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

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Pith's one-line read The paper seeks to disprove Boston's conjecture by showing that just-infinite branch pro-$p$ groups $G_n$ admit only zero-dimensional branch actions on the $p$-adic tree.

desk verdict A substantial paper that likely kills Boston's 25-year-old conjecture, but the disproof leans on unstated results from the author's earlier paper and on one uncited structural fact; the referee should check those before signing off. read the letter →

arxiv 2507.22507 v1 pith:C4X7TY5G submitted 2025-07-30 math.GR

Jorge Fariña-Asategui This is my paper · ORCID
classification math.GR MSC 03C0720E0828A7805C6320F6537F10
keywords Boston'sconjecturebranchgroupsweaklyactionsrigidityHausdorffdimensionstructuregraphfirst-ordertheoryp-adicautomorphisms
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 claims that Boston's conjecture from 2000 is false: a just-infinite branch pro-$p$ group need not embed into the group of $p$-adic automorphisms with positive Hausdorff dimension. The counterexamples are the groups $G_n$ constructed in the author's earlier work, whose closures in $W_p$ are just-infinite branch pro-$p$ groups with Hausdorff dimension zero. The paper's main new step is to show that each $G_n$ has a rigid action on a tree $T_n$ obtained from the $p$-adic tree by deleting levels, and that this rigidity forces every branch action of $G_n$ on the $p$-adic tree to have zero Hausdorff dimension. Along the way it generalizes the structure graph to weakly branch groups, proves first-order definability of the congruence topology and the structure graph, and gives an easy rigidity criterion for fractal groups of $p$-adic automorphisms. A reader should care because Boston's conjecture was a basic structural question about just-infinite branch pro-$p$ groups, and the counterexamples show that positive dimension is a property of embeddings, not of the abstract group.

What carries the argument

The central object is the structure graph of a weakly branch group, a graph whose vertices are equivalence classes of basal subgroups, where two basal subgroups are equivalent if they have non-trivial intersection and equal normalizers. A basal subgroup is one with finitely many conjugates whose normal closure is the direct product of its distinct conjugates; rigid vertex stabilizers are the standard examples. The structure graph encodes all weakly branch actions of a group on spherically homogeneous rooted trees. Rigidity is characterized through $T$-filtrations: a weakly branch group is $T$-rigid exactly when the family $\mathcal{F}_T$ of subgroups appearing in $T$-filtrations coincides with the set of vertex stabilizers of the action. The disproof of Boston's conjecture applies this characterization to prove $T_n$-rigidity of $G_n$; the proof is carried by a dichotomy that forces any finite-index subgroup of $G_n$ containing a deep vertex stabilizer to contain a level stabilizer.

What would settle it

Exhibit a branch action $\chi: G_n \to W_p$ with positive Hausdorff dimension, or equivalently a finite-index subgroup $H \leq G_n$ with $|G_n:H| \leq p^{\ell_n}$ that contains a vertex stabilizer at level $t_n^2$ of $T_p$ but not the level stabilizer $\mathrm{St}_{\rho_n}(t_n^1)$. Lemma 6.3 rules out such $H$, and this dichotomy is the step that transfers rigidity into the index equalities that force dimension zero.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that positive Hausdorff dimension is not an intrinsic feature of just-infinite branch pro-$p$ groups. For every $n \geq 1$, the closure in $W_p$ of the just-infinite branch group $G_n$ is a counterexample to Boston's conjecture: it is branch and just-infinite, and every branch action of $G_n$ on the $p$-adic tree has Hausdorff dimension zero. The proof establishes $T_n$-rigidity of $G_n$, meaning the induced action on the deleted-level tree $T_n$ is unique up to conjugation, and then compares level-stabilizer indices along the deleted levels $t_n^k$: rigidity makes the index sequence of any branch action on $T_p$ coincide with that of the original action, whose Hausdorff dimension was already known to be zero. Hence no embedding of $G_n$ into $W_p$ with positive Hausdorff dimension exists.

