# Verification report — AMR-109-0082 (Farb, Question 7.6: is the multiplicity of spec(I_g(k)) bounded for k ≥ 3?)

Verification date: 2026-10-11.

**Verdict.** The note is an **observation**: Question 7.6 of the source is **answered in the negative** for every
genus g ≥ 2 and every k, and every statement the note makes is proved or, where it says so, computer-assisted.
- **Settled.** For every g ≥ 2, k ≥ 1 and N there are N pseudo-Anosov classes in I_g(k) with one dilatation which
  are pairwise non-conjugate in Mod_g, also up to inversion and in the extended mapping class group
  (Theorem 1.1). So spec(I_g(k)) has unbounded multiplicity for every k. The general theorem is proved without
  computation, given three facts quoted from Leininger (Geom. Topol. 8 (2004), Section 3.2).
- **Not new mathematics in its mechanism.** The negative answer follows from the mechanism which the source
  itself gives for the Torelli group and the Johnson kernel (Thurston's representation of the group generated by
  the twists about two filling curves, equal traces of non-conjugate elements, the two curves chosen
  separating), applied to a two-generator subgroup of a deep term of the lower central series of that group,
  which the source itself places in I_g(k) (in the proof of its Theorem 5.10, with a reference to Farb,
  Leininger and Margalit). What
  the note adds is the comparison of conjugacy in the Fuchsian subgroup with conjugacy in the mapping class
  group, which the source leaves implicit also for k ≤ 2, written out with proofs, and explicit instances.
- **Computer-assisted.** The explicit instances of Section 8 (for example 64 classes in I_2(4) with one
  dilatation λ, where λ + 1/λ is an integer with 5116 digits) rest on exact integer computations.
- **Not answered.** Question 7.7 of the source (simple length spectrum); the growth of the multiplicities;
  whether the classes are primitive (not treated in general).
- **Novelty.** The answer is apparently not recorded in the literature accessible to us. It may well be known
  to specialists. No priority is claimed. A search that finds nothing is not a proof of novelty.

The note is unrefereed.

## Statement checked
- **Primary source.** B. Farb, "Some problems on mapping class groups and moduli space", in: Problems on Mapping
  Class Groups and Related Topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc. 2006, pp. 11–55,
  doi:10.1090/pspum/074/2264130.
  - Read in the arXiv version arXiv:math/0606432v1 (54 pages; the only version; file of 575,893 bytes, sha256
    `d00c32daa884331bf9f85ba4f676adb2fd4d5e22b6837b08b95696774acc0956`): Section 1.2 (pages 6–7), Theorem 5.10
    with its proof (pages 33–34), Sections 7.1–7.3 (pages 44–47). Page numbers below refer to this version; the
    note itself cites sections, statements and formulas.
  - Verification run A compared Section 7.2 and the paragraph on the Johnson filtration with the author's
    public file of the volume (the URL of the corpus record): the wording is the same; the question is on
    printed page 51 there. The third run found the same numbers there for formula (5), Theorem 5.10 and
    Question 7.6. The printed volume was not accessible.
  - Question 7.6 (p. 46) is one sentence: "Does spec(I_g(k)) have bounded multiplicity for k ≥ 3?"
  - Definitions: Γ_0 = Γ, Γ_(i+1) = [Γ, Γ_i], Γ = π_1(Σ_g); I_g(k) = ker(Mod_g → Out(Γ/Γ_k)) (p. 6, formula (5));
    I_g(1) is the Torelli group, I_g(2) the group generated by the twists about separating curves (p. 7). In
    the usual numbering of the lower central series, I_g(k) = ker(Mod_g → Out(π/γ_(k+1)(π))). spec(Mod_g) is
    the set of the numbers log λ(f), f pseudo-Anosov; spec(H) is the part coming from the pseudo-Anosov
    elements of a subgroup H (p. 44).
  - Multiplicity (p. 46) is defined for spec(Mod_g) only: unbounded multiplicity means that for every N some
    value is log λ(f_i) for at least N pseudo-Anosov classes which are different conjugacy classes of Mod_g.
  - What the source states as known (p. 46): unbounded multiplicity for spec(Mod_g) (hyperbolic surfaces
    isometrically embedded in moduli space); for the group generated by the twists about two filling curves
    (Thurston's representation; its spectrum is, with the qualifier "essentially", the length spectrum of the
    quotient of the hyperbolic plane); hence, with the two curves chosen separating, for spec(I_g) and
    spec(I_g(2)).
  - Proof of Theorem 5.10 (p. 34): an element of the k-th level of the lower central series of a free group
    generated by the twists about two separating curves lies in I_g(k) (reference to Farb–Leininger–Margalit).
- **Corpus record.** ulamai/UnsolvedMath, AMR-109-0082 (dataset version 1.6.0; upstream status `open`). Its
  statement joins two sentences of the source. Only the first is Question 7.6. The second introduces the simple
  length spectrum and belongs to Question 7.7, which the note does not answer.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Multiplicity of a value r of spec(H) = number of conjugacy classes of Mod_g which contain a pseudo-Anosov element of H with log λ = r (the reading under which the source's argument for I_g and I_g(2) makes sense: it passes from a subgroup to a larger group) | unbounded for H = I_g(k), every g ≥ 2, every k ≥ 1 | Theorem 1.1 |
| Classes counted up to inversion (unoriented closed geodesics) | unbounded | Theorem 1.1 |
| Conjugacy in the extended mapping class group | unbounded | Theorem 1.1 |
| Multiplicity counted by conjugacy classes of H itself | infinite for every value, for a trivial reason (I_g(k) is normal of infinite index, centralisers of pseudo-Anosov classes are virtually cyclic); not the question | Section 1.1 |
| "for k ≥ 3": for all k, or for some k; all genera g ≥ 2 | answered for every g ≥ 2 and every k ≥ 1 | Theorem 1.1 |
| Question 7.7 (simple length spectrum; second sentence of the corpus record) | **not answered**; the closed geodesics of the classes constructed are not simple | Section 1.2, Remark 7.2 |
| Primitive classes only | not treated in general; in all 24 instances the classes are primitive (computations of run A for d ≤ 3, m ≤ 3 and of the third run for all instances) | Remark 7.1 |

