The Minkowski dimension of the image of an arboreal Galois representation

Chifan Leung; Clayton Petsche

REVIEW 3 major objections 3 minor 19 references

The Minkowski dimension of the image of an arboreal Galois representation

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Arboreal Galois groups have Minkowski dimension below 1 whenever the base point is postcritical or periodic or the generic image is not maximal, and dimension 0 for power, Chebyshev, and Lattès maps.

desk verdict Solid new theorems on Minkowski dimension of arboreal Galois groups; the abstract promises results that the body does not contain, so fix that before acceptance. read the letter →

arxiv 2512.18825 v2 pith:VBQNGDYI submitted 2025-12-21 math.NT

Chifan Leung , Clayton Petsche This is my paper · ORCID

classification math.NT MSC 11R3237P0537P15

keywords arborealGaloisrepresentationMinkowskidimensionrationalmapspreimagetreepost-criticalpointsperiodicChebyshevLattès

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes the Minkowski dimension of an arboreal Galois group G_{f,α} as a single quantitative measure of how large such a group is, and it proves the one-sided parts of a conjectural dichotomy. On the large side, it shows that if the dimension is maximal (equal to 1), then the base point α cannot be post-critical, cannot be periodic, and the generic function-field image must be the full tree automorphism group; each of those three conditions alone forces the dimension strictly below 1, in two cases with an explicit gap of 1-d^{-m}. On the small side, it proves that power maps, Chebyshev maps, and Lattès maps give dimension 0 for every base point, and that any abelian subgroup of a rooted-tree automorphism group has dimension 0. Because dimension 0 is also preserved under sharing an iterate, and because the generic arithmetic and geometric monodromy groups are shown to have the same dimension, the paper gives a common framework that reduces several previously separate large-image and small-image questions to a single numerical threshold. If its dimension-0 conjecture is true, the paper shows a known abelian-arboreal classification conjecture follows.

What carries the argument

The central object is the infinite d-ary rooted tree T_{f,α} whose level-n vertices are the points of f^{-n}(α), with the arboreal Galois group G_{f,α} acting by tree automorphisms. The workhorse identity expresses lower and upper Minkowski dimension as dim(G) = (d-1)/log(d!) · limsup_{n→∞} (1/d^n) log |r_n(G)|, where r_n(G) is the restriction of G to the first n levels of the tree. This converts a covering-number definition into a field-degree growth rate, making a missing level at depth m visible as a factor d^{-m} in the dimension. A second key input is a structural bound on abelian tree automorphism groups: an abelian group of automorphisms of a finite d-ary tree has order at most q^{I(T

What would settle it

Compute dim(G_{f,α}) for f(x)=x^2-1 at the periodic base point α=0 using Proposition 8: if the limsup of d^{-n} log[K(f^{-n}(0)):K] reaches the threshold that gives dimension 1, then Theorem 17's bound 1-d^{-2}=3/4 is violated and the central dichotomy fails. A cheaper check of the metric dependence: recompute dim(Aut(T)) for an incomplete binary tree under the alternative normalization ε_n = 1/|Aut(T_n)|; if it reaches 1, Proposition 5 collapses.

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Extended reading notes

Core claim

The central discovery is that Minkowski dimension behaves like a sharp phase indicator for arboreal Galois groups. For a rational map f of degree d over a number field and a base point α, the dimension of G_{f,α} is computed from the normalized logarithmic growth rates of [K(f^{-n}(α)):K]. The paper shows that the occurrence of a critical point in some f^{-m}(α) (post-critical base point), the occurrence of α in a periodic cycle of length m, or a proper generic group G_{f,t} ⊊ Aut(T_{f,t}) each force dim G_{f,α} below 1, with explicit bounds 1-d^{-m} in the first two cases, so dimension 1 can only occur when none of these obstructions is present. On the small side, power maps, Chebyshev maps

Load-bearing premise

The proofs that a missing level in the tree forces dimension below 1 rely on measuring each level by the automorphism-group size of the complete d-ary tree rather than by the actual, possibly incomplete tree; if that normalization were changed, the same missing level might no longer push the dimension down.

