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On a question of Ab\'ert and Vir\'ag

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For self-similar positive-dimensional tree groups, nontrivial normal closed subgroups always have Hausdorff dimension 1.

desk verdict A strong counterexample construction and a genuinely interesting self-similar theorem, but the proof of Theorem B has a concrete gap that leaves the main result unsupported as written. read the letter →

arxiv 2505.23142 v1 pith:6BLTT7UN submitted 2025-05-29 math.GR

Jorge Fariña-Asategui This is my paper
classification math.GR MSC 20E0828A7820E18
keywords Hausdorffdimensionself-similargroupsweaklybranchspectraiteratedwreathproductsrigidstabilizersnormalsubgroupstreeautomorphisms
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 takes up a question from 2005: for a level-transitive closed subgroup of an iterated wreath product with positive Hausdorff dimension, must every nontrivial normal closed subgroup have Hausdorff dimension 1? The paper shows the answer is no in general, by constructing counterexamples that are level-transitive and positive-dimensional but not weakly branch, with a nontrivial normal subgroup of dimension 0. It then shows the answer is yes inside the class of self-similar groups: a closed self-similar level-transitive subgroup with positive Hausdorff dimension is weakly branch, its full Hausdorff spectrum is $[0,1]$, and every nontrivial normal closed subgroup has Hausdorff dimension 1. A supporting theorem, used in the self-similar argument, states that every closed subgroup of an iterated wreath product has the same Hausdorff dimension as its commutator subgroup.

What carries the argument

The machinery has two parts. The construction for the counterexamples is the diagonal embedding $D_m(K)$, the preimage under $\psi$ of the diagonal copy of $K$ in the direct power $W_H^{\times m}$; the group $G_K=\langle H,D_m(K)\rangle$ has dimension $\operatorname{hdim}_{W_H}(K)/m$, is not weakly branch, and has $H$ as a finite normal subgroup of dimension 0. For the positive result, the central objects are the rigid stabilizers $\operatorname{rist}_G(v)$ and $\operatorname{Rist}_G(n)$: the first consists of elements that fix everything outside the subtree at $v$, the second is their product over level-$n$ vertices. Lemma 5.1 proves $\operatorname{hdim}_G(\operatorname{rist}_G(v))\geq m^{-l(v)}$ using self-similarity and the strong Hausdorff dimension of self-similar groups; Lemma 5.3 sums these bounds to get $\operatorname{hdim}_G(\operatorname{Rist}_G(k))=1$. The proof of Theorem B, which upgrades commutators to full dimension, uses the uniform bound $|G:G'|\leq 2^{\#X-1}$ for finite permutation groups to control abelian quotients along orbit partitions.

What would settle it

Take a weakly regular branch self-similar group and compute the Hausdorff dimension of $\operatorname{rist}_G(v)$ for a level-1 vertex; Lemma 5.1 predicts at least $1/m$, so any value strictly below $1/m$ would falsify the proof. Separately, for Theorem B, exhibit a closed $G$ and levels $n>k$ where the abelianization of the action on the fully branched orbits at level $n$ is not equal to the abelianization of the action on the predecessor orbits at level $n-k$; that would break the chain.

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Extended reading notes

Core claim

The central claim is Theorem C: if $G\leq_c W_H$ is closed, self-similar, level-transitive, and has positive Hausdorff dimension in $W_H$, then $G$ is weakly branch; the full Hausdorff spectrum is complete, $\operatorname{hspec}(G)=[0,1]$; and every nontrivial normal closed subgroup $N$ of $G$ has Hausdorff dimension 1 in $G$. In particular, the normal Hausdorff spectrum is $\{0,1\}$. This answers the 2005 question affirmatively for self-similar groups and answers the companion question of whether such groups are weakly branch for all positive dimensions in this class. The negative side is Theorem A: for every $m\geq 2$, every transitive $H\leq \operatorname{Sym}(m)$, and every $\alpha\in[0,1/m]$, there is a level-transitive closed subgroup with strong Hausdorff dimension $\alpha$ that is not weakly branch and has a nontrivial normal closed subgroup of Hausdorff dimension 0. Theorem B, the engine of the positive side, says $\operatorname{hdim}_{W_H}(G)=\operatorname{hdim}_{W_H}(G')$ for every closed subgroup $G$.

Load-bearing premise

The self-similar theorem inherits its force from Theorem B, and Theorem B rests on the assertion that the action on the orbits at level $n$ that branch into the full $m^k$ suborbits is completely determined by the action on the predecessor orbits at level $n-k$; the text does not justify that equality, and if it fails the proof that commutator subgroups have the same Hausdorff dimension, and with it the main theorem, collapses.

