Non-abelian arboreal Galois groups associated to PCF rational maps

TL;DR

This paper proves that arboreal Galois groups associated with certain rational maps are non-abelian, providing new cases supporting a conjecture and indicating the rarity of potential counterexamples.

Contribution

It establishes non-abelian nature of arboreal Galois groups for PCF rational maps, advancing understanding of Galois groups in dynamical systems.

Findings

Arboreal Galois extensions are never abelian for specified maps.

New cases supporting the Andrews-Petsche conjecture.

Counterexamples, if any, are likely sparse.

Abstract

We prove that arboreal Galois extensions of number fields are never abelian for post-critically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of Andrews-Petsche. Together with a result of Ferraguti-Ostafe-Zannier, this result implies that counterexamples to the conjecture, if they exist, are sparse. We also prove an auxiliary result on places of periodic reduction for rational maps, which may be of independent interest.