Arboreal representations for rational maps with few critical points

TL;DR

This paper investigates the arboreal representations of quadratic and cubic rational maps, proving a version of Jones' Conjecture under certain conjectures and providing explicit examples of maps with finite index arboreal representations.

Contribution

It proves a version of Jones' Conjecture for quadratic and cubic maps assuming major conjectures and constructs explicit examples of such maps with finite index representations.

Findings

Proves a version of Jones' Conjecture assuming the abc- and Vojta's Conjectures.

Provides explicit examples of degree 2 rational maps with finite index arboreal representations.

Constructs degree 3 polynomial maps with finite index arboreal representations.

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.