Editorial extensions

If this is right

  • If Theorem C is right, every closed self-similar level-transitive subgroup of $W_H$ with positive Hausdorff dimension is weakly branch; in particular, profinite groups with nontrivial center admit no such faithful action, since the presence of a nontrivial center would be incompatible with this structure.
  • The full Hausdorff spectrum of such a group is $[0,1]$: every dimension between 0 and 1 is realized by some closed subgroup of $G$.
  • Every nontrivial normal closed subgroup of such a $G$ has the same Hausdorff dimension as $G$ itself, so normal subgroups cannot have intermediate relative size inside the group.
  • Any self-similar subgroup of $W_H$ that satisfies a group law has Hausdorff dimension zero in $W_H$.
  • Weakly regular branch closed subgroups satisfy Theorem C's hypotheses, so their full and normal Hausdorff spectra are $[0,1]$ and $\{0,1\}$, respectively.

Reading between the lines

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

  • A boundary question the paper leaves open is whether self-similarity can be weakened for dimensions close to 1; the counterexamples only reach dimension $1/m$, and the key lemma genuinely fails without self-similarity when the dimension is below 1.
  • Because Theorem B removes a generation assumption that appeared in earlier work, perfectness of the Hausdorff dimension may be a general feature of profinite groups whose congruence quotients grow exponentially in level; testing it on other level-filtered profinite groups would clarify how much is special to wreath products.
  • The normal-subgroup full-dimension property gives a practical criterion: for a self-similar group, the Hausdorff dimension can be computed from any nontrivial normal closed subgroup, which is often simpler to work with than the whole group.
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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 / 5 minor

Summary. The paper studies two questions of Abért and Virág about Hausdorff dimension in iterated wreath products W_H acting on rooted regular trees. Theorem A constructs, for every transitive H ≤ Sym(m) and every α ∈ [0, 1/m], a level-transitive closed subgroup G ≤ W_H with strong Hausdorff dimension α, a non-trivial normal closed subgroup of Hausdorff dimension 0, and trivial first rigid level stabilizer; this gives negative answers to Questions 1 and 2 in the general setting and yields a counterexample to a conjecture of Bartholdi, Grigorchuk and Šunić about the kernel of the canonical branch action. The rest of the paper aims at positive answers within the self-similar class. Theorem B asserts that hdim_{W_H}(G) = hdim_{W_H}(G') for every closed G ≤ W_H, and Theorem C asserts that every closed self-similar level-transitive G ≤ W_H with positive Hausdorff dimension is weakly branch, has full Hausdorff spectrum [0, 1], and has every non-trivial normal closed subgroup of Hausdorff dimension 1 in G. The proofs of Theorems B and C are the core of the paper.

Significance. If established, the results would be substantial. Theorem A is a clean construction that answers the general versions of Abért and Virág's two questions and disproves the Bartholdi–Grigorchuk–Šunić conjecture on the kernel of the canonical branch action; the diagonal-extension construction and its dimension computation are convincing. Lemmas 5.1 and 5.3, showing that rigid stabilizers of self-similar positive-dimensional groups have full dimension in G, are elegant and potentially useful independently. The paper also performs a service by identifying a gap in the original proof of [2, Theorem 5] and by pointing to the recent sublinear bound of [18]. However, the validity of the main positive result currently depends on the proof of Theorem B, and that proof contains a load-bearing gap. Theorem C is therefore not established as it stands; the counterexample part of the paper appears sound.

