# Verification report — KOU-17.16 (Kourovka Notebook 17.16, V. G. Bardakov: is Aut(A) linear for every Artin group A of finite type?)

Verification date: 2026-10-22.

**Verdict.** The note gives a **partial answer: Aut(A) is linear for every Artin group A of finite type without
components of types E_6, E_7, E_8, F_4, H_3, H_4 (type D_5 relying on an unrefereed preprint), and the question
reduces to linearity of Aut(A/Z(A)) for these six irreducible types.** Every statement of the note is proved,
given the cited results; the scope is as follows.
- **Settled (Theorem A).** For A = A_1 × … × A_k of finite type, Aut(A) is linear over C if and only if every
  Aut(A_i) is, if and only if every Aut(A_i/Z(A_i)) is; for irreducible A, finiteness of Out(A/Z(A)) suffices.
- **Settled (Theorem B).** For the irreducible types A_n, B_n, D_n and I_2(m), Aut(A) has a subgroup of finite
  index isomorphic to A/Z(A) (A_n with n ≥ 2, D_n, I_2(m) with m odd) or to A/Z(A) × Z (B_n with n ≥ 3, I_2(m)
  with m even); Aut(A) has order 2 for A_1. Hence Aut(A) and Aut(A/Z(A)) are linear over C. For D_5 this uses the
  unrefereed preprint of Paris and Soroko (arXiv:2606.26082v2); without it, D_5 joins the open list.
- **Corollary C.** Aut(A) is linear over C for every finite-type A without components of the six exceptional types
  (D_5 via the preprint), and the question, in its reading over C, is equivalent to the linearity of Aut(A/Z(A)) for
  the six irreducible groups of types E_6, E_7, E_8, F_4, H_3, H_4 (seven types if the preprint is set aside).
- **Not new mathematics.** Theorem A assembles known facts: a linear group modulo its centre is linear (Wehrfritz,
  Thm 6.4, as cited by Paris–Soroko 2025 and Soroko 2020; proof included); the lift of Charney–Crisp 2005, Prop 8;
  parabolic closures (Cumplido–Gebhardt–González-Meneses–Wiest 2019); Cumplido–Paris 2022, Prop 6(1), (4) and Thm 4(1);
  standard facts on direct factors of centreless groups (Lemma 4.3, proofs included).
  Theorem B is a deduction from published computations of Aut(A), by the argument of Bardakov–Bryukhanov for braid
  groups; the cases B_n, D_n, I_2(m) may be regarded as folklore. Not found in print: the assembled reduction (the
  reducible case, the "only if" directions) and its application to the recent computations.
- **Open.** The types E_6, E_7, E_8, F_4, H_3, H_4. Representations in positive characteristic are not considered.

The note is unrefereed.

## Statement checked
- **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka
  Notebook, No. 21, arXiv:1401.0300, Problem 17.16 (17th issue, 2010), proposed by V. G. Bardakov. Read in version
  48 (6 October 2026; p. 113 of the PDF) and version 49 (20 October 2026); the text is the same in both. It
  recalls that A is linear (Bigelow 2001, Krammer 2002, Digne 2003, Cohen–Wales 2002), asks whether Aut(A) is also
  linear, and records the affirmative answer for the braid groups B_n, n ≥ 2 (Bardakov–Bryukhanov, Vestnik
  Novosibirsk. Univ. 7 (2007), in Russian). The problem is not marked as solved and has no further comment.
- **Corpus record.** ulamai/UnsolvedMath, KOU-17.16 (dataset version 1.8.0): status `open`; the statement
  reproduces the Notebook text without truncation; proposer V. G. Bardakov, 2010.
- **Conventions.** "Finite type": the Coxeter group is finite; reducible groups are included (the Notebook does
  not restrict to irreducible A). "Linear": isomorphic to a subgroup of GL_r(K) for some field K. The note proves
  linearity over C, which for these countable groups is the same as linearity over a field of characteristic 0.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Aut(A) linear (over some field) for every A of finite type | partial: yes for all A without components E_6, E_7, E_8, F_4, H_3, H_4 (D_5 via a preprint); open for those six types | Corollary C(1), Section 6 |
| The same, over C or over a field of characteristic 0 | equivalent to the linearity of Aut(A/Z(A)) for the six irreducible exceptional types | Theorem A, Corollary C(2) |
| Irreducible A only | the same answer (Theorem B for the classical types) | Theorem B |
| Reducible A, with isomorphic or rank-one factors | covered; no indecomposability or uniqueness of decompositions needed | Section 4 |
| Out(A) instead of Aut(A) | not the question; not substituted | — |
| Linearity over a field of positive characteristic | not considered | Section 1.2 |

