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A just-infinite iterated monodromy group without the congruence subgroup property

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Pith's one-line read The paper proves that $\mathrm{IMG}(z^2+i)$ is just-infinite and fails the congruence subgroup property, establishing the first polynomial iterated monodromy group with this combination and giving a counterexample to the converse of a…

desk verdict Solid new result on IMG(z^2+i) with a reproducibility gap in two load-bearing GAP computations that should be patched before publication. read the letter →

arxiv 2505.19649 v1 pith:6QURZ4CB submitted 2025-05-26 math.GR

Santiago Radi This is my paper · ORCID
classification math.GR MSC 20E0819B3720F6520F1068Q4520F05
keywords groupsactingontreescongruencesubgrouppropertyiteratedmonodromyself-similarautomatabranchjust-infiniteregular
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

This paper proves that the iterated monodromy group of the quadratic polynomial $z^2+i$ is just-infinite and does not have the congruence subgroup property. That combination is new for polynomial iterated monodromy groups, and it makes the group a counterexample to the converse of the known statement that tree groups with the congruence subgroup property are just-infinite. The proof is carried by the maximal regular branching subgroup $K$: five explicit generators of $K$, each of order 4, are found, and their abelianization is shown to be the five-fold direct product of cyclic groups of order 4. Because $K$ also contains a level stabilizer, the paper can compute the rigid, branch, and congruence kernels of the group, giving a concrete picture of why the congruence topology is strictly coarser than the profinite topology.

What carries the argument

The carrying object is the maximal regular branching subgroup $K$, with its five-element generating set $S_K=\{x,y,z,t,w\}$, together with a parity morphism $e$ from words over $S_K$ to $\mathbb{Z}/2\mathbb{Z}$ that descends to a homomorphism on $K$. The Reidemeister algorithm converts the known $L$-presentation of $G$ into an $L$-presentation of $K$, and the parity function verifies that every relation has even length. The subgroup $\widetilde{K}=\{(g_0,g_1)\in K_1:e(g_0)=e(g_1)=0\}$, where $K_1$ is the first geometric power of $K$ (elements whose two sections lie in $K$), is used to rule out the two remaining candidate relations $ty^{-2}$ and $wz^{-2}$. A stabilization lemma, fed by a level-6 computer check, fixes $\pi_n(K)_{\mathrm{ab}}\cong C_4^3$ for all $n\ge 5$, so the only possible extra relations in $K/K'$ are those two; excluding them proves $K/K'\cong C_4^5$.

What would settle it

Recompute the two finite assertions with independent software or a complete hand calculation: the equality $[\pi_3(G):\pi_3(K)]=16$ and the inclusion $\pi_6(\mathrm{St}_G(5))\le \pi_6(K')$; if the inclusion fails, Lemma 2.4 no longer fixes $\pi_n(K)_{\mathrm{ab}}\cong C_4^3$ for $n\ge 5$. Separately, compute $K/K'$ directly and check that it is exactly the five-fold direct product of cyclic groups of order 4; any extra relation would refute Theorem B, and hence Theorem A.

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Extended reading notes

Core claim

Let $G=\mathrm{IMG}(z^2+i)$, the group generated by the automaton states $a=(1,1)\sigma$, $b=(a,c)$, and $c=(b,1)$ acting on the binary tree. The central claim is Theorem A: $G$ is just-infinite and fails the congruence subgroup property. The mechanism is Theorem B: its maximal regular branching subgroup $K=\langle [a,b],[b,c]\rangle^G$ satisfies $K/K'\cong C_4^5$, generated by $x=[a,b]$, $y=[b,c]$, $z=aya$, $t=cxcy^{-1}x^{-1}$, and $w=ata$, all of order 4. Since a regular branch group is just-infinite exactly when the abelianization of its branching subgroup is finite, Theorem B yields just-infiniteness. For the congruence subgroup property, the paper shows that $K'$ has finite index in $G$ but contains no level stabilizer $\mathrm{St}_G(n)$, so $K'$ witnesses a finite-index subgroup that is not congruence; the congruence kernel is nontrivial and in fact isomorphic to $C_4[[\partial T]]$, while the rigid kernel is trivial.

Load-bearing premise

The load-bearing premise is that two specific finite checks performed with computer algebra are correct: the index computation $[G:K]=[\pi_3(G):\pi_3(K)]=16$, and the inclusion of the image of the level-5 stabilizer into the image of $K'$ at level 6; if either check is wrong or uses different conventions, the proof's stabilization step and the no-congruence-subgroup conclusion are unsupported.

