Finite index theorems for iterated Galois groups of cubic polynomials

Let K be a number field or a function field. Let f is an element of K(x) be a rational function of degree d >= 2, and let beta is an element of P-1((K) over bar). For all n is an element of N boolean OR {infinity}, the Galois groups G(n)(beta) = Gal(K(f(-n)(beta))/K(beta)) embed into 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 [Aut(T-infinity) : G(infinity)(beta)] < infinity. When f is a cubic polynomial and K is a function field of transcendence degree 1 over an algebraic extension of 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 P-1 x P-1. We also use our approach to solve some natural variants of the finite index problem for modified trees.