Arboreal Galois groups with colliding critical points

Abstract: Let $f\in K(z)$ be a rational function of degree $d\geq 2$ defined over a field $K$ (usually $\mathbb{Q}$), 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.

number theory

Audience: researchers in the topic

Comments: Joint BU Number Theory and Dynamics seminar. Note the non-standard room.

Boston University Number Theory Seminar