Editorial extensions

If this is right

  • The group $\mathrm{IMG}(z^2+i)$ is the first polynomial iterated monodromy group shown to be just-infinite without the congruence subgroup property.
  • It is a counterexample to the converse of the theorem that, for finitely generated groups acting on rooted trees, the congruence subgroup property implies just-infiniteness.
  • The failure of the congruence subgroup property is measured exactly: the congruence kernel and the branch kernel are both isomorphic to $C_4[[\partial T]]$, and the rigid kernel is trivial.
  • The abelianization theorem constrains the subgroup structure of $K$: $K'$ is a finite-index subgroup of $G$ that contains no level stabilizer, which is the precise obstruction to the congruence subgroup property.

Reading between the lines

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

  • If the two computer-assisted checks are made fully reproducible, the same strategy should apply to other post-critically finite quadratic iterated monodromy groups: exhibit order-4 generators for the branching subgroup, build a parity function on its words, and compare the full abelianization with the stabilized level-$n$ abelianizations.
  • The equality of the congruence and branch kernels suggests that, in regular branch groups whose branching subgroup contains a level stabilizer, the congruence subgroup problem reduces to computing the branch kernel; this may provide a shortcut for other examples.
  • Because $K/K'$ is a 2-group, the pro-2 completion of this group is a natural place to look for a just-infinite pro-2 branch group with a nontrivial congruence kernel, extending the paper's conclusion to profinite settings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proves that the iterated monodromy group G = IMG(z^2+i) is just-infinite, regular branch, and does not have the congruence subgroup property (Theorem A). The strategy is to analyze the maximal branching subgroup K, establish that K/K' is isomorphic to C4^5 (Theorem B) by combining an L-presentation of G from [23], the Reidemeister algorithm, a parity function on K, and two finite GAP computations (Lemmas 3.3 and 6.2), and then derive the failure of the congruence subgroup property. The paper also computes the rigid, branch, and congruence kernels (Theorem C).

Significance. If correct, the result provides the first example of an iterated monodromy group of a complex polynomial that is just-infinite and lacks the congruence subgroup property, and it gives a counterexample to the converse of Françoeur's theorem that groups with the congruence subgroup property are just-infinite. The overall strategy is coherent and the non-computational parts are carefully argued, including the descent of the parity function and the use of Lemma 2.4 to propagate abelianizations. However, the proof of the no-CSP conclusion depends on two undocumented GAP computations and an unproved base abelianization, so the main claim is currently conditional on those computational verifications.

major comments (2)
  1. [Section 6, Lemma 6.2] The lemma asserts that for n0=5, the inclusion π6(St_G(5)) ≤ π6(K') is verified by GAP, and then concludes that πn(K)^ab ≃ π5(K)^ab ≃ C4^3 for all n ≥ 5, with the relations [π5(t)] = [π5(y)]^2 and [π5(w)] = [π5(z)]^2. The first statement (the GAP-verified inclusion) only feeds Lemma 2.4 to transfer the isomorphism class from level 5 to all higher levels; it does not establish that π5(K)^ab is C4^3. The base case is a separate computational claim that is asserted without proof or description. Since Theorem B and hence the no-CSP part of Theorem A rest on this base abelianization, this is a load-bearing gap. Please provide the GAP code, session output, or a detailed hand proof showing that π5(K)^ab ≃ C4^3 with the stated relations.
  2. [Sections 3 and 6, Lemmas 3.3 and 6.2] The two finite computations used in the proof are not reproducible from the manuscript. Lemma 3.3 states that GAP verifies [G:K] = [π3(G):π3(K)] = 16, and Lemma 6.2 states that GAP verifies π6(St_G(5)) ≤ π6(K'). No code, session log, or table of computed values is provided, even though the paper cites GAP [31]. Because Lemma 6.2 is the linchpin of the argument (it fixes the abelianization of πn(K) for all n ≥ 5), an independent referee cannot check this step without reimplementing the automaton group from scratch. The paper should be revised to include the GAP script and its output, or at least a precise specification of the objects and maps that were checked, so that the computation is fully verifiable.
minor comments (6)
  1. [Section 6, Lemma 6.1] The statement reads "[K : K'] ≤ 45" but from the context it should be 4^5; please correct the typesetting of the exponent.
  2. [Section 4, paragraph after Theorem 4.2] The paper states in the introduction's strategy that it finds an L-presentation for K via the Reidemeister algorithm, but the actual L-presentation is never written out. Although the subsequent proof of the parity descent in Section 5 avoids needing the full presentation, displaying it would make the verification more transparent.
  3. [Section 5, Lemma 5.6] In the proof, the finite verification that each listed word R' has β(R') = 0 is not shown; the values are simply asserted. A short table listing the seven words and their β-values would remove any doubt.
  4. [Section 6, Lemma 6.3] The sentence "In particular, for any α ∈ SG and β ∈ SK, we have e(αβα) = 1" is imprecise because e is not yet known to be well-defined on K at that stage of the paper. It should be phrased more carefully, e.g., "for each generator β in SK".
  5. [Section 6, Proposition 6.4] The list of commutators in the proof contains "[z,t] = 1" twice; the second occurrence should presumably be a different commutator, for instance [w,t].
  6. [Throughout] There are several OCR/formatting artifacts in the abstract and body (e.g., "z2+i", "L SG" not formatted as a free monoid). Please ensure the source compiles correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is independent of its conclusion, with only an incidental self-citation and finite GAP checks that are not fitted predictions.

