Abstract: Let $f\in K(z)$ be a rational function of degree $d\geq 2$ defined over a field $K$ (usually $Q$), and let $x_0\in K$. The backward orbit of $x_0$, which is the union of the iterated preimages $f^{-n}(x_0)$, has the natural structure of a $d$-ary rooted tree. Thus, the Galois groups of the fields generated by roots of the equations $f^n(z)=x_0$ are known as arboreal Galois groups. In 2013, Pink observed that when $d=2$ and the two critical points $c_1,c_2$ of $f$ collide, meaning that $f^m(c_1)=f^m(c_2)$ for some $m\geq 1$, then the arboreal Galois groups are strictly smaller than the full automorphism group of the tree. We study these arboreal Galois groups when $K$ is a number field and $f$ is either a quadratic rational function (as in Pink's setting over function fields) or a cubic polynomial with colliding critical points. We describe the maximum possible Galois groups in these cases, and we present sufficient conditions for these maximum groups to be attained.
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