Non-abelian arboreal Galois groups associated to PCF rational maps

2024/07/24 by Leung, Chifan, Petsche, Clayton · 2 citations
#37P05 37P15 11S82 11R32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.17415

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.

Cited by

• Cyclotomic integral points for affine dynamics
• Iterates of post-critically finite polynomials of the form \boldsymbolx^d+c

Related

• Arboreal Galois groups of rational maps with nonreal Julia sets
• Arboreal Galois groups for quadratic rational functions with colliding critical points
• Constraining images of quadratic arboreal representations
• Galois theory of quadratic rational functions with periodic critical points
• Arboreal Galois groups of postcritically finite quadratic polynomials: The periodic case