Editorial extensions

If this is right

If the base point α is f-post-critical with a critical point appearing at level m, then dim G_{f,α} ≤ 1 - d^{-m}; in particular large image is impossible.
If α is periodic of period m, the same explicit bound holds, so periodic base points never yield dimension 1.
If the generic group G_{f,t} over the function field is not the full tree automorphism group, then every specialization α has dimension <1; large image can only occur over a generic image that is maximal.
Power maps, Chebyshev maps, and Lattès maps all have dim G_{f,α} = 0 for every α, and any iterate f^m has dimension <1, so these are the only currently known sources of dimension 0.
If the paper's dimension-0 conjecture is correct, then the abelian-arboreal classification conjecture follows as a consequence, because abelian groups have dimension 0 and the dimension-0 maps in the polynomial case are exactly the ones listed in that classification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit 1-d^{-m} bounds suggest a testable strengthening: the dichotomy might hold with a uniform positive gap below 1 in all non-maximal cases, not merely a strict inequality; the paper does not conjecture this.
The metric normalization is doing real work. Recomputing the key estimates under a metric normalized by the actual (possibly incomplete) tree's automorphism counts would clarify whether the 0/1 phase transition is intrinsic or an artifact of the chosen normalization.
The genus-1 vanishing result comes from elliptic-curve torsion bounds, hinting that semiconjugacy to abelian varieties or other algebraic groups could yield analogous dimension-zero results for higher-dimensional dynamical systems.
Iterate-sharing invariance means the dimension-0 conjecture only needs to be verified on one representative of each iterate-sharing class; maximal representatives that are not themselves nontrivial iterates are the natural testing ground.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Core of this paper is in good shape. The main new results are Theorem 7, extending Abért–Virág's abelian-dimension-zero result from complete p-ary trees to arbitrary d-ary trees; Theorems 16–18, giving the 'only if' direction of Conjecture 2; and Theorems 20–22, proving that power, Chebyshev, and Lattès maps have dimension zero. The reduction of Andrews–Petsche to Conjecture 3 (Theorem 25) is clean. I checked the logic in the main chain: Proposition 2 gives the dimension formula from the metric, Theorem 16 follows immediately from Proposition 5, Theorem 17 is the most intricate and appears correct, and Theorem 18's specialization argument is standard. No load-bearing gap found.

What gave me pause is the abstract. It announces existence of Minkowski dimension for G_f^arith and G_f^geom and a dichotomy theorem, but the full text never defines these groups or proves those statements. That is a genuine mismatch and needs to be resolved—either add the material or cut the claims. A referee should not have to guess whether the body accidentally dropped a section.

Two smaller things. Theorem 24 for Lattès maps omits the details for |Γ|=3,4,6; the 'we omit the details' line is too much hand-waving for a statement in the final section, even if the |Γ|=2 case is representative. And the metric in §2 is normalized by |Aut(T_n^com)| even for incomplete trees. That is explicit and internally consistent, and the dimension formula in Proposition 2 depends on it. It means Theorems 16 and 17 are in part statements about the chosen metric rather than purely Galois-theoretic facts. I don't think that's an error—the authors defend it—but readers should be told more clearly that non-maximality for postcritical/periodic base points is tied to this normalization.

Overall: this is a real contribution, not a gimmick. The conjectures are clearly stated and the partial results are meaningful. Presentation needs fixing before acceptance; content deserves a serious referee.

Referee Report

3 major / 3 minor

Summary. The paper introduces lower and upper Minkowski dimensions for arboreal Galois groups G_{f,α} associated to rational maps of P^1 over number fields, using the metric on Aut(T) in §2. It proves several non-large-image results: Theorem 16 (post-critical base point implies dim<1), Theorem 17 (periodic base point implies dim<1), and Theorem 18 (failure of the generic group G_{f,t} to be full implies dim<1 for every specialization). Together these give the 'only if' direction of Conjecture 2. In the opposite direction, Theorems 20–22 prove that power maps, Chebyshev maps, and Lattès maps always have dimension zero. The paper also proves Theorem 7 that abelian subgroups of tree automorphism groups have dimension zero, and Theorem 25 that Conjecture 3 implies the Andrews–Petsche/Ferraguti–Ostafe–Zannier conjecture. Theorems 23 and 24 establish invariance of both sides of Conjecture 3 under iterate-sharing equivalence.

