{"id":"2780e718-0d6c-424c-b8a1-50668683669d","arxiv_id":"2505.23142","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For self-similar level-transitive groups with positive Hausdorff dimension in iterated wreath products, every nontrivial normal closed subgroup has full dimension; in general, counterexamples exist.","lead":"A 20-year-old question about the Hausdorff dimension of normal subgroups in groups acting on rooted trees gets a two-sided answer. The author constructs counterexamples in general and proves a positive result for self-similar groups, with consequences for branch groups and their Hausdorff spectra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The B_n step in the proof of Theorem B is false: for G=W_H with m=2, B_n is one orbit with abelianization of size 2^n, so log|G_n^{B_n}:(G_n^{B_n})'| cannot be bounded by the orbit count #B_n=1 used in the proof.","rationale":"The reader identified the A_n/P_{n-k} equality as the weak point; our analysis shows that this equality is actually defensible when A_n and P_{n-k} are interpreted as vertex unions, because each fully branched G_n-orbit projects bijectively to its parent orbit, so the induced action on A_n is isomorphic to the action on the parent vertices. The more serious failure is the subsequent B_n bound. The proof defines B_n as a set of orbits and proves that the number of such orbits is o(m^n), but (4.2) requires a partition of the vertex set, and (4.1) then bounds the abelianization by the number of vertices. For G=W_H, the single non-full orbit at level n has size m^n, so the vertex count does not tend to zero; using the orbit count gives a false inequality. This is load-bearing because Theorem B is the tool that lets Proposition 5.4 replace commutators of rigid stabilizers by the stabilizers themselves. The claim of Theorem C therefore rests on an invalid proof step. Since the underlying theorem may be repairable along the lines of the original Abért-Virág argument, and since the rest of the paper (Theorem A, Lemmas 5.1 and 5.3) appears sound, a conditional verdict remains appropriate: the central self-similar results should not be accepted until Theorem B is given a correct proof. No judgment is made about the authors; the issue is purely mathematical.","tokens_in":10324,"tokens_out":56545,"duration_ms":543901,"concrete_test":"Evaluate the disputed B_n inequality for G=W_H (the full group of p-adic-like automorphisms), m=2, k=1, at a fixed n>=2. Compute B_n: since O_n has a single orbit, B_n consists of that single orbit. Then check whether log|G_n^{B_n}:(G_n^{B_n})'| <= #B_n holds under the interpretation used in the proof. Under the vertex-union reading, #B_n=2^n and the inequality is true but yields no subexponential decay; under the orbit-set reading, #B_n=1 and the inequality asserts log_2|H_n:H_n'| <= 1, which fails because |H_n:H_n'| = 2^n for H=S_2. Either interpretation invalidates the proof's conclusion that the B_n contribution is o(m^n).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4, A_n and B_n are first defined as subsets of the orbit set O_n, and the proof shows #B_n/m^n -> 0 by counting orbits. However, Equation (4.2) requires A_n and B_n to be G_n-invariant subsets of the vertex set L_n; then (4.1) gives log|G_n^{B_n}:(G_n^{B_n})'| <= |B_n|-1, where |B_n| is the number of vertices, not the number of orbits. If B_n is the set of orbits, the bound log <= #B_n does not control the quantity appearing in (4.2), and the action on the orbit set is trivial. If B_n is the vertex union, the inequality log <= #B_n is false. Concretely, take G=W_H, m=2, k=1, and any n>=2. Then O_n has one orbit, so B_n is the entire nth level; #B_n=1, but for H=S_2 the n-fold wreath product H_n has |H_n:H_n'| = 2^n, so log|G_n^{B_n}:(G_n^{B_n})'| = n, contradicting the bound by #B_n. Thus the B_n contribution is not controlled by the orbit count that tends to zero. Since Theorem C uses Theorem B to replace Rist_G(n)' by Rist_G(n), this gap leaves the main self-similar theorem unsupported. The statement of Theorem B may still be true---Abért and Virág proved it for W_p using a sublinear bound on abelian quotients of transitive p-groups---but the argument given here does not establish it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10727,"tokens_out":21295,"duration_ms":245799,"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":[{"comment":"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.","section":"Section 4, proof of Theorem B"},{"comment":"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.","section":"Section 5.2, Proposition 5.4"}],"minor_comments":[{"comment":"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.","section":"Section 4, after Theorem 4.2"},{"comment":"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.","section":"Section 5.1, Lemma 5.1"},{"comment":"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.","section":"Section 5.2, Proposition 5.4 proof"},{"comment":"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.","section":"Section 1, Corollary 5"},{"comment":"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.","section":"Section 1, Theorem C"}],"recommendation":"major_revision","confidential_remarks":"The gap in Section 4 is substantial and is not a routine repair: the proof appears to conflate sets of orbits with sets of vertices, and the claimed simplification of [2, Theorem 5] is therefore invalid. I would ask for a complete rewrite of the proof of Theorem B, possibly using the sublinear bound from [18], before further consideration. The counterexample part of the paper (Theorem A and the discussion of the branch-structure kernel) seems solid and could be published on its own. The spectrum part of Theorem C is conditional on the author's unpublished preprint [10], which should be disclosed prominently in the introduction and at the point of use."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper falls into two very different halves. Theorem A, the general counterexample to Abert and Virag's question, is clean and convincing: the construction G_K = <H, D_m(K)> is simple, the dimension computation is transparent, and the byproduct about Bartholdi–Grigorchuk–Sunic's Conjecture 3 is a nice bonus. That half deserves credit. The paper also deserves credit for Remark 4.1, which honestly identifies a genuine gap in the 2005 Abert–Virag proof and correctly points to the more recent sublinear bound for abelian quotients of transitive groups. That is good scholarship.