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Arboreal representations of linear groups

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Pith's one-line read Linear groups over any integral domain have strong zero Hausdorff dimension in every embedding into the automorphism group of a bounded rooted tree.

desk verdict A serious and partially successful attack on the Abért-Virág conjecture, but Theorem A is not proven: the reduction to level-transitive case is missing and the lifting step has a false subdirect-product claim. read the letter →

arxiv 2506.12745 v1 pith:KB4PXP46 submitted 2025-06-15 math.GR

Jorge Fariña-Asategui This is my paper · ORCID
classification math.GR MSC 20E0828A7805C2520F6520G2520E18
keywords Hausdorffdimensionarborealrepresentationslineargroupsoverintegraldomainsweaklybranchnon-commutingofgraphsboundedrootedtreesgrouplawspro-p
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 to settle the 2005 conjecture that every embedding of a linear group over a pro-$p$ domain into the group of $p$-adic tree automorphisms has zero Hausdorff dimension. It proves the stronger statement that over any integral domain, every embedding of a linear group into the automorphism group of a bounded rooted tree has strong zero Hausdorff dimension. The result matters because Hausdorff dimension measures how much of the full tree automorphism group a subgroup occupies; the paper shows that linear groups are too constrained to occupy a positive share. It also yields corollaries: groups satisfying a law have no positive-dimensional arboreal representations on bounded trees, and positive-dimensional just-infinite subgroups of tree automorphism groups can only have finite images in linear groups.

What carries the argument

The proof rests on three tools. First, Theorem 2.1: in a bounded tree, a level-transitive closed subgroup with positive upper-box dimension has all sufficiently deep vertex projections weakly branch, where weakly branch means a level-transitive action in which the rigid stabilizer of every vertex is nontrivial; boundedness enters through a product of factors $(M-1)/M$ along an infinite path. Second, non-commuting representations of graphs: for the graph $V_n$ consisting of $n$ disjoint edges, a non-commuting representation in $\mathrm{GL}_k(R)$ yields $n$ linearly independent $k \times k$ matrices over $R$, so $n \le k^2$, meaning linear groups over integral domains cannot represent $V_n$ for $n > k^2$. Third, the lifting lemma: from a subdirect product inside a product of conjugates of a subgroup, a nontrivial normal piece in one coordinate extends uniquely to lifts in the other coordinates, so a non-commuting representation in a weakly branch projection lifts to one in the original subgroup. The contradiction then forces zero Hausdorff dimension.

What would settle it

Search for a bounded rooted tree $T$ and an injection of a linear group $G \le \mathrm{GL}_n(R)$, with $R$ an integral domain, into $\mathrm{Aut}\,T$ whose image has positive lower-box dimension, equivalently positive Hausdorff dimension; the theorem predicts none exists. Since the supplied proof begins by assuming the image is level-transitive, the most informative test is a positive-dimensional non-level-transitive action: if any such action is linear over an integral domain, Theorem A as stated is false, and if all such actions reduce to level-transitive ones, the missing reduction would be the place to look.

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Extended reading notes

Core claim

The central claim is Theorem A: if $R$ is an integral domain, $G \le \mathrm{GL}_n(R)$, and $T$ is a bounded rooted tree, then any embedding of $G$ into $\mathrm{Aut}\,T$ has strong zero Hausdorff dimension. The proof works by showing that positive upper-box dimension is incompatible with linearity over an integral domain. A positive-dimensional level-transitive subgroup of $\mathrm{Aut}\,T$ has almost all of its vertex projections weakly branch; weakly branch groups contain non-abelian direct products, giving non-commuting representations of arbitrarily large graphs. But a subgroup of $\mathrm{GL}_k(R)$ can admit such representations only for $n \le k^2$, by a linear-independence argument over an integral domain. A lifting lemma transfers a non-commuting representation from a deep projection back to the original group, producing the contradiction that forces zero dimension.

Load-bearing premise

The proof assumes the acting subgroup of $\mathrm{Aut}\,T$ is level-transitive before applying Theorem 2.1, and it does not show that an arbitrary positive-dimensional embedding can be reduced to a level-transitive one; the theorem as stated for every embedding rests on that missing reduction.