Significance. If the results stand, the paper gives a substantial and usable quantitative measure of the size of arboreal Galois images, with clean statements and explicit bounds. The proofs of the core theorems (Theorems 16–18 and 20–22) are detailed, parameter-free, and internally consistent. The paper also provides a new reduction from a conjecture on abelian arboreal Galois groups to a dimension-minimality conjecture, which is a useful structural insight. The authors are transparent about the crucial metric normalization in §2, and the main derivation is not circular. However, the abstract advertises theorems about G_f^arith and G_f^geom that are absent from the body, and the proof of Theorem 24 omits the Lattès cases with |Γ|=3,4,6. These gaps do not undermine the proved central theorems, but they do affect the manuscript as a submitted research article.

major comments (3)

[Abstract; §1, Conjectures 1–3] The abstract states three results that do not appear anywhere in the body: (i) existence of Minkowski dimension for G_f^arith and G_f^geom; (ii) equality of their dimensions; and (iii) a dichotomy theorem for these two groups. None of these objects is defined after the abstract, and Conjecture 1 in the body only conjectures existence of dim(G_{f,α}). The abstract's final sentence about a 'conjecture on dimension minimality for quadratic polynomials' also does not match Conjecture 3, which is stated for general rational maps and all degrees. The manuscript must either add the promised material on profinite iterated monodromy groups or revise the abstract to describe exactly what is proved. As submitted, the advertised contributions are unsupported.
[§8, proof of Theorem 24 (Lattès maps)] In the proof of the Claim for Lattès maps, the case |Γ|=2 is treated, and then the text says 'The same argument works with slight variations in the remaining cases ... we omit the details.' The cases |Γ|=3,4,6 are precisely where the descent of the Lattès diagram from C to \bar K is least routine, and the theorem requires the semi-conjugacy to be defined over \bar K. As written, Theorem 24 is therefore not fully proved. Please supply the omitted cases or restate the theorem with the missing cases explicitly excluded.
[§2, display (3); Theorem 16] The choice ε_n = 1/|Aut(T_n^com)| for every tree, complete or not, is explicit and internally consistent, but it makes the non-maximality results metric-dependent. Proposition 5 and Theorem 16 follow by comparing a non-complete level with the complete-tree normalization; if one instead normalized by |Aut(T_n)| for the actual tree, dim(Aut(T_{f,α})) would not in general be <1. The authors should state this caveat more prominently, since Theorem 16 is presented as an arboreal Galois non-large-image result rather than as a consequence of the chosen metric. I do not regard this as an error, but the framing should not suggest that the invariant is independent of the normalization.

minor comments (3)

[§2, proof of Proposition 5] In the application of Proposition 2, the text writes 'dim(G)' although G has not been introduced; it should be 'dim(Aut(T))'.
[§1, Conjecture 3] The phrase 'only if direction' is ambiguous. The theorem proved in §7 is the implication 'f is a power/Chebyshev/Lattès map ⇒ dim(G_{f,α})=0'. Please reword to avoid logical confusion.
[§6, Theorem 17 proof] After equation (34), the displayed bound uses α_i where β_i is meant; please correct the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central dimension theorems are derived from the paper's explicit metric definitions and independent field-theoretic results, while the one conditional reduction relying on a same-author classification is not used to prove the unconditional claims.

full rationale

The derivation chain is self-contained rather than circular. Propositions 2–5 fix the metric explicitly in §2 (display (3)) and derive the dimension formulas; Theorems 16–18 are then direct consequences of those formulas and standard Galois/field facts, not of the conjectures. Theorems 20–22 prove vanishing dimension for power, Chebyshev, and Lattès maps by independent reductions: Kummer-theoretic degree bounds for x^±d, a semiconjugacy from Chebyshev to the power map, and the genus-1 abelian-torsion argument for Lattès maps. Theorem 25 is explicitly conditional: it assumes Conjecture 3 and then uses published classification results [4, Thm. 12,13] to conclude Conjecture 4; although one author of [4] is a co-author, this is an external, published special-case classification and not an unverified premise that the paper renames as a prediction. The metric normalization ε_n = 1/|Aut(T_n^com)| is an explicit modeling convention, not a hidden fitted input. Presentation gaps exist—the abstract promises results on G_f^arith and G_f^geom that do not appear in the body, and Theorem 24 omits some |Γ|=3,4,6 Lattès descent details—but these are not circularity. No equation is equal to its input by construction, no fitted parameter is renamed a prediction, and no load-bearing citation reduces to the paper's own conclusions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities; the paper is pure mathematics. The list records the external theorems and structural assumptions the proofs rest on.

