational function of degree $$d\ge 2$$, and let $$\beta \in {\mathbb {P}}^1(\overline{K})$$. For all $$n\in \mathbb {N}\cup \{\infty \}$$, the Galois groups $$G_n(\beta )={{\mathrm{Gal}}}(K(f^{-n}(\beta ))/K(\beta ))$$ embed into $${{\mathrm{Aut}}}(T_n)$$, the automorphism group of the $$d$$-ary rooted tree of level $$n$$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $$[{{\mathrm{Aut}}}(T_\infty ):G_\infty (\beta )]<\infty$$. When $$f$$ is a cubic polynomial and $$K$$ is a function field of transcendence degree 1 over an algebraic extension of $${\mathbb {Q}}$$, we resolve this problem by proving a list of necessary and sufficient conditions for finite index. This is the first result that gives necessary and sufficient conditions for finite index, and can be seen as a dynamical analog of the Serre Open Image Theorem. When $$K$$ is a number field, our proof is conditional on both the abc conjecture for $$K$$ and Vojta’s conjecture for blowups of $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$. We also use our approach to solve some natural variants of the finite index problem for modified trees.

##### MSC:
 37P15 Dynamical systems over global ground fields 11G50 Heights 11R32 Galois theory 14G25 Global ground fields in algebraic geometry 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
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