full rationale

The main theorems are derived from an L-presentation and branching-subgroup structure taken from prior work [23], together with standard branch-group results (Lemma 2.4, Proposition 2.5, Lemma 2.6). The proof shows K/K' is C4^5 by first proving an upper bound from five explicit generators, then ruling out the only possible additional relations using the parity function e and the subgroup K. This is a genuine derivation, not a renaming or a fitted-input prediction. The single self-citation, [11], appears only in the introduction and is not used in any proof, so it is not load-bearing. The two GAP-dependent lemmas (3.3 and 6.2) are finite verifications of concrete inclusions and indices; they are not parameters tuned to force the conclusion, and they are not equivalent to the target theorem by construction. Even though their reproducibility is not documented, that is a correctness or rigor concern, not circularity. No quoted step exhibits Eq. X reducing to Eq. Y by definition, and no cited result is used solely because it is the author's own prior theorem. Accordingly, the circularity score is 0.

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

The central claim rests on the L-presentation and branching structure from [23], plus two GAP computations that are stated but not documented. No new entities or fitted parameters are introduced. The proof is a derivation from these inputs using standard group-theoretic tools.

assumptions (5)
  • standard math The L-presentation of IMG(z^2+i) given in [23, Theorem 3.1] (reproduced here as Theorem 3.4).
    The paper's computation of the presentation of K and the descent of the parity function e assume this presentation is correct.
  • standard math IMG(z^2+i) is regular branch over the maximal branching subgroup K = <[a,b],[b,c]>^G, and K is of finite index 16 in G.
    Quoted from [23, Theorem 2.5] and Lemma 3.3; the regular branch property is the framework for all subsequent arguments.
  • domain assumption The finite index equality [G:K] = [π3(G):π3(K)] = 16, verified by GAP in Lemma 3.3, implying St_G(3) ≤ K.
    Computational claim not backed by shipped code; needed for Lemma 2.3/2.4 and the no-CSP argument.
  • domain assumption π_6(St_G(5)) ≤ π_6(K') as verified by GAP in Lemma 6.2.
    Computational claim not documented; enables Lemma 2.4 to stabilize the abelianization π_n(K)ab for all n ≥ 5.
  • standard math Criterion that a regular branch group G over K with K/K' finite is just-infinite (Proposition 2.5, credited to [1, Prop 3.5]).
    Used to derive just-infiniteness from Lemma 6.1.

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Pith. "Pith review of A just-infinite iterated monodromy group without the congruence subgroup property." pith.science (2026). https://pith.science/paper/6QURZ4CB

@misc{pith2026250519649,
  author       = {Pith},
  title        = {Pith review of: A just-infinite iterated monodromy group without the congruence subgroup property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6QURZ4CB}},
  note         = {Machine review of arXiv:2505.19649}
}
abstract

We prove that the iterated monodromy group of the polynomial $z^2+i$ is just-infinite, regular branch and does not have the congruence subgroup property. This yields the first example of an iterated monodromy group of a polynomial with these properties. Additional information is provided about the congruence kernel, rigid kernel and branch kernel of this group.

Figures

Figures reproduced from arXiv: 2505.19649 by the authors.

Figure 1
Figure 1. Automaton corresponding to IMG(z 2 + i). where the limit runs over all normal subgroups N of finite index in G. Since G is branch, the family of rigid stabilizers is a subset of the family of normal subgroups of finite index of G, and therefore we have a well-defined surjective homomorphism ψ2 : Gb → G. e The kernel of ψ2 is called the branch kernel. Composing the last two maps, we obtain an epimorphism ψ3 = ψ1 ◦ ψ2… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Groups of finite type: classification and structural properties

    math.GR 2025-09 conditional novelty 7.0 of 10

    For many groups of finite type, topological finite generation, just-infiniteness and strong completeness are equivalent, and new algorithms classify the groups up to isomorphism on binary and ternary trees.

Reference graph

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