assumptions (5)

standard math Field automorphisms commute with f and fix α, inducing the tree action
Basic property of Galois action on preimage tree, used throughout.
standard math Riemann-Hurwitz: a degree-d map on P^1 with a fiber of size < d^n contains a critical point
Used in Theorem 16 to bound the number of level-m vertices of the preimage tree.
standard math Galois specialization: for β not post-critical, [L(f^{-m}(β)):L] ≤ [L(t)(f^{-m}(t)):L(t)]
Load-bearing in Theorem 18; cited to Lang Ch. VII Thm 2.9.
standard math Density zero of incomplete vertices in a d-ary tree (Abért-Virág Lemma 8.2)
Used in Theorem 7 to show that abelian subgroups have dimension 0.
domain assumption Zdunik's theorem: a rational map is Lattès iff its canonical measure is absolutely continuous w.r.t. Lebesgue
Used in Theorem 24 to lift the Lattès property from C to \bar{K}.

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Pith. "Pith review of The Minkowski dimension of the image of an arboreal Galois representation." pith.science (2026). https://pith.science/paper/VBQNGDYI

@misc{pith2026251218825,
  author       = {Pith},
  title        = {Pith review of: The Minkowski dimension of the image of an arboreal Galois representation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBQNGDYI}},
  note         = {Machine review of arXiv:2512.18825}
}

abstract

We consider the Minkowski dimension of the arboreal Galois group $G_{f,\alpha}$ associated to a rational map $f:\mathbb{P}^1\to\mathbb{P}^1$ and a base point $\alpha\in\mathbb{P}^1(K)$. This is a subgroup of the automorphism group of the infinite $d$-ary rooted tree whose vertices are indexed by the backward orbit $f^{-\infty}(\alpha)$. We show that the Minkowski dimension exists for the profinite iterated monodromy groups $G_f^\mathrm{arith}$ and $G_f^\mathrm{geom}$, and that these two groups have the same dimension. We prove a dichotomy theorem stating that $G_f^\mathrm{arith}$ and $G_f^\mathrm{geom}$ are either the full tree automorphism group or else have non-maximal dimension. We identify several cases of interest in which dimension non-maximality $\overline{\dim}(G_{f,\alpha})<1$ holds, including the cases of postcritical base point, the case of periodic base point, the case in which $f$ is a nontrivial iterate, and the postcritically finite case. We identify several cases of interest in which dimension minimality $\dim(G_{f,\alpha})=0$ holds, including the power, Chebyshev, Latt\`es, and abelian cases. We formulate a conjecture on dimension minimality for quadratic polynomials, which if true would imply the $d=2$ case of a conjecture of Andrews-Petsche on abelian arboreal Galois groups.

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Works this paper leans on

19 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

A. G. Abercrombie , Subgroups and subrings of profinite rings , Math. Proc. Cambridge Philos. Soc., 116 (1994), pp. 209--222

1994

[2] [2]

Ab\'ert and B

M. Ab\'ert and B. Vir\'ag , Dimension and randomness in groups acting on rooted trees , J. Amer. Math. Soc., 18 (2005), pp. 157--192

2005

[3] [3]

Ahmad, R

F. Ahmad, R. L. Benedetto, J. Cain, G. Carroll, and L. Fang , The arithmetic basilica: a quadratic PCF arboreal G alois group , J. Number Theory, 238 (2022), pp. 842--868

2022

[4] [4]

Andrews and C

J. Andrews and C. Petsche , Abelian extensions in dynamical G alois theory , Algebra Number Theory, 14 (2020), pp. 1981--1999

2020

[5] [5]

Barnea and A

Y. Barnea and A. Shalev , Hausdorff dimension, pro- p groups, and K ac- M oody algebras , Trans. Amer. Math. Soc., 349 (1997), pp. 5073--5091

1997

[6] [6]

R. L. Benedetto, X. Faber, B. Hutz, J. Juul, and Y. Yasufuku , A large arboreal G alois representation for a cubic postcritically finite polynomial , Res. Number Theory, 3 (2017), pp. Art. 29, 21

2017

[7] [7]

R. L. Benedetto and J. Juul , Odoni's conjecture for number fields , Bull. Lond. Math. Soc., 51 (2019), pp. 237--250

2019

[8] [8]

C. J. Bishop and Y. Peres , Fractals in probability and analysis , vol. 162 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2017

2017