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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.

Finite index theorems for iterated Galois groups of cubic polynomials