Arboreal Galois groups for quadratic rational functions with colliding critical points

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 In unpublished work from 2013, Pink described a certain proper subgroup \(M_{\ell }\) $M_{\ell }$ that this so-called arboreal Galois group \(G_{\infty }\) $G_{\infty }$ must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