Editorial extensions

If this is right

  • If Theorem A is correct, the 2005 conjecture follows as the special case of a pro-$p$ domain and the $p$-adic tree: every embedding of a linear group into the $p$-adic automorphism group has zero Hausdorff dimension.
  • Corollary 2 rules out positive-dimensional arboreal representations for every topological group satisfying a law, including all solvable-by-finite linear groups.
  • Corollary 3 says a positive-dimensional just-infinite closed subgroup of a bounded-tree automorphism group cannot have an infinite linear representation over an integral domain.
  • Consequently, any positive-dimensional arboreal representation on a bounded tree must be supported by a group that is not linear over any integral domain and does not satisfy a law, consistent with weakly branch examples.

Reading between the lines

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

  • The method suggests a testable extension to trees with unbounded branching: the proof's product bound uses boundedness to force dimension decay, so one could check whether slowly growing branching allows positive-dimensional linear subgroups.
  • The lifting lemma isolates a general mechanism: any subgroup of $\mathrm{Aut}\,T$ with almost all deep projections weakly branch inherits arbitrarily large non-commuting graph representations, so its non-commutativity is uniform in a way that may be measurable by the growth of the minimal matrix size.
  • If a missing reduction to the level-transitive case can be supplied, the same three-step argument would also rule out positive-dimensional embeddings of linear groups into automorphism groups of spherically homogeneous trees that are only locally bounded, broadening the class of trees covered.
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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 / 3 minor

Summary. The paper claims a proof of a generalization of the Abért–Virág conjecture: for every integral domain R, every linear group G ≤ GL_n(R), and every bounded rooted tree T, any embedding of G into Aut T has image with strong zero Hausdorff dimension. The proof strategy has three ingredients: Theorem 2.1 shows that positive-dimensional level-transitive closed subgroups of Aut T have weakly branch projections from some level on; Lemma 3.2 gives a lower bound on the matrix size needed to realize the non-commuting graph V_n over an integral domain; and Lemma 3.3 is a lifting lemma intended to transfer non-commuting representations from weakly branch projections up to the original group. These are combined in Section 3.3 to show that any positive-dimensional level-transitive closed subgroup of Aut T contains non-commuting representations of V_n for all n, contradicting linearity. Corollaries are drawn for groups satisfying a law and for just-infinite arboreal groups. The paper is clearly organized and several auxiliary results are potentially useful, but the proof of Theorem A contains two load-bearing gaps.

Significance. If the main theorem were valid, it would settle the 2005 Abért–Virág conjecture, extend it to arbitrary integral domains and bounded trees, and connect to the Fontaine–Mazur and Boston conjectures. Theorem 2.1 is a promising structural statement about positive-dimensional level-transitive groups, and Lemma 3.2 is a clean linear-algebra argument. However, the significance is conditional: the proof as written does not establish the theorem for arbitrary embeddings, and the central lifting step in Section 3.3 is based on a false containment assertion. The paper therefore does not currently deliver its advertised main result.

major comments (2)
  1. [§3.3, proof of Theorem A; Corollary 2] The proof of Theorem A begins with 'Let G ≤c Aut T be a level-transitive closed subgroup', but the theorem is stated for every embedding of a linear group into Aut T. No reduction from an arbitrary closed image to a level-transitive subgroup is given, and such a reduction is not automatic: a positive-dimensional subgroup can be supported on a single branch of T, fixing all other branches, and this subgroup is not level-transitive. Since Theorem 2.1 has level-transitivity as a hypothesis, the argument does not cover this case. The same gap appears in the proof of Corollary 2, where ρ(G) need not be level-transitive yet Theorem 2.1 is invoked without comment.
  2. [§3.3, paragraph defining L and h_i] The assertion 'H ≤ L × L^{h_1} × ... × L^{h_k} is a subdirect product' is false in general. For h ∈ H, membership in the product would require that each section h|_{v_i} be realizable by an element of H that fixes all vertices outside T_{v_i}; this is not implied by level-transitivity or by weak branchness of the projections. For example, let B be a weakly branch group of positive upper-box dimension on the binary tree and let G = ⟨(g,g) : g ∈ B, τ⟩, where τ swaps the two children of the root. Then G is closed, level-transitive and has positive upper-box dimension, but St_G(1) is the diagonal copy of B, and the product L × L^{τ} is contained in the intersection of H with the image of the first-level rigid stabilizer, which is a proper subgroup of H for any non-trivial weakly branch B. Hence H is not contained in the product and Lemma 3.3 cannot be applied. This invalidates the proof even in the level-transitive case.
minor comments (3)
  1. [§3.2 heading] The heading 'Lifing non-commuting representations' contains a typo; it should read 'Lifting non-commuting representations'.
  2. [§3.3 proof of Theorem A] The letter N is used both for the level supplied by Theorem 2.1 and for the normal subgroup produced by Lemma 3.3; this makes Equation (3.1) difficult to parse. Rename one of these objects, for example use M for the normal subgroup.
  3. [§2.3, Corollary 3] Corollary 3 is described as an immediate corollary to Theorem A, but since Theorem A is stated for embeddings while Corollary 3 concerns a fixed closed subgroup G ≤ Aut T, the statement should clarify that the corollary applies to the natural inclusion G → Aut T and depends on a repaired version of Theorem A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the theorem is proved by a contradiction argument whose inputs (linearity, non-commuting representation bounds, weak branching) do not encode the zero-dimensional conclusion.

