Arboreal Galois groups for quadratic rational functions with colliding critical points

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 In unpublished work from 2013, Pink described a certain proper subgroup M that this so-called arboreal Galois group G must lie in if f is quadratic and its two critical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