  Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational
function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all
$n\in\mathbb{N}\cup\{\infty\}$, the Galois groups
$G_n(\beta)=\text{Gal}(K(f^{-n}(\beta))/K)$ embed into $\text{Aut}(T_n)$, 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
$[\text{Aut}(T_\infty):G_\infty]<\infty$. 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.