## Results in the paper
- **Theorem A (reduction)** and Theorem 3.7: the embedding φ ↦ (φ mod Z(A), φ_* on H_1(A;Q)) of Aut(A)
  (Proposition 3.5); the lift of Charney–Crisp for irreducible A with the uniqueness of the sign (d ≥ 3 in rank at
  least 2; Proposition 3.6); the virtual centre (Lemma 4.1 = Cumplido–Paris, Prop 6(1)); no finite normal subgroups
  in A/Z(A) (Corollary 4.2; for irreducible A its key step is Cumplido–Paris, Prop 6(4)); direct factors of
  centreless groups (Lemma 4.3, standard); at most 2^{n−k} direct factors of
  A/Z(A) (Lemma 4.4, via Selberg's lemma and cd(A) ≤ n); restriction to the components (Proposition 4.5).
  Remark 4.6: direct indecomposability of A_i/Z(A_i) from Cumplido–Paris, Thm 4(1), a second route.
- **Theorem B**: principle (P1) (finite Out(A) gives linearity; for braid groups the argument of
  Bardakov–Bryukhanov, Corollaries 1–2) and transvections (Lemma 5.1); Lemma 5.2 (A(I_2(2k)) = ⟨a, y | ay^k = y^k a⟩
  with y = ab, A/Z(A) ≅ Z * Z/k, and η^k = T_λ with λ = (1, −1)); Table 2 lists the published descriptions of Aut(A)
  used. Remark 5.3: the shorter route through finiteness of Out(A/Z(A)) for B_n and I_2(2k).
- **Corollary C** and Section 6 (what is known for the exceptional types, and where the known methods stop).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program written with the note** (`check_note.py`, standard library, 0.07 s; ALL CHECKS PASSED: True). For the
  45 types A1–A8, B2–B8, D4–D8, E6–E8, F4, G2, H3, H4, I2(m) with 3 ≤ m ≤ 20, in exact arithmetic: N, a reduced word
  of w0 of length N, w0 = −1 and (separately) w0 central, the diagram permutation, the classes of generators against
  the odd-label components, letters per class against positive roots per orbit, [ζ], d, d ≥ 3 in rank ≥ 2, and the
  transvection data (λ(ζ) = −2 solvable exactly for A1 and B2; the lattice λ(ζ) = 0 for B_n generated by
  (1, −(n−1))); all values agree with Table 1. In the free group, for all even m ≤ 20: the identities of Lemma 5.2,
  the free-group identities behind the centrality of Δ = (ab)^{m/2}, and γηγ = η⁻¹, εηε = η.
- **Program with which the values were first computed** (`first_version/coxeter_facts.py`, 0.08 s): the same data
  for types up to rank 9 and I2(m), m ≤ 16 (w0 = −1 for I2(m) set by parity there), and η^j for j ≤ 12.
- **Verification run A** (`independent_runs/run_A/`): free-group identities for η^j, j ≤ 20, and η^{m/2} for all
  even m ≤ 40; N, w0 = −1, classes and d for 40 types in floating point; the composition law of transvections.
- **Verification run B** (`independent_runs/run_B/`): exact Coxeter data for 44 types (no mismatch with the first
  version on any common type); an exact solver of the word problem (Garside normal forms) and 55 tests inside the
  Artin groups (centrality of Δ or Δ²; η, ψ, η^{m/2}, γηγ = η⁻¹, εηε = η for m = 4..16 even; the automorphism τ of
  Charney–Crisp for B3, B4, B5 and the relations τTτ = T⁻¹, εTε = T, τε = ετ for B3, B4; graph automorphisms of
  D4, D5, D6 and I2(m) induced or not by W); 8 of 8 deliberately wrong inputs detected.
- **Third verification run** (`independent_run_2/`, on the text of the note): Table 1 for 48 types from explicit root
  systems in standard coordinates (exact arithmetic in Q, Q(√5) and cyclotomic fields; not from Cartan matrices),
  agreeing with `check_note.py` on all 45 common types; Lemma 5.2 in F(a, b) for all even m ≤ 20 and inside
  A(I2(m)), 3 ≤ m ≤ 20, with a word-problem solver based on the central quotient (Z * Z/k or Z/2 * Z/m) and the
  length, with γηγ = η⁻¹, εηε = η and the order 2m of the quotient used for I2(m); 5 wrong inputs rejected.
- **Re-runs.** On 2026-10-22 all programs with a recorded output were run again from the package, and by the third
  run from an extracted copy of the archive; all outputs were reproduced byte for byte (`reproducibility/README.md`).