Load-bearing premise

The disproof rests on the author's earlier results that the groups $G_n$ are just-infinite branch pro-$p$ groups and that their closures in $W_p$ have Hausdorff dimension zero, together with the formula computing Hausdorff dimension from level-stabilizer indices; if any of those imported results fails, the counterexample to Boston's conjecture collapses.

Editorial extensions

If this is right

  • Boston's conjecture is false: there exist just-infinite branch pro-$p$ groups with no positive-dimensional embedding into $W_p$.
  • Every branch action of $G_n$ on the $p$-adic tree is zero-dimensional, so zero Hausdorff dimension is a feature of the abstract group, not merely of one chosen embedding.
  • The congruence topology and the structure graph are first-order definable in any weakly branch group, and level stabilizers are first-order definable whenever the action is rigid.
  • A fractal weakly branch group $G \leq W_p$ is $T_p$-rigid exactly when $G/\mathrm{St}_G(2) \ncong C_p \times C_p$; in particular, Hausdorff dimension larger than $1/p$ forces rigidity, and the threshold is sharp.
  • A weakly branch group is $T$-rigid exactly when its congruence completion is $T$-rigid, so rigidity passes between a group and its profinite completion for these actions.

Reading between the lines

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

  • Going beyond the paper: if the answer to its Question 8 is yes, then every weakly branch group would have some rigid action on a deleted-level tree, turning the proof into a general recipe for reducing questions about arbitrary branch actions to a single rigid action.
  • Going beyond the paper: the rigidity-versus-dimension threshold at $1/p$ suggests that Hausdorff dimension could serve as a finer invariant separating non-isomorphic branch groups; the paper gives one such separation, and the sharp example at the threshold indicates the boundary is exactly where non-rigidity lives.
  • Going beyond the paper: the first-order definability results imply that the model-theoretic type of a weakly branch group records its structure graph and, in the rigid case, its level-stabilizer filtration; this may allow elementary equivalence to detect dynamical data such as monodromy actions, a direction the paper opens but does not develop.
  • Going beyond the paper: the counterexamples do not obviously disturb the arithmetic motivation behind Boston's conjecture, because the paper recalls that positive-dimensional subgroups of $W_p$ still satisfy the generalized $p$-adic representation expectation; a testable next step is whether deleted-level rigidity can be used to force zero-dimensionality in the Galois representations arising in th
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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

3 major / 6 minor

Summary. The paper develops tools for studying weakly branch actions on spherically homogeneous rooted trees: it extends Wilson's structure graph to weakly branch groups, proves first-order definability of the congruence topology and the structure graph, gives a characterization of rigidity in terms of T-filtrations, and shows equivalence between rigidity of a group and its congruence completion. For fractal groups of p-adic automorphisms, it gives a complete rigidity criterion and relates rigidity to Hausdorff dimension. The main application is a disproof of Boston's conjecture: the author claims that the closures of the just-infinite branch groups G_n constructed in his earlier work admit no positive-dimensional embedding into W_p, using a newly proved rigidity theorem for actions on trees obtained by deletion of levels.

Significance. If the main theorems are correct, the paper resolves a 25-year-old conjecture and provides substantial new structural tools. The extension of the structure graph to weakly branch groups, the rigidity criterion in Theorem 3, the fractal characterization in Theorem 5, and the first connection between Hausdorff dimension and rigidity (Corollary 6) are significant contributions. The proof of Theorem 7, if fully justified, would show that the zero-dimensional just-infinite branch pro-p groups constructed in [20] indeed violate Boston's conjecture. The manuscript is generally careful and contains detailed arguments, and the new rigidity arguments in Section 6 appear coherent. However, the disproof rests on one unsupported assertion about non-branch embeddings and on several implicit technical steps that need clarification before the central claim can be accepted.