\n\nThe trouble is in the proof of Theorem B, the perfectness theorem. The stress-test note is right: A_n and B_n are defined as subsets of the orbit set O_n, but then they are used as subsets of the vertex set L_n when bounding log|G_n^{B_n}:(G_n^{B_n})'|. If B_n is a set of orbits, the action is trivial and the bound tells you nothing about the quantity in (4.2). If B_n is the union of vertices in those orbits, the bound log <= #B_n can be exponentially false; the concrete case G = W_H, m = 2, with one orbit at every level, gives #B_n = 1 but log|G_n^{B_n}:(G_n^{B_n})'| = n. So the sublinearity of #B_n/m^n does not control the abelian quotient, and the proof collapses at that step. Proposition 5.4 also misstates the ambient group: the conclusion should be Hausdorff dimension 1 in G, not in W_H, and the proof itself shows the in-G statement. That is a smaller error, but it matters for the theorem statements.\n\nThe self-similar positive answer (Theorem C) is a plausible and important result, and Lemmas 5.1 and 5.3 are good steps toward it. But since Theorem C relies on Theorem B to replace Rist_G(n)' by Rist_G(n), the gap in Theorem B leaves the main theorem unsupported. The spectrum part also leans on the unpublished preprint [10], so the full weight is on unverified material.\n\nRecommendation: this paper deserves a serious referee, not a desk reject. The counterexample half is solid. The self-similar half may well be true, but the proof needs a real fix — either a correct argument for the B_n contribution or a different approach to perfectness. I would engage with it, and I would ask the author to fix the proof before publication.","headline":"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.","tokens_in":785,"tokens_out":2071,"would_cite":true,"duration_ms":60237,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E08","28A78","20E18"],"pacs":[],"model":"deepseek-v4-flash","headline":"For self-similar positive-dimensional tree groups, nontrivial normal closed subgroups always have Hausdorff dimension 1.","keywords":["Hausdorff dimension","self-similar groups","weakly branch groups","Hausdorff spectra","iterated wreath products","rigid stabilizers","normal subgroups","tree automorphisms"],"falsifier":"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.","tokens_in":10158,"feed_emoji":"🌳","tokens_out":10723,"duration_ms":101032,"temperature":0.7,"pith_summary":"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.","feed_headline":"Self-similar tree groups: normal subgroups have full dimension","feed_subtitle":"For self-similar groups, every nontrivial normal subgroup has Hausdorff dimension 1; the general case has counterexamples.","key_machinery":"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.","core_discovery":"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 $G$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Raises the question and provides the 1-dimensional case and the methods that the paper generalizes.","marker":"[2]"},{"why":"Supplies the strong Hausdorff dimension result for self-similar groups used in Lemma 5.1.","marker":"[9]"},{"why":"Provides the criterion that full-dimensional rigid level stabilizers force the full and normal Hausdorff spectra; Theorem C leans on it.","marker":"[10]"},{"why":"Supplies the fact that a nontrivial normal subgroup contains the commutator of a rigid level stabilizer, used in Proposition 5.4.","marker":"[12]"},{"why":"Supplies the bound $|G:G'|\\leq 2^{\\#X-1}$ that powers the proof of Theorem B.","marker":"[3]"},{"why":"Gives the result that weakly branch closed subgroups satisfy no group law, used in Corollary 5.","marker":"[1]"}],"fun_headline_variants":["Self-similar tree groups: normal subgroups are full dimensional","Positive dimension tree groups: normal subgroups have dimension 1","Self-similar groups: normal subgroups are 1-dimensional","Normal subgroups of self-similar tree groups are 1-dimensional"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Self-similar tree groups: normal subgroups are full dimensional","Positive dimension tree groups: normal subgroups have dimension 1","Self-similar groups: normal subgroups are 1-dimensional","Normal subgroups of self-similar tree groups are 1-dimensional"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001344,"raw_usage":{"total_tokens":5499,"prompt_tokens":1022,"completion_tokens":4477,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":4409}},"tokens_in":638,"tokens_out":4477,"duration_ms":30225,"temperature":1.0,"reasoning_tokens":4409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:55:32.285626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ab´ ert and B","cited_arxiv_id":null,"evidence_quote":"Raises the question and provides the 1-dimensional case and the methods that the paper generalizes."},{"cited_title":"Fari˜ na-Asategui, Restricted Hausdorff spectra ofq-adic automorphisms,Adv","cited_arxiv_id":null,"evidence_quote":"Supplies the strong Hausdorff dimension result for self-similar groups used in Lemma 5.1."},{"cited_title":"On finitely generated Hausdorff spectra of groups acting on rooted trees","cited_arxiv_id":"2504.11948","evidence_quote":"Provides the criterion that full-dimensional rigid level stabilizers force the full and normal Hausdorff spectra; Theorem C leans on it."},{"cited_title":"Garrido and J","cited_arxiv_id":null,"evidence_quote":"Supplies the fact that a nontrivial normal subgroup contains the commutator of a rigid level stabilizer, used in Proposition 5.4."},{"cited_title":"Aschbacherand and R","cited_arxiv_id":null,"evidence_quote":"Supplies the bound $|G:G'|\\leq 2^{\\#X-1}$ that powers the proof of Theorem B."},{"cited_title":"Ab´ ert, Group laws and free subgroups in topological groups,Bull","cited_arxiv_id":null,"evidence_quote":"Gives the result that weakly branch closed subgroups satisfy no group law, used in Corollary 5."}],"review_version":1}