## Independent verification runs
The results were first obtained, with proofs and with `coxeter_facts.py`. Two independent verification runs, both
AI-assisted, followed (2026-10-22), each with its own programs: run A (statement, proofs, sources) and run B
(independent computation, sources, literature, novelty). A third run, on the text of the note, is described in the
next section.

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, degenerate cases (S = ∅, rank one, reducible A) against the Notebook (v48) and the record | CONFIRMED | CONFIRMED |
| Lemmas 3.1–3.4, Proposition 3.5 | CONFIRMED (re-derived) | used; no gap found |
| Proposition 3.6 (lift), Theorem 3.7 | CONFIRMED_WITH_FIXES (uniqueness of the sign: d ≥ 3) | no gap found |
| Lemma 4.1, Corollary 4.2 | CONFIRMED (parabolic closures checked in the source) | CONFIRMED; Lemma 4.1 is Cumplido–Paris, Prop 6(1) |
| Lemmas 4.3, 4.4, Proposition 4.5, Theorem A | CONFIRMED (re-derived; isomorphic and rank-one factors examined) | CONFIRMED (re-derived); indecomposability is in Cumplido–Paris, Thm 4(1) |
| Theorem B, every cited description of Aut(A) at the place cited | CONFIRMED (Charney–Crisp, Soroko 2021, Castel–Paris, Paris–Soroko 2025 and 2026) | CONFIRMED (also Crisp–Paris 2002 preprint, Bell–Margalit, Soroko 2020) |
| Lemma 5.2 (η), index computations (4 for B_n, 2m for I_2(2k)) | CONFIRMED (own programs) | CONFIRMED (own solver) |
| Corollary C | CONFIRMED | CONFIRMED_WITH_FIXES (D_5 caveat on the second statement) |
| Novelty | braid-group mechanism (P1) is Bardakov–Bryukhanov's | no source beyond braid groups; Lemma 4.1 and indecomposability in print |

Neither run found a mathematical error.

**Corrections required by the runs**, all applied in the note:
1. (Run A) Proposition 3.6 states that the sign is unique because d ≥ 3 in rank ≥ 2, and that d = 1 only for A = Z.
2. (Runs A and B) Bardakov–Bryukhanov is read and cited in the English version arXiv:math/0506114v3; principle (P1)
   is credited to their Corollaries 1–2 (Hol(B_n) linear); "not available" was removed.
3. (Run A) The second-hand citation for A/Z(A) ≅ Z * Z/(m/2) (m even) was replaced by the derivation of Lemma
   5.2(a) (y = ab, relation [a, y^{m/2}] = 1).
4. (Run A) Footnote: Castel and Paris write ζ for the graph automorphism; the note uses ζ for the central generator.
5. (Run A, optional) Remark 5.3: the shorter route through finiteness of Out(A/Z(A)) for B_n and I_2(m), m even.
6. (Run A, optional) Bell–Margalit 2007 (Thm 6 of the arXiv version) as a second source for Aut(A(B_n)).
7. (Run B) Cumplido–Paris, Rev. Mat. Iberoam. 38 (2022): Proposition 6(1) is cited for Lemma 4.1, Theorem 4(1) for
   the indecomposability (Remark 4.6); the novelty statement on Theorem A was corrected accordingly.
8. (Run B) The D_5 caveat is on the second statement of Corollary C, in the abstract, the conclusion and in every
   summary.
9. (Run B) Precise citations: Charney–Crisp Thm 1(ii)(a) is stated for n ≥ 3 (A_2 via I_2(3)); the rank-two
   Section 5 of Crisp–Paris is cited as the 2002 preprint; for D_4 the immediate route through Soroko 2020, Cor 2
   with Soroko 2021, Cor 6 is mentioned (neither paper states it).