major comments (3)
  1. [Section 1 and Section 6.4] The disproof of Boston's conjecture requires showing that every embedding of G_n into W_p has zero-dimensional closure, not only those whose images are branch. The manuscript states in Section 1 that 'non-branch just-infinite pro-p subgroups of Wp are known to be zero-dimensional', but no reference or proof is given. This assertion is load-bearing: Theorem 7 and Eq. (6.5) only control branch actions, and if a non-branch just-infinite pro-p subgroup of W_p admitted a positive-dimensional embedding, the conclusion that each G_n is a counterexample would not follow. Please provide a precise citation or a proof of this assertion.
  2. [Theorem 6.4 proof] The step 'If Hu ≥ stτn+|u|(w) ... then H ≥ stτn(uw)' is stated to follow from Lemma 5.1 and Lemma 5.2, but the implication is not immediate. One needs a kernel argument: the kernel of the section map φ_u on St(u) is contained in St(w_k) because w_k is below u, so a lift of a section in St(w) can be adjusted by an element of H ∩ St(w_k) to obtain the full preimage in H ∩ St(uw). This argument is not written and is essential for the induction producing FTn = Sτn. Please expand this step.
  3. [Lemma 6.3 proof] The equality stρn(w) = ker φ_i|Stρn(tn1) relies on the assertion that every non-trivial element of A_{n+1} moves every vertex at level t_{n+1}^1. This is stated without proof or citation. The equality is used to derive Eq. (6.3), which is essential for the dichotomy and the subsequent index computation. Please justify this claim directly from the definition of A_{n+1} or supply a reference.
minor comments (6)
  1. [Page 1] The running title contains a typo: 'WEAKL Y BRANCH' should read 'WEAKLY BRANCH'.
  2. [References] Reference [36] spells 'Apects'; it should be 'Aspects'.
  3. [Proof of Theorem 4] The line 'ristH(v) = H ∩ rist(v) ≤ H ∩ rist(v) = ristH(v)' is garbled, presumably missing overlines; please rewrite the closure argument clearly.
  4. [Proof of Theorem 7] The notation hdim_{Wp}(χn(Gn)) is used even though χn(Gn) may not be closed; the Hausdorff dimension is defined for closed subgroups. Please write hdim_{Wp}(\overline{χn(Gn)}) or clarify that the same formula computes the dimension of the closure.
  5. [Lemma 6.2] The notation eAn for \tilde A_n is confusing, especially because H1 is initially defined as a subgroup of G_n/St(tn1) = A_n and later used as a subgroup of \tilde A_n. Please spell out the identification H1 ≤ \tilde A_n explicitly and use \tilde A_n consistently.
  6. [Section 5.4] In the definition of s_n(G), the expression 'p log_p |StG(n − 1) : StG(n)| − log_p |StG(n) : StG(n + 1)|' should be parenthesized as p·log_p(...) to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step found: Theorem 7's reduction via Eq. (6.5) is a genuine rigidity argument; reliance on [20] is independent prior work, and the uncited 'known' claim on non-branch embeddings is a correctness gap, not a circularity.

full rationale

Walking the derivation chain, the central disproof (Theorem 7) is not circular in the rubric sense. The new content is Theorem 6.4, Tn-rigidity of Gn, proved from Proposition 6.1 and Lemmas 6.2–6.3; the final step compares an arbitrary branch action χn on Tp with the distinguished action ρn via the induced action on Tn. Equation (6.5) is a consequence of conjugation in Aut Tn, not a definitional identity, and the inequality hdimWp(χn(Gn)) ≤ liminf along the deleted levels is a standard subsequence argument. The facts that Gn is just-infinite branch and that the distinguished embedding has zero Hausdorff dimension are imported from the author's prior paper [20, Props. 6.11, 6.12], as is the generating-function formula Theorem 5.6; these are parameter-free published theorems with stated assumptions, so under the rubric they are independent support, not circularity. I do flag, as a non-circular correctness gap, the introduction's uncited assertion 'Since non-branch just-infinite pro-p subgroups of Wp are known to be zero-dimensional', on which the treatment of non-branch embeddings in Theorem 7 depends; this is an omitted proof/reference, not a reduction of the conclusion to the assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters fitted to data and no invented entities. The main axioms are standard mathematics plus a set of load-bearing theorems imported from the author's prior work [20] and from the classification of just-infinite pro-p groups.

