Arboreal Galois representations of rational functions: fixed-point proportion and the extension problem

TL;DR

This paper explicitly describes the extension of iterated Galois groups of rational functions, solves the extension problem in certain cases, and explores fixed-point proportions, providing new examples and counterexamples in the field.

Contribution

It offers a complete solution to the extension problem for certain Galois groups, characterizes when positive fixed-point proportion occurs, and constructs new fractal, branch groups with unique fixed-point properties.

Findings

Complete solution to the extension problem when Galois groups are branch.

Sufficient condition for positive fixed-point proportion in iterated Galois groups.

Constructed new fractal, branch groups acting on the d-adic tree with positive probability fixed points.

Abstract

We give an explicit description of the arithmetic-geometric extension of iterated Galois groups of rational functions. This yields a complete solution to the extension problem when either the arithmetic or the geometric iterated Galois group is branch, answering a question of Adams and Hyde. Furthermore, we obtain a sufficient condition for the arithmetic iterated Galois group of a rational function to have positive fixed-point proportion, which further applies in many instances to the specialization to non strictly post-critical points. In particular, this holds for all unicritical polynomials of odd degree, which greatly generalizes a result of Radi for the polynomial $z^d+1$. Lastly, we obtain the first family of groups acting on the $d$-adic tree whose fixed-point process becomes eventually $d$ for any $d\ge 2$ with positive probability. What is more, these groups are fractal and branch and thus positive-dimensional; hence they yield the first family of counterexamples to a conjecture of Jones for every $d$-adic tree.