Arboreal Galois groups for quadratic rational functions with colliding critical points

Let $K$ be a field, and let $f\in K(z)$ be rational function. The preimages of a point $x_0\in P^1(K)$ under iterates of $f$ have a natural tree structure. As a result, the Galois group of the resulting field extension of $K$ naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup $M_{\mathrm{\ell }}$ that this so-called arboreal Galois group $G_{\mathrm{\infty }}$ must lie in if $f$ is quadratic and its two critical points collide at the $\mathrm{\ell }$-th iteration. After presenting a new description of $M_{\mathrm{\ell }}$ and a new proof of Pink’s theorem, we state and prove necessary and sufficient conditions for $G_{\mathrm{\infty }}$ to be the full group $M_{\mathrm{\ell }}$.