# Verification report — OWR-1274-007 (Bartholdi: closures of the Aleshin and Bellaterra groups)

Verification date: 2026-10-22.

**Verdict.** The note gives a **complete negative answer** (a complete proof): the conjecture is false: the closure
of the Bellaterra group is 2-adic analytic but is not isomorphic to a pro-2 Sylow subgroup of SL_2(Z_2) (nor of
PSL_2(Z_2) or PGL_2(Z_2)), and the closure of the Aleshin group is not 2-adic analytic.
- **Theorem B.** The closure of the Bellaterra group G(B) in Aut(T_2) is isomorphic, as a topological group, to
  Γ′ = {g ∈ GL_2(Z_2) : g ≡ 1 mod 2, det g = ±1}/{±1}, an open subgroup of index 12 of PGL_2(Z_2); so it is a
  2-adic analytic group of dimension 3.
- **Corollary B.** This closure is not isomorphic to any subgroup of SL_2(Z_2) (it has three distinct
  involutions, SL_2(Z_2) only one), nor, as a topological group, to a pro-2 Sylow subgroup of PSL_2(Z_2)
  (minimal numbers of generators 3 against 2; abelianizations) or of PGL_2(Z_2) (at least 4 generators).
  Corollary B does not use the Möbius model.
- **Theorem A.** The closure of the Aleshin group G(A) is not 2-adic analytic (it has infinite rank); it has index
  2 in the closure J of ⟨G(B), τ⟩ (τ the complement map), and no closed subgroup of Aut(T_2) containing the
  closure of G(B) and τ is 2-adic analytic.
- **Framing.** Read literally, the Sylow sentence fails by the involution count; the substance is in Theorems A
  and B. The closure of G(B) is commensurable with SL_2(Z_2), and its level quotients have the same orders
  2^(3n−5) as the groups induced by the Möbius action of the Sylow subgroup of PSL_2(Z_2).
- **Open.** A further description of the closure of G(A) (for instance its Hausdorff dimension) is not attempted.

The note is unrefereed.

## Statement checked
- **Primary source.** L. Bartholdi, "Profinite groups generated by automata", extended abstract in Oberwolfach
  Report No. 25/2006, *Pro-p Extensions of Global Fields and pro-p Groups* (workshop of 21–27 May 2006, organised
  by N. Boston, J. Coates, F. Grunewald), Oberwolfach Rep. 3(2), pp. 1485–1489 of the report pp. 1463–1536,
  doi:10.4171/OWR/2006/25 (Crossref: issued 2007-03-31).
  - Read in the publisher's open PDF (74 pages; text extracted, pages 1488–1489 rendered and read). Convention (1),
    p. 1486: a vertex v acts by v(x_1x_2…) = (x_1+n_1)(x_2+n_2)…, following the edges labelled by the input letters;
    so the vertex label says whether the state changes the current letter.
  - p. 1488: the "Aleshin" (A) and "Baby-Aleshin" (B) automata are said to be "respectively the right and left
    automata" of the figure; then G(A) is free of rank 3 (Vorobets) and G(B) ≅ C2∗C2∗C2 (Muntyan and Savchuk,
    [Nekrashevych, Theorem 1.10.2]).
  - p. 1489: the closures (written with overlines) are far from the pro-2 completions ("of G(A) and G(A)", a typo);
    Bartholdi conjectures that these closures are Z_2-analytic and, more precisely, that the closure of G(B) is a
    pro-2 Sylow subgroup of SL_2(Z_2). The Ramanujan conjecture that follows is a different problem.
- **The figure (p. 1488), decoded.** Two pictures with the same graph; vertices a (bottom right), b (left),
  c (top right); edges a –0→ b, a –1→ c, b –0→ c, b –1→ b (loop), c –0,1→ a; vertex labels (n_a, n_b, n_c) =
  (1, 1, 0) in the left picture and (0, 0, 1) in the right one. Hence left: a = (b,c)σ, b = (c,b)σ, c = (a,a);
  right: a = (b,c), b = (c,b), c = (a,a)σ. The right automaton has one active state, so its states are
  involutions and cannot generate a free group: the text's "right and left" clashes with the stated facts. The
  facts and the names force A = left (the complement-conjugate of the Aleshin automaton of Vorobets–Vorobets) and
  B = right (literally the Bellaterra automaton of Kharlampovich–Vdovina (2.1)). Part (i) is decided for both
  automata; under the text's assignment part (ii) would be false by Theorem A.
- **Corpus record.** ulamai/UnsolvedMath, OWR-1274-007 (dataset version 1.8.0, snapshot d371ee9; status `open`),
  title "2-Adic Analyticity of the Groups G(A) and G(B)"; statement: the closures of G(A) and G(B) are
  Z_2-analytic groups, and more precisely the closure of G(B) is a pro-2 Sylow subgroup of SL_2(Z_2). The
  statement matches the source (overlines lost). The record's second literature link (Bartholdi–Bush, J. Number
  Theory) concerns 3-class towers and is unrelated.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (i) the closures of G(A) and G(B) are 2-adic analytic (Lazard; for pro-2 groups equivalently finite rank) | false: closure of G(B) analytic of dimension 3, closure of G(A) not analytic | Theorems B, A |
| (ii) closure of G(B) isomorphic (as a topological group) to a pro-2 Sylow subgroup of SL_2(Z_2) | false, even abstractly: three involutions against one | Corollary B(a) |
| (ii) read through the Möbius action (Sylow subgroup of PSL_2(Z_2)) | false: d = 3 against 2; abelianization elementary abelian against exponent ≥ 8 | Corollary B(b) |
| (ii) read in PGL_2(Z_2) (its Sylow subgroup, the Iwahori subgroup) | false: d ≤ 3 against ≥ 4 | Corollary B(c) |
| the text's assignment (A = right, B = left) | (i) unchanged; (ii) would claim the closure of the Aleshin group is a Sylow subgroup of SL_2(Z_2): false (not analytic) | Section 1.2, Theorem A |
| other conventions (Vorobets–Vorobets, AutomGrp: conjugates by the complement map) | closures change by a conjugation in Aut(T_2); nothing changes | Section 1.2 |