**Changes made when the note was written.** The status wording is the same in the abstract, the introduction, the
conclusion and the summaries; the novelty statement is limited as above; the published descriptions are collected
in Table 2; `check_note.py` was written and re-checks the data and identities used.

## Third independent verification run (on the text of the note, 2026-10-22)
A third run, also AI-assisted, examined the text of the note itself, as if nothing had been checked (the two runs
above examined the first written version of the results). It wrote its own programs (`reproducibility/
independent_run_2/`) and produced a written report.

| Item | Third run |
|---|---|
| Statement: Kourovka v48 and v49 (text of 17.16 identical; v49 is the latest arXiv version), corpus record (verbatim the Notebook text, status open) | CONFIRMED |
| (F1)–(F6), Lemmas 3.1–3.4, Propositions 3.5, 3.6, Theorem 3.7 (re-derived; Charney–Crisp Prop 8 read again) | CONFIRMED |
| Lemma 4.1, Corollary 4.2, Lemmas 4.3, 4.4, Proposition 4.5, Remark 4.6, Theorem A (re-derived; Cumplido–Gebhardt–González-Meneses–Wiest and Cumplido–Paris read again) | CONFIRMED_WITH_FIXES (credit: Cumplido–Paris Prop 6(4) is the key step of Corollary 4.2; Lemma 4.3 is standard) |
| Theorem B type by type, every description of Aut(A) at the place cited (Charney–Crisp, Crisp–Paris, Bell–Margalit, Soroko 2020, 2021, Castel–Paris, Paris–Soroko 2025, 2026), Lemmas 5.1, 5.2, the indices 4 and 2m, Remark 5.3 | CONFIRMED_WITH_FIXES (precision of the Crisp–Paris credit in Lemma 5.2(a)) |
| Corollary C, D_5 caveat | CONFIRMED |
| Section 6 | CONFIRMED_WITH_FIXES (list of known cases in Castel–Paris, §1; methods of the computations) |
| Computations (own programs; package re-run from the archive) | CONFIRMED; one wrong count in the description of run A (40 types, not 41) |
| Novelty, references (all 20 DOIs checked with Crossref) | no prior answer found; CONFIRMED_WITH_FIXES (credit list in the abstract; attribution of Lemma 3.3; issue number of Cumplido–Paris) |

It found no mathematical error. **Corrections required by the third run**, all applied in the note:
1. Run A's floating-point recomputation covers 40 types (Verification paragraph and this report).
2. Section 6: Castel–Paris, §1, list A_n, B_n, I_2(m) and D_4 as known and treat D_n, n ≥ 6; Paris–Soroko 2026, §1,
   list these with D_n, n ≥ 6, and add D_5.
3. Section 1.3: Lemma 4.3 (direct factors of centreless groups) is no longer listed among what was not found in
   print; it is introduced as standard, with proofs included.
4. Cumplido–Paris, Prop 6(4) (A/Z(A) injects in its abstract commensurator) is credited for the key step of
   Corollary 4.2 for irreducible A (Section 1.3, after Corollary 4.2, Scope and priority).
5. Lemma 5.2(a): the presentation with generators ab and b and the isomorphism A/Z(A) ≅ Z/k * Z are credited to
   Crisp–Paris, proof of Theorem 5.1.
6. Abstract and Zenodo description: Crisp–Paris added to the list of published computations of Aut(A).
7. Lemma 3.3 is called classical; the reference to Wehrfritz, Thm 6.4, is given as cited by Paris–Soroko 2025 and
   Soroko 2020, and the bound e ≤ r² is attributed to the proof given.
8. Section 6: the description of the methods (mapping class groups, rigidity, classifications of homomorphisms) is
   restricted to the papers that use them; in rank two Crisp–Paris use the free-product structure of A/Z(A).
9. This run is recorded in the Verification paragraph, Section 7, this report and `reproducibility/independent_run_2/`.
10. Bibliography: issue number 2 of Rev. Mat. Iberoam. 38 for Cumplido–Paris.