major comments (2)
  1. [Section 4, proof of Theorem B] The sets A_n and B_n are defined as subsets of O_n, the set of G_n-orbits on L_n, but they are then used as if they were G_n-invariant subsets of L_n. Equation (4.2) is valid only for a decomposition of the set on which G acts, and the bound log|G_n^{B_n}:(G_n^{B_n})'| ≤ #B_n is the bound (4.1) for a permutation group on a set of size #B_n. If B_n is the set of orbits, then the action on B_n is trivial, so this quantity does not control the contribution of the action on the corresponding vertices to |G_n:G_n'|. If B_n is the union of those vertices, then the earlier proof only gives #B_n/m^n → 0 for the number of orbits, not for the number of vertices, and for the level-transitive group G = W_H with m = 2 the vertex union is the whole level, so #B_n = 2^n and the limit #B_n/m^n → 0 is false. In the orbit reading, the same example gives #B_n = 1 while log|G_n:G_n'| = n, so the bound by #B_n is false. The claim that the action of G_n on A_n is completely determined by the action of G_{n-k} on the predecessor orbit set P_{n-k} is also not justified: the permutation action on a set of predecessor orbits is trivial, and movement of descendants depends on the sections of elements of G_n. Hence the inequality log|G_n:G_n'|/m^n ≤ 1/m^k + #B_n/m^n and the conclusion lim log|G_n:G_n'|/m^n = 0 are not established. Since Proposition 5.4 and Theorem C use Theorem B to replace Rist_G(n)' by Rist_G(n), this gap also leaves the main self-similar theorem unsupported.
  2. [Section 5.2, Proposition 5.4] Proposition 5.4 states that every non-trivial normal closed subgroup N has Hausdorff dimension 1 in W_H. The proof, however, only establishes hdim_G(Rist_G(n)) = 1 and hence hdim_G(N) ≥ 1, which gives hdim_G(N) = 1. If d = hdim_{W_H}(G) < 1, then Lemma 2.1 gives hdim_{W_H}(N) = d · 1 = d, not 1. The statement should read 'dimension 1 in G', as it does in Theorem C(iii); in the present form the proposition is false for d < 1 and should be corrected.
minor comments (5)
  1. [Section 4, after Theorem 4.2] The sentence 'by taking logarithms in the inequality in Proposition 4.2' refers to Theorem 4.2, not Proposition 4.2; the cross-reference should be corrected.
  2. [Section 5.1, Lemma 5.1] In the lower bound for |ker ψ_¬v|, the replacement of |W_H:St_{W_H}(n)| by |W_H:St_{W_H}(n-k)|^{m^k} ignores a constant factor depending on k and H. This can be absorbed by enlarging N_ε and shrinking ε, so the argument is repairable, but the displayed inequality is not literally correct as written.
  3. [Section 5.2, Proposition 5.4 proof] The proof refers to 'Proposition 2.1' for the equality hdim_G(Rist_G(n)') = hdim_G(Rist_G(n)); the intended reference is Lemma 2.1, and the equality requires both Theorem B and that lemma.
  4. [Section 1, Corollary 5] Corollary 5 is stated for all self-similar G ≤ W_H without the level-transitive hypothesis present in Theorem C. If a reduction to the level-transitive case is intended, that reduction should be stated and proved; as written the corollary does not follow directly from Theorem C.
  5. [Section 1, Theorem C] Part (ii) of Theorem C depends on the unpublished preprint [10, Theorem 3.5]. The paper cites this preprint, but the theorem statement presents (ii) as unconditional; the dependence should be stated explicitly at the point where Theorem 5.5 is invoked.

Circularity Check

1 steps flagged · score 4.0 of 10

Self-similar branch theorem is mostly independent, but the Hausdorff-spectrum conclusions reduce to the author's own unpublished preprint [10].

  1. self citation load bearing [Section 5.2, 'Proof of the main results', Theorem 5.5 and Theorem C]
    "Theorem 5.5 (see [10, Theorem 3.5]). Let G ≤_c W_H be a closed and level-transitive subgroup such that for all n ≥ 1 we have hdim_G(Rist_G(n)) = 1. Then the Hausdorff spectrum of G is given by hspec(G) = [0,1], and the normal Hausdorff spectrum of G is given by hspec⊴(G) = {0,1}.... Proof of Theorem C. The result follows from Lemmata 5.1 and 5.3, Proposition 5.4 and Proposition 5.5."

    The two spectral conclusions of Theorem C, namely hspec(G)=[0,1] and hspec⊴(G)={0,1}, are not derived in this paper; they are exactly [10, Theorem 3.5] with the hypothesis hdim_G(Rist_G(n))=1 verified by the author's own Lemmas 5.1 and 5.3. Reference [10] is an unpublished preprint by the same authors (Fariña-Asategui, Garaialde Ocaña, Uria-Albizuri). The only new ingredient added is Theorem B, used to 'drop the assumption (ii) in [10, Theorem 3.5]', but the spectrum statement itself is imported from the self-citation. Thus the headline spectral part of Theorem C is load-bearingly self-citational rather than independently derived here, even though the weakly-branch and normal-subgroup-dimension parts carry independent proof content.