## Results in the paper
- **Lemma 2.1.** a, b, c are three distinct non-trivial involutions; the Aleshin states are τ ∘ a, τ ∘ b, τ ∘ c and
  τ inverts them; ⟨G(A), τ⟩ = ⟨G(B), τ⟩, and the closure of G(A) has index 2 in its closure J (τ ∉ G(A)_4).
- **Lemmas 2.2, 2.3.** Möbius maps of Z_2 with c even are isometries; prefix contractions; congruence subgroups
  K(k) of PGL_2(Z_2); lifting lemma: if N is closed, m ≥ 2 and |N_{m+1}| = 8|N_m|, then N ⊇ K(m).
- **Section 3 (Möbius model, after Kharlampovich–Vdovina).** Lemma 3.1: six identities in the quaternion algebra
  (−1,−3) over Q(α), α² + 3α − 2 = 0, and the values tr²/nrd = (1−α)/2 ≠ (5+α)/2 of η_0, η_1. Lemma 3.2: the
  2-adic splitting, the residues of the matrices M_a, M_b, M_c (mod 8) and h_0, h_1 (mod 16), the projective
  identities. Proposition 3.3: the coding κ of the tree by 2-adic balls, equivariant for a, b, c (proved in full).
  Corollary 3.4: the closure of G(B) is isomorphic to the closure H of ⟨M_a, M_b, M_c⟩ in PGL_2(Z_2).
- **Theorem 4.1.** H = Γ = {c ≡ 0 mod 4, χ = ±1} (|H_3| = 32, |H_4| = 256 and the lifting lemma with m = 3);
  conjugation by diag(2,1) gives Γ′; index 12. Remark 4.2: |G(B)_n| = 2^(3n−5) for n ≥ 4.
- **Corollary B, Remarks 5.1, 5.2** as in the verdict.
- **Section 6.** Lemma 6.1: τ′ = κτκ^{-1} is an isometry with τ′h_0 = h_1τ′. Lemma 6.2: τ′ is not differentiable at
  the fixed point p_0 of h_0, because the multipliers of h_0 and h_1 at their fixed points differ (λ_1/λ_0 ≡ 377
  mod 2^10). Facts (F1) (closed subgroups and homogeneous spaces of p-adic Lie groups) and (F2) (continuous
  homomorphisms are analytic), with the consequences (U1) (uniqueness of the analytic structure) and (U2) (an
  open subgroup of index 2). Lemma 6.3: an analytic closed subgroup of Aut(T_2) containing the closure of G(B) acts
  on Z_2 by differentiable maps. Theorem A follows. Remark 6.4 and Table 1: layer ranks (evidence only).