## Relation to the literature, novelty and scope
- **Searches (21 and 22 October 2026).** arXiv API (automorphisms and endomorphisms of Artin groups of spherical,
  finite or exceptional type; holomorphs of braid groups; linearity of automorphism groups; "Kourovka" with
  "Artin"; author searches; the newest listings, latest relevant entry on 22 October arXiv:2610.13904 of
  9 October 2026), zbMATH Open API, OpenAlex (works citing Charney–Crisp 2005 (31), Castel–Paris 2025 (2), Soroko
  2021 (5), Bardakov–Bryukhanov (2), Paris–Soroko 2026 (0); titles read), Crossref, Semantic Scholar (works citing
  arXiv:math/0506114 (3); a keyword search was refused with HTTP 429), five web searches. The third run repeated
  the arXiv, zbMATH, OpenAlex and Crossref searches on 22 October and made one of the web searches.
  - No printed answer to the question beyond braid groups was found; no computation of Aut(A) for E_6, E_7, E_8,
    F_4, H_3, H_4 was found (Castel–Paris 2025, §1, list A_n, B_n, I_2(m) and D_4 as known and treat D_n, n ≥ 6;
    Paris–Soroko 2026, §1, list these with D_n, n ≥ 6, and add D_5).
  - Closest results: Bardakov–Bryukhanov (Hol(B_n) linear); the computations of Aut(A) in Table 2; for D_4, Soroko
    2020 with Soroko 2021.
- **What was read.** In the arXiv versions (numbering used; journal versions not compared): Charney–Crisp
  math/0408412v1 completely; Castel–Paris 2307.02880v3 §§1–2; Paris–Soroko 2409.12552v2 §2; Paris–Soroko
  2606.26082v2 §§1–3; Soroko 2009.10471v4 §§1–3; Soroko 1903.12567v1 §1; Cumplido–Paris 1904.09461v2 §§1, 3;
  Crisp–Paris math/0210438v1 §5; Bell–Margalit math/0501051v2 §1; Bardakov–Bryukhanov math/0506114v3 (introduction,
  Theorem 1, Corollaries 1–3); Cumplido–Gebhardt–González-Meneses–Wiest 1712.06727v3 (Theorem 1.1, Definition 6.7,
  Propositions 6.8 and 7.2, Lemma 8.1, Theorem 8.2); the third run read these again at the places cited, with
  Cumplido–Paris, Prop 6(4). Not read: Dyer–Grossman 1981, Gilbert–Howie–Metaftsis–Raptis
  2000 and Wehrfritz 1973 (used as quoted by Charney–Crisp, Paris–Soroko 2025 and Soroko 2020; the statements of
  Gilbert et al. used are also proved in the Crisp–Paris preprint). The classical results (Brieskorn–Saito,
  Brieskorn, Deligne, Andreotti–Frankel, Selberg's lemma in Alperin's account) and the linearity theorems are used as
  stated. Bibliographic data: Crossref (20 DOIs), arXiv API (12 records), zbMATH (Alperin; Bardakov–Bryukhanov).
- **Caveats.** D_5 rests on an unrefereed preprint. The equivalences concern linearity over C. A search that finds
  nothing is not a proof of novelty; no priority is claimed.
- **Scope.** A partial answer; the question stays open for the six exceptional types.

## 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/