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

Jorge Fariña-Asategui

TL;DR

The work develops a robust, self-similar extension framework for arboreal Galois representations, yielding a complete positive solution to the extension problem when either the arithmetic or geometric iterated Galois group is branch. It introduces the fixed-point proportion and bad monodromy concepts to show that positive fixed-point proportions arise broadly, including for unicritical odd-degree polynomials, independent of the full extension or specialization problems. A central achievement is constructing the intermediate group $G_H$ to produce fractal, branch groups with fixed-point processes that can converge to $d$ with positive probability, providing counterexamples to Jones’ conjecture on the $d$-adic tree. The results unify algebraic and geometric perspectives, yield concrete bounds in the unicritical case, and illuminate specialization behavior through monodromy and branching analyses with potential broad impact on Arboreal Galois representations and related dynamics.

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.

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

TL;DR

The work develops a robust, self-similar extension framework for arboreal Galois representations, yielding a complete positive solution to the extension problem when either the arithmetic or geometric iterated Galois group is branch. It introduces the fixed-point proportion and bad monodromy concepts to show that positive fixed-point proportions arise broadly, including for unicritical odd-degree polynomials, independent of the full extension or specialization problems. A central achievement is constructing the intermediate group to produce fractal, branch groups with fixed-point processes that can converge to with positive probability, providing counterexamples to Jones’ conjecture on the -adic tree. The results unify algebraic and geometric perspectives, yield concrete bounds in the unicritical case, and illuminate specialization behavior through monodromy and branching analyses with potential broad impact on Arboreal Galois representations and related dynamics.

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 . Lastly, we obtain the first family of groups acting on the -adic tree whose fixed-point process becomes eventually for any 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 -adic tree.
Paper Structure (15 sections, 18 theorems, 125 equations)

This paper contains 15 sections, 18 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Self-similar extensions
  3. Self-similar groups
  4. Intermediate self-similar groups
  5. Fixed-point proportion
  6. Bad monodromy
  7. Branchness, regular branchness and groups of finite type
  8. Proof of \ref{['Theorem: arbitrary fixed point process']}
  9. Arboreal Galois representations of rational functions
  10. Arithmetic and geometric iterated Galois groups
  11. Self-similarity and fractality
  12. Base field extensions
  13. Extension problem
  14. Fixed-point proportion and bad monodromy
  15. An example: unicritical polynomials

Key Result

Theorem 2

Let $K$ be a field, $f\in K(z)$ a rational function and set $G:=G_\infty(K,f,t)$ and $H:=G_\infty(K^{\mathrm{sep}},f,t)$. Then

Theorems & Definitions (31)

  • Theorem 2
  • Conjecture 3
  • Corollary 4
  • Theorem 6
  • Corollary 7
  • Theorem 9
  • Definition 2.1: The groups $G_H$
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 21 more