Algebraic Dynamics Seminar: Arboreal Galois groups with colliding critical points
ALGEBRAIC DYNAMICS, SEMINARS
Rob Benedetto - Amherst College
Let f(z) be a rational function of degree d>1 over a field K (usually K=C(t) or K=Q), and let x_0 be a point in P^1(K). The Galois groups of the equations f^n(z)=x_0 are known as arboreal Galois groups because they induce an action on a d-ary rooted tree. In 2013, Pink observed that when d=2 and the two critical points c_1, c_2 collide, meaning that f^m(c_1)=f^m(c_2) for some m>0, then the arboreal Galois groups are strictly smaller than the full automorphism group of the tree. We study these arboreal Galois groups when f is either a quadratic rational function or a cubic polynomial. When the critical points collide, we describe the maximum possible Galois groups in these cases, and we find sufficient conditions for these maximum groups to be attained.
For more information, please see: Algebraic Dynamics Seminar at Harvard