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Profinite Iterated Monodromy Groups of Unicritical Polynomials

Ophelia Adams, Trevor Hyde

TL;DR

The paper develops explicit recursive presentations for the geometric profinite iterated monodromy groups $igl| ext{overline{Arb}} figr|$ of unicritical PCF polynomials $f(x)=ax^d+b$ by embedding these groups into the iterated wreath product $[C_d]^ ty$ and classifying the inertia generators according to the post-critical orbit. A semirigidity phenomenon shows that these inertia-generated recurrences determine the entire group up to conjugacy, yielding model groups $ ext{A}(d,n)$ (periodic) and $ ext{B}(d,m,n,oldsymbol{ extomega})$ (preperiodic) that are conjugate to $igl| ext{overline{Arb}} figr|$ in all cases. The authors then analyze finite-level truncations, Hausdorff dimensions, and normalizers of these model groups, obtaining precise information on constant-field extensions via the outer action of $ ext{Gal}(igl| ext{K}_figr|/K)$, with sharp descriptions in terms of cyclotomic fields. A key innovation is the branch-cycle lemma–style reduction of the constant-field problem to a purely group-theoretic criterion involving inertia at infinity and the cycle structure of odometers, enabling exact determinations of $igl| ext{K}_{f,oldsymbol{ ho}}igr|$ in PCI, periodic, and most preperiodic settings. Overall, the results connect arithmetic dynamics, profinite group theory, and ramification theory to give a coherent, computable picture of profinite iterated monodromy for unicritical polynomials and their constant-field extensions.

Abstract

Let $f(x) = ax^d + b \in K[x]$ be a unicritical polynomial with degree $d \geq 2$ which is coprime to $\mathrm{char} K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this group, and use this analysis to determine the constant field extension in $K(f^{-\infty}(t))/K(t)$.

Profinite Iterated Monodromy Groups of Unicritical Polynomials

TL;DR

The paper develops explicit recursive presentations for the geometric profinite iterated monodromy groups of unicritical PCF polynomials by embedding these groups into the iterated wreath product and classifying the inertia generators according to the post-critical orbit. A semirigidity phenomenon shows that these inertia-generated recurrences determine the entire group up to conjugacy, yielding model groups (periodic) and (preperiodic) that are conjugate to in all cases. The authors then analyze finite-level truncations, Hausdorff dimensions, and normalizers of these model groups, obtaining precise information on constant-field extensions via the outer action of , with sharp descriptions in terms of cyclotomic fields. A key innovation is the branch-cycle lemma–style reduction of the constant-field problem to a purely group-theoretic criterion involving inertia at infinity and the cycle structure of odometers, enabling exact determinations of in PCI, periodic, and most preperiodic settings. Overall, the results connect arithmetic dynamics, profinite group theory, and ramification theory to give a coherent, computable picture of profinite iterated monodromy for unicritical polynomials and their constant-field extensions.

Abstract

Let be a unicritical polynomial with degree which is coprime to . We provide an explicit presentation for the profinite iterated monodromy group of , analyze the structure of this group, and use this analysis to determine the constant field extension in .

Paper Structure

This paper contains 33 sections, 65 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Related work
  4. Directions for future work
  5. Overview
  6. Acknowledgments
  7. Preliminaries
  8. Conjugacy in wreath products
  9. Iterated wreath products
  10. Systems of recurrences
  11. Iterated wreath products of cyclic groups
  12. Heisenberg group
  13. Profinite iterated monodromy groups
  14. Monodromy groups
  15. Iterated preimage extensions
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $f(x) = ax^d + b \in K[x]$ where $d$ is coprime to $\operatorname{char} K$. There exists a labeling of $T_d^\infty$ such that $\operatorname{\overline{Arb}} f \subseteq [C_d]^\infty$, and

Theorems & Definitions (152)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 142 more