full rationale

The derivation chain does not encode the theorem's conclusion in its hypotheses. Theorem A is proved by contradiction: assuming a level-transitive closed subgroup with positive upper-box dimension, Theorem 2.1 (itself proved by a dimension product argument independent of the target) yields weakly branch projections; these projections admit non-commuting representations of V_n; the lifting lemma transfers them to the original group; and Lemma 3.2 contradicts linearity over an integral domain. None of these steps defines a quantity in terms of the desired zero-dimensionality or fits a parameter to the target. The self-citations ([13], [14]) are contextual or standard structural facts and are not the load-bearing justification of the central claim: [14, Proposition 4.2] (level-transitivity of sections) is elementary and independent of the conclusion, while [13] is used only for examples and sharpness. Separate mathematical gaps identified in the text—the asserted containment 'H ≤ L × L^{h1} × ... is a subdirect product' and the absence of a reduction from an arbitrary embedding to a level-transitive one—are correctness risks, not circularity: they do not make an input logically equivalent to the target result. No honest circular step can be quoted with a specific reduction of the kind required by the criteria.

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

The paper introduces no fitted constants and no new mathematical entities. Its central claim rests on standard background results and, crucially, on an unproven reduction to level-transitive actions.

assumptions (6)
  • standard math R is a commutative unital integral domain with zero ideal prime, used for cancellation in Lemma 3.2.
    The proof of Lemma 3.2 cancels a nonzero scalar to conclude that aibj = bjai, which requires the integral domain property.
  • domain assumption T is bounded, with vertex degree at most M at every level.
    Theorem 2.1 uses boundedness to make the product ((M-1)/M)^n tend to zero, forcing zero upper-box dimension.
  • ad hoc to paper The proof reduces Theorem A to the level-transitive case.
    The proof of Theorem A starts with a level-transitive closed subgroup and gives no argument that an arbitrary positive-dimensional embedding can be reduced to this case. This is the critical missing premise.
  • standard math Weakly branch groups do not satisfy a group law, cited as [1, Theorem 1].
    Used in Corollary 2 and in the proof of Theorem A to show projections cannot be weakly branch if they satisfy a law.
  • standard math Weakly branch groups admit non-commuting representations of V_n for every n, cited as [2, Corollary 7].
    Used to produce the graph representations that contradict linearity in the proof of Theorem A.
  • standard math A nonzero normal subgroup of a group acting on a rooted tree contains derived subgroups of rigid vertex stabilizers, formalized as Lemma 3.4.
    This is used to show that the subgroup N obtained from the lifting lemma contains non-abelian rigid stabilizers.

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Pith. "Pith review of Arboreal representations of linear groups." pith.science (2026). https://pith.science/paper/KB4PXP46

@misc{pith2026250612745,
  author       = {Pith},
  title        = {Pith review of: Arboreal representations of linear groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KB4PXP46}},
  note         = {Machine review of arXiv:2506.12745}
}
abstract

Ab\'ert and Vir\'ag conjectured in 2005 that any embedding of a linear group over a pro-$p$ domain into the group of $p$-adic automorphisms $W_p$ should be zero-dimensional. We prove their conjecture in greater generality, namely for embeddings of linear groups over any integral domain into the automorphism group of a bounded rooted tree.

Figures

Figures reproduced from arXiv: 2506.12745 by the authors.

Figure 1
Figure 1. The graph Vn. Remark 3.1. A non-commuting representation of Vn in G corresponds to no more than a direct product of n non-abelian 2-generated subgroups in G. Weakly branch groups are precisely groups admitting such direct products as subgroups for every n ≥ 1, while linear groups over a field do not admit such direct products for arbitrarily big n ≥ 1. This was used by Ab´ert in [2, Corollary 7] to prove that weakly… 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. Weakly branch actions: first-order theory, rigidity and Boston's conjecture

    math.GR 2025-07 conditional novelty 8.0 of 10

    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...

Reference graph

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