## Computations (sanity checks; programs and outputs in reproducibility/)
The proofs use four finite computations: the residues of Lemma 3.2; |H_3| = 32 and |H_4| = 256; τ ∉ G(A)_4 (only
for the index 2); and ⟨T, w⟩ = P_4 or the exponent 8 of the abelianization of P_4 (Corollary B(b)).
- **The exact program of the note** (`check_note.py`, standard library, about 1 s): 60 checks in nine parts, all
  passed: Lemma 3.1 exactly; the splitting and the identities as matrix identities; the 2-adic data (α, z mod
  2^200) and the residues; Proposition 3.3 for words of length ≤ 10 with a negative control; |H_3| = 32,
  |H_4| = 256, H_k = Γ_k (k = 3, 4, 5), |PGL_2(Z/8)| = 384; for n = 3..6, G(B)_n from the automaton equals the group
  induced by Γ_n through κ (orders 2^5, 2^7, 2^10, 2^13; involutions 15, 43, 167, 647; kernel {1, z_n}); the data
  of Corollary B and Remark 5.1 (characters, ⟨T, w⟩ = P_4, P_4^ab of order 16 and exponent 8, d(G(B)_3) = 3, four
  characters of I); Lemma 2.1 on level 6 and τ ∉ G(A)_4; the multipliers, λ_1/λ_0 ≡ 377 mod 2^10, τ′(p_0) = p_1,
  difference quotients ≡ 3, 11 mod 16 alternately.
- **Programs of the first version** (`original/`, s1–s8): the same statements on larger ranges (coding n ≤ 14;
  H_k = Γ_k for k ≤ 7; d and abelianizations of Γ_k, P_k, I_k for k ≤ 6) and the level orders of Table 1
  (G(B) for n ≤ 12, G(A) for n ≤ 14).
- **Evidence (not used in proofs).** Layer ranks of G(A): 1, 1, 2, 3, 3, 4, 7, 13, 23, 45, 87, 174, 344, 686
  (n ≤ 14; log2 |G(A)_14| = 1393); |G(B)_n| = 2^(3n−5) for 4 ≤ n ≤ 20 (proved for all n in Remark 4.2).
- **Programs of verification run 2** (`independent_run_2/`, seven programs written for the text of the note;
  standard library and SymPy; about one minute in all): see the section on that run below.
- **Re-runs.** On 2026-10-22 all programs of the package were run again from a copy, except the two runs for
  G(A)_14 (about 15 and 22 minutes); outputs identical up to running times (`reproducibility/RERUN_LOG.txt`).
  Verification run 2 ran all of them again from an extracted copy of `source.zip`, the two runs for G(A)_14
  included (892 s and 1355 s): all outputs agree with the recorded ones up to running times.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two independent
verification runs, both AI-assisted, examined this first written version. Each read the source in the
publisher's open PDF and decoded its figure (run A from the drawing commands of the PDF, run B from a rendering of
the page), re-derived all proofs line by line and wrote its own programs.

| Item | Run A | Run B |
|---|---|---|
| Statement, figure, conventions | CONFIRMED (figure decoded from the PDF operators; per-state labels requested) | CONFIRMED (figure rendered; Vorobets–Vorobets and Kharlampovich–Vdovina conventions compared) |
| Möbius model (Lemmas 3.1, 3.2, Proposition 3.3) | CONFIRMED (exact; residues mod 2^160; coding n ≤ 12) | CONFIRMED (exact; coding n ≤ 16 against own automaton code; negative controls) |
| Theorem B | CONFIRMED (H_k = Γ_k, k ≤ 6; lifting lemma re-derived) | CONFIRMED (H_k = Γ_k, k ≤ 64; level orders 2^(3n−5), n ≤ 20) |
| Corollary B | CONFIRMED (d = 3, 2, 4; abelianizations) | CONFIRMED (table of orders, d, abelianizations, involutions; model-free) |
| Theorem A (Lemmas 2.1, 6.1–6.3) | CONFIRMED given textbook p-adic Lie theory; theorem numbers requested | CONFIRMED; references requested |
| Evidence (layer ranks of G(A)) | n ≤ 8 recomputed (SymPy) | n ≤ 14 recomputed (own sifting; validated) |
| Novelty | not established (left to run B) | no prior determination found; two citing works unread |

