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arxiv: 2605.22466 · v1 · pith:A275MJBPnew · submitted 2026-05-21 · 🧮 math.NT · math.GR

Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points

\"Ozlem Ejder , Zofia Go{\l}aska , Yasemin Kara , Leonie Nienhaus , \"Ozge \"Ulkem This is my paper

Pith reviewed 2026-05-22 03:20 UTC · model grok-4.3

classification 🧮 math.NT math.GR
keywords arboreal Galois groupsiterated monodromy groupspost-critically finite mapsquadratic rational functionsHausdorff dimensionpreimage treesnumber fields
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The pith

For the quadratic rational map f(x) = 2/(x-1)^2 with strictly pre-periodic critical points, the arithmetic iterated monodromy group has Hausdorff dimension zero and arboreal Galois maximality is verifiable at level four.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the arithmetic and geometric iterated monodromy groups for a post-critically finite quadratic rational function over a number field, where both critical points are strictly pre-periodic. It supplies explicit recursive descriptions of the topological generators for the geometric group and establishes that the arithmetic group has Hausdorff dimension zero. An explicit criterion is developed to identify base-field elements that make the associated arboreal Galois group attain its largest possible size, with the additional result that this maximality can already be confirmed by examining the fourth level of the preimage tree. The work also computes the intersection of the constant field of the arithmetic group with the 2-power cyclotomic extension, yielding a complete arithmetic description for this specific map.

Core claim

For the post-critically finite quadratic rational function f(x) = 2/(x-1)^2 defined over a number field k whose critical points are both strictly pre-periodic, the geometric iterated monodromy group admits explicit recursive topological generators, the arithmetic iterated monodromy group has Hausdorff dimension zero, an explicit criterion determines the values a in k for which the arboreal Galois group achieves maximum size, this maximality is detectable already at level four, and the constant field of the arithmetic group intersects k(μ_{2^∞}) in a manner that gives the first full study of a PCF quadratic map with non-abelian constant field.

What carries the argument

The arboreal Galois group of the infinite preimage tree under iteration of f, which encodes the Galois action on all backward orbits and is studied via its geometric and arithmetic iterated monodromy subgroups.

If this is right

  • Maximality of the arboreal Galois group reduces to a finite computation at the fourth level of the preimage tree.
  • Hausdorff dimension zero for the arithmetic iterated monodromy group implies that its profinite closure grows slower than the full automorphism group of the tree.
  • The constant-field intersection with the 2-power cyclotomic extension completely determines the arithmetic constants for this map.
  • The recursive generator descriptions allow explicit computation of finite-level images of the monodromy groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The level-four maximality test may extend to other quadratic PCF maps whose critical points are strictly pre-periodic.
  • Zero Hausdorff dimension constrains the possible density of splitting primes in the preimage fields.
  • This explicit case supplies a template for determining constant fields in arithmetic dynamics of rational maps with non-abelian monodromy.

Load-bearing premise

The rational map must be post-critically finite with both critical points strictly pre-periodic.

What would settle it

For a concrete choice of a in k, compute the Galois group acting on the level-four preimages under f and check whether its order equals the predicted maximum size given by the level-four criterion.

read the original abstract

We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function $f(x)=\frac{2}{(x-1)^2}$ defined over a number field $k$, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of $f$ and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values $a\in k$ for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. Finally, we determine the intersection of the constant field of the arithmetic iterated monodromy group with $k(\mu_{2^{\infty}})$, providing the first full study of a PCF quadratic map with non-abelian constant field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the arithmetic and geometric iterated monodromy groups of the post-critically finite quadratic rational function f(x) = 2/(x-1)^2 over a number field k, where both critical points are strictly pre-periodic. It furnishes explicit recursive descriptions of the topological generators for the geometric iterated monodromy group, establishes that the arithmetic iterated monodromy group has Hausdorff dimension zero, and provides an explicit criterion for determining when the associated arboreal Galois group attains its maximal possible size. Notably, it demonstrates that this maximality can be verified already at level four and computes the intersection of the constant field of the arithmetic iterated monodromy group with k(μ_{2^∞}).

Significance. Assuming the derivations are correct, this paper offers a comprehensive analysis of arboreal Galois groups for a concrete PCF quadratic map with non-abelian constant field, which is the first such full study. The explicit recursive generator descriptions and the level-four maximality criterion represent significant computational advances in the field. The determination of the constant field intersection further strengthens the work by addressing potential arithmetic extensions. The stress-test concern about missing higher-level constraints does not land here, since the explicit computation of the intersection with k(μ_{2^∞}) ensures control over the arithmetic part of the group.

minor comments (3)
  1. The abstract and introduction would benefit from a short statement on the computational feasibility of the level-four check to highlight its practicality.
  2. The recursive descriptions of the generators could include a small example computation at level 3 to illustrate the recursion.
  3. A few instances of unclear notation in the definition of the Hausdorff dimension for the group could be clarified with a reference to the standard definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We appreciate the recognition that this constitutes the first full study of a PCF quadratic map with non-abelian constant field, along with the value placed on the explicit recursive generators and the level-four maximality criterion.

Circularity Check

0 steps flagged

No circularity: explicit constructions and level-four criterion rest on standard monodromy theory for PCF maps

full rationale

The paper derives recursive generator descriptions directly from the given PCF quadratic map with strictly pre-periodic critical points, applies standard iterated monodromy group theory to obtain the Hausdorff dimension zero result for the arithmetic group, and states an explicit computational criterion for maximality verifiable at level four. These steps use the map's post-critical finiteness as an input assumption rather than defining the output in terms of itself. The constant-field intersection is presented as an independent computation. No self-citations, fitted parameters renamed as predictions, or ansatzes are load-bearing in the central claims, and the derivation remains self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background from arithmetic dynamics and Galois theory together with the given properties of the chosen map; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of iterated monodromy groups and arboreal Galois representations for rational maps over number fields.
    Invoked throughout to define the geometric and arithmetic groups and their generators.
  • domain assumption The rational map f(x) = 2/(x-1)^2 is post-critically finite with both critical points strictly pre-periodic.
    Stated in the setup and used to obtain the recursive descriptions and level-four criterion.

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