Arboreal representations for rational maps with few critical points

Abstract

Jones conjectures the arboreal representation of a degree two rational map will have finite index in the full automorphism group of a binary rooted tree except under certain conditions. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the $abc$-Conjecture and Vojta's Conjecture. We also exhibit a family of degree $2$ rational maps and give examples of degree $3$ polynomial maps whose arboreal representations have finite index in the appropriate group of tree automorphisms.