Constraining images of quadratic arboreal representations

Abstract

In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence towards Jones' finite index conjecture. On the other hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite. Finally, we combine our results with local class field theory to prove that if f ∈ Q[x] is quadratic and α ∈ Q, then the dynamical Galois group attached to the pair (f, α) is abelian if and only if (f, α) is Q-conjugated to either (x 2 , ±1) or (x 2 − 2, β), where β ∈ {±2, ±1, 0}. This improves on recent results of Andrews and Petsche.