full rationale

The derivation chain for Theorem C starts with Lemma 5.1, which uses the author's published [9] (strong Hausdorff dimension for self-similar groups) together with an epsilon argument to prove hdim_G(rist_G(v)) ≥ m^{-l(v)}. This is independent content: [9] is a published prior theorem and the epsilon computation does not presuppose weak branchness or the spectra. Lemma 5.3 sums the rigid-stabilizer dimensions; its equality log|Rist_G(k):St(n)| = Σ_v log|rist_G(v):St(n)| is exact by the direct-product structure of Rist_G(k), not an input assumption. Proposition 5.4 combines these with Theorem B (proved in Section 4) and the standard normal-subgroup fact [12, Lemma 4]; replacing Rist_G(n)' by Rist_G(n) via Theorem B is legitimate if Theorem B's proof is valid, but that is a correctness question, not a circularity. The only load-bearing self-citation is Theorem 5.5, which is taken essentially verbatim from the author's own unpublished preprint [10]; the paper only checks the hypothesis using its lemmas. There is also a secondary dependence on [10, Lemma 2.4] for a dimension-transfer step in Proposition 5.4, but that is auxiliary. No fitted-parameter or prediction-by-construction pattern occurs anywhere in the paper, and no quantity is defined in terms of the target conclusion. A separate mathematical gap exists in the proof of Theorem B, where the equality between the abelianization of the A_n-action and that of the P_{n-k}-action is asserted without proof and appears false for W_H with m=2; this is a correctness risk rather than a circularity, so it does not raise the circularity score on its own. Overall, the central self-similar answer to the Abert-Virag question has substantial independent content, but the spectral claims rely on the authors' prior unpublished work, giving partial circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new entities. Its results rest on a chain of externally cited theorems, two from the author's own prior work, one of which ([10]) is an unpublished preprint. No target conclusion is assumed as an axiom.

assumptions (6)
  • standard math Lemma 2.1 (product formula for Hausdorff dimensions in a filtration; [10, Lemma 2.4])
    Used in Section 5 to translate dimensions between W_H and a subgroup G; taken as background.
  • domain assumption Self-similar subgroups of W_H have strong Hausdorff dimension ([9, Theorem B])
    Invoked in Lemma 5.1 and elsewhere; it is the author's own earlier theorem and the main reason self-similarity enters.
  • domain assumption Closed subgroups K of W_H realizing every strong Hausdorff dimension in [0,1] exist ([9, Theorem A])
    Needed for the full range of dimensions in Theorem A.
  • standard math For a finite permutation group on X, |G:G'| ≤ 2^{#X-1} ([3, Theorem 2])
    Used repeatedly in the proof of Theorem B to bound abelian quotients; the paper relies on the stated generality.
  • domain assumption Every nontrivial normal subgroup of a weakly branch group contains the commutator subgroup of some rigid level stabilizer ([12, Lemma 4])
    Essential in Proposition 5.4 to lower-bound the dimension of an arbitrary normal subgroup by a rigid stabilizer.
  • domain assumption If hdim_G(Rist_G(n))=1 for all n, then the Hausdorff spectra are hspec(G)=[0,1] and hspec_normal(G)={0,1} ([10, Theorem 3.5])
    Gives the spectrum conclusions of Theorem C; the preprint is not yet published.

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Pith. "Pith review of On a question of Ab\'ert and Vir\'ag." pith.science (2026). https://pith.science/paper/6BLTT7UN

@misc{pith2026250523142,
  author       = {Pith},
  title        = {Pith review of: On a question of Ab\'ert and Vir\'ag},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BLTT7UN}},
  note         = {Machine review of arXiv:2505.23142}
}
abstract

Ab\'ert and Vir\'ag proved in 2005 that the Hausdorff dimension of a non-trivial normal subgroup of a level-transitive 1-dimensional subgroup of the group of $p$-adic automorphisms $W_p$ is always 1. They further asked whether the same holds replacing 1-dimensional with positive dimensional. On the one hand, we provide a negative answer in general by giving counterexamples where the non-trivial normal subgroups are not all 1-dimensional. Furthermore, these counterexamples are pro-$p$ subgroups of $W_p$ with positive Hausdorff dimension in $W_p$ but with non-trivial center, and thus not weakly branch. On the other hand, we restrict ourselves to the class of self-similar groups and answer the question of Ab\'ert and Vir\'ag in the positive in this case. Along the way, we generalize a result of Ab\'ert and Vir\'ag on the closed subgroups of $W_p$ being perfect in the sense of Hausdorff dimension to closed subgroups of any iterated wreath product $W_H$ and show that self-similar positive-dimensional subgroups of $W_H$ do not satisfy any group law.

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Forward citations

Cited by 2 Pith papers

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

  1. Weakly branch actions: first-order theory, rigidity and Boston's conjecture

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    Just-infinite branch pro-p groups introduced earlier by the author are shown to be rigid on trees obtained by deleting levels, forcing every branch action on the p-adic tree to be zero-dimensional and disproving Bosto...

  2. Arboreal representations of linear groups

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    The author proves that positive-dimensional level-transitive subgroups of bounded-tree automorphism groups are not linear over an integral domain, which is a restricted version of the claimed theorem.

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