**Corrections required by the runs**, all applied in the note:
1. (A1, B2) references for the facts of p-adic Lie theory in Lemma 6.3 and the two uniqueness statements: the
   facts are stated as (F1), (F2) with references to Lazard, Bourbaki Ch. III, Serre Part II Ch. IV–V, Dixon–du
   Sautoy–Mann–Segal Ch. 8–9 and Schneider Ch. II–III; (U1) and (U2) are stated and proved from (F2). The books
   were not accessible, so theorem numbers are not given (chapter titles checked through Crossref);
2. (A2) the per-state vertex labels and the decoded edge list of the figure (Section 1.2);
3. (A3) "g ≡ scalar mod 2" in PGL_2(Z_2), and the GL_2 description of Γ′ (Theorem B, Theorem 4.1);
4. (A4) the closedness of {x : x² = 1} in the abelianization argument (Corollary B(b));
5. (A5) the framing: the literal Sylow sentence fails by the involution count; Theorems A and B carry the
   substance (abstract, Section 1.4);
6. (A5, B1) the novelty caveats: Siegenthaler's thesis and Grigorchuk's survey were not read (abstract,
   Section 1.4, "Scope and priority");
7. (B3) Crossref dates the report 2007; the workshop date (May 2006) is given;
8. (B4) Corollary B does not depend on the AI-assisted preprint (Remark 5.1);
9. (B5) commensurability with SL_2(Z_2) (Remark 5.2).
None of them changes a theorem.

## Independent verification run 2 (of the text of the note)
A further independent verification run, also AI-assisted, examined the text of the note itself (12 pages, as
it stood before its final revision), line by line and as if nothing had been checked before. Its report is
summarized here; its programs and recorded outputs are in `reproducibility/independent_run_2/`.

- **Statement.** It read the source again in the publisher's open PDF (pp. 1485–1489; pages 1488–1489 rendered)
  and decoded the figure once more from the drawing operators of the PDF page (circles, edge paths, arrowheads,
  label positions), with a rendering as a check: the same edge list and vertex labels as in Section 1.2. It
  compared every item taken from Kharlampovich–Vdovina ((2.1), (3.1), Section 4.1, (4.4), (4.6), (4.8), (4.9),
  (4.11), (6.1)–(6.3), (6.6)–(6.13), Lemma 6.1, Proposition 6.2, the acknowledgement; version 1 is the only
  version on 22 October 2026), from Vorobets–Vorobets (arXiv:math/0604328v1, Sections 1, 3, 6;
  arXiv:math/0601231v2, Sections 1, 3) and the corpus record with these texts.
- **Proofs.** It re-derived every lemma, proposition and theorem; it found no mathematical error and no gap.
- **Own programs** (seven, written before the package's programs were run or read): Lemma 3.1 in exact
  arithmetic over Q(α) (multiplication table derived from the relations); the 2-adic data of Lemma 3.2 and the
  identities (4) modulo 2^204; the coding of Proposition 3.3 for all words of length ≤ 14, with negative controls;
  H_k = Γ_k for k ≤ 7, |PGL_2(Z/8)| = 384, the identification δΓδ⁻¹ = Γ′ at levels 3–6 and the index 12;
  the squaring step of Lemma 2.3(a) for 3 ≤ j ≤ 8 and its failure for j = 2; d and the abelianizations of Γ_k,
  P_k, I_k (k ≤ 6; P_k and I_k enumerated from their definitions), ⟨T, w⟩ = P_4, the characters of Corollary B(b),
  (c) on all pairs, the level groups G(B)_n = κ⁻¹Γ_nκ (n = 3..6) with the kernels {1, z_n} and the involution
  counts, d(G(B)_3) = 3, τ ∉ G(A)_4 and [J_n : G(A)_n] = 2; the multipliers (λ_1/λ_0 ≡ 377 mod 2^10) and the
  difference quotients of τ′; the level orders of G(A) and G(B) for n ≤ 10 with SymPy (Table 1 for n ≤ 10).
  All checks passed.
- **Re-run.** It extracted `source.zip` and ran every program of the package again, including the two runs for
  G(A)_14 (892 s and 1355 s); all outputs agree with the recorded ones up to running times.
- **Literature.** It repeated the search (arXiv API, OpenAlex, zbMATH Open, Crossref, one web search) and read the
  full text of arXiv:0902.3220 (Bartholdi–Siegenthaler–Zalesskii, one of the five works citing the report): it
  cites the report only for the Hanoi towers group (p. 1477). No determination of either closure was found.
- **Corrections required by this run**, all made in the note: (1) the Bellaterra automaton is in Section 6 of
  the arXiv version 1 of Vorobets–Vorobets, not Section 5 (Section 1.2; "Scope and priority" now says that the
  section numbers refer to that version); (2) the difference quotients of τ′ along h_0^j(1) alternate between 3
  and 11 mod 16 from j = 2 on (q_1 ≡ q_2 ≡ 3; remark after Lemma 6.2); (3) Lemma 3.1 cites KV26 (4.6) as well,
  and (3) is marked as computed here; (4) "the Möbius action … on P¹(Q_2)" instead of "on the boundary"
  (Section 1.4); (5) the acknowledgement of KV26 is described completely (editing assistance and help in finding
  known results for its Theorem 8.1; Section 1.4); (6) this run is recorded in the note and in this report, with
  its programs in the package; (7) the search list in "Scope and priority" is updated (six web searches in all).
  None of them changes a statement or a proof.