assumptions (4)
  • domain assumption The groups G_n from [20] are just-infinite branch pro-p groups whose closure in W_p has Hausdorff dimension zero (Prop 6.1 and 6.12 of [20]).
    Theorem 7 (the disproof of Boston's conjecture) uses these groups as counterexamples; the properties are imported from the author's prior paper [20], not re-proved in this manuscript.
  • domain assumption The Hausdorff dimension generating function result: for self-similar G ≤ W_p, hdim_Wp(G) = 1 - S_G(1/p), with s_n(G) non-negative (Theorem B of [20]).
    Used in the proof of Corollary 6 and Remark 5.7; cited from [20].
  • domain assumption Non-branch just-infinite pro-p subgroups of W_p have zero Hausdorff dimension.
    Invoked in the introduction to reduce the disproof to branch actions; credited to the classification literature [2,30].
  • standard math Standard ZFC and group theory; basic properties of profinite groups, trees, Hausdorff dimension, and first-order logic.
    Background for all sections.

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Pith. "Pith review of Weakly branch actions: first-order theory, rigidity and Boston's conjecture." pith.science (2026). https://pith.science/paper/C4X7TY5G

@misc{pith2026250722507,
  author       = {Pith},
  title        = {Pith review of: Weakly branch actions: first-order theory, rigidity and Boston's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C4X7TY5G}},
  note         = {Machine review of arXiv:2507.22507}
}
abstract

We disprove a well-known conjecture of Boston (2000), which claims that a just-infinite pro-$p$ group is branch if and only if it admits a positive-dimensional embedding in the group of $p$-adic automorphisms. This is obtained as a result of a comprehensive study of the rigidity of branch actions. Firstly, we generalize the notion of the structure graph, introduced by Wilson in 2000, to weakly branch groups and use it to prove several results on the first-order theory of weakly branch groups, extending previous results of Wilson on branch groups. Secondly, we completely characterize the rigidity of weakly branch and branch actions on arbitrary spherically homogeneous rooted trees, extending previous partial results (for branch actions) by Hardy, Garrido, Grigorchuk and Wilson. Moreover, we prove that rigidity of a weakly branch group is equivalent to rigidity of its closure in the full automorphism group. Thirdly, we extend greatly the sufficient conditions $(*)$ and $(**)$ of Grigorchuk and Wilson, which leads to a complete and very easy-to-check characterization of the rigidity of the weakly branch actions of a fractal group of $p$-adic automorphisms. We further establish the first connection in the literature between the Hausdorff dimension of a weakly branch action and its rigidity. Lastly, we put everything together to show that the zero-dimensional just-infinite branch pro-$p$ groups introduced recently by the author admit rigid branch actions on a tree obtained by deletion of levels. This, together with previous results of the author, shows that these groups are indeed counterexamples to the aforementioned conjecture of Boston.

Figures

Figures reproduced from arXiv: 2507.22507 by the authors.

Figure 1
Figure 1. An illustration of the proof of Theorem 6.4. At each step, we apply Lemma 6.3 to climb one level up in Tn and obtain a larger vertex stabilizer inside the subgroup H. 6.4. Disproval of Boston’s conjecture. As remarked in [20], to prove Theo￾rem 7, it is enough to show that every branch action of Gn on Tp (as a subgroup of Wp) is zero-dimensional. We conclude the paper by using the Tn-rigidity of the group Gn to prov… view at source ↗

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Forward citations

Cited by 2 Pith papers

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  1. Automorphism-invariant refinements of weakly branch actions via overlap functions

    math.GR 2026-07 accept novelty 7.0 of 10

    For every finitely generated weakly branch action on a rooted tree there is a canonical refinement tree on which the automorphism group acts weakly branch and equals the normalizer of the group.

  2. 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.

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