  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 prove that quadratic polynomials over global
fields with abelian dynamical Galois group are necessarily post-critically
finite, and we combine our results with local class field theory to classify
quadratic pairs over $\mathbb Q$ with abelian dynamical Galois group, improving
on recent results of Andrews and Petsche. On the other 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.