## Relation to the literature, novelty and scope
- **Credits.** The conjecture: Bartholdi (2006 report). The Aleshin automaton: Aleshin (1983); freeness of the group:
  M. and Y. Vorobets, Geom. Dedicata 124 (2007). The Bellaterra automaton and G(B) ≅ C2∗C2∗C2: Muntyan and
  Savchuk, 2004 summer school at the Autonomous University of Barcelona (Vorobets–Vorobets, Groups Geom. Dyn. 4
  (2010), Theorem 1.4; cited by Bartholdi as Nekrashevych, Self-similar groups, Theorem 1.10.2). The Möbius model:
  O. Kharlampovich and A. Vdovina, "The Bellaterra graphs are Ramanujan", arXiv:2610.25324v1 (18 October 2026),
  unrefereed; its acknowledgement states that it was written with the editing assistance of an AI system; it
  contains no statement about closures. All identities taken from it are verified again by exact computation,
  and the coding is proved in full. p-adic Lie theory: Lazard; Bourbaki; Serre; Dixon–du Sautoy–Mann–Segal;
  Schneider.
- **Searches (22 October 2026).** arXiv API (Aleshin, Bellaterra, Siegenthaler, bireversible automata, automaton
  groups with closure / analytic / PGL, self-similar p-adic analytic groups); full texts or abstracts of
  arXiv:2610.25324, math/0601231, math/0604328, 1212.0605, 2312.05427, 2203.12963, 1407.4600, 2602.22356,
  0806.4801, 1707.00215, 1012.1531, 1812.09921, 2002.02053, 2012.00919, 2303.14852; OpenAlex (the five works citing
  the report; keyword searches); zbMATH Open; Semantic Scholar; Crossref; five web searches. Repeated by
  verification run 2: arXiv API (Bellaterra, Aleshin, closure / profinite / Sylow / bireversible / p-adic analytic
  with automaton groups, Ramanujan; version check of arXiv:2610.25324), the full text of arXiv:0902.3220
  (Bartholdi–Siegenthaler–Zalesskii, one of the five citing works: it cites the report only for the Hanoi towers
  group), OpenAlex, zbMATH Open, Crossref and a sixth web search. No printed answer to the question was found.
- **Not read.** O. Siegenthaler, Discrete and Profinite Groups Acting on Regular Rooted Trees (PhD thesis,
  Göttingen 2010, doi:10.53846/goediss-3487; the repository refused automated access, which was not circumvented)
  and R. I. Grigorchuk, Some topics in the dynamics of group actions on rooted trees, Proc. Steklov Inst. Math. 273
  (2011) 64–175 (closed access). Both cite the report; if either determines these closures, the novelty statement
  must be revised. No proof depends on them. Aleshin (1983), Nekrashevych's book, Lazard's memoir and the textbooks
  were not consulted.
- **Caveat.** A search that finds nothing is not a proof of novelty; no priority is claimed.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/