## Results in the paper
- **Theorem 1.1.** For g ≥ 2, k ≥ 1, N ≥ 1 there are pseudo-Anosov classes f_1, …, f_N in I_g(k) with the same
  dilatation such that f_i is conjugate neither to f_j nor to f_j^(−1) in Mod_g^± for i ≠ j. They can be chosen
  in γ_d(⟨T_a, T_b⟩) ⊂ I_g(2d), d = max(k, 2), for separating simple closed curves a, b which fill Σ_g.
- **Lemmas 2.1, 2.2.** [I(k), I(l)] ⊂ I(k + l) (with the inclusion [γ_i, γ_j] ⊂ γ_(i+j) proved by the three
  subgroups lemma); twists about separating curves lie in I(2). Classical; proofs included.
- **Lemma 3.1, Lemma 3.2, Proposition 3.3.** The square-tiled model S(n, τ, τ′); the seeds M_2 (8 squares) and
  M_3 (12 squares); sums at good points; for every g ≥ 2 an admissible model of genus g. Known in substance
  (Aougab–Taylor, Lemma 2.2; Jeffreys, Theorem 1.2).
- **Lemmas 4.1–4.4.** The two affine twists with derivatives A = (1 n; 0 1), B = (1 0; n 1) (τ_a = T_a^(−1),
  τ_b = T_b for left-handed twists T_c); ping-pong: ⟨A, B⟩ is free and every element not conjugate to a power
  of A or B is hyperbolic (n ≥ 3); the lattice Λ of the holonomy vectors of saddle connections and
  D(Aff^±(q)) ⊂ PGL(Λ); conjugacy in Mod^±(S) of affine pseudo-Anosov classes implies conjugacy of the
  derivatives in PGL(Λ) (in PSL(Λ) for Mod(S)). Known in substance: the flat structure with the two affine
  twists is Thurston's construction as described by Leininger (Section 5.1), and the statement on the elements
  of ⟨T_a, T_b⟩ is Leininger's Lemma 6.3 and Proposition 6.4; the note says so.
- **Lemmas 5.1, 5.2.** Fricke polynomials and the reversal identity (known: Kapovich–Levitt–Schupp–Shpilrain,
  Proposition 5.3 of the arXiv version; Anderson's survey); a family of 2^m positive words with equal traces
  which are pairwise non-conjugate, also up to inversion, with proof.
- **Lemma 6.1, Corollary 6.2.** A group acting on a tree with finite vertex stabilisers: a conjugacy class of
  elements of infinite order meets a finitely generated subgroup G in boundedly many classes of G; applied to
  PSL(2,Z) = Z/2 * Z/3.
- **Section 8 (computer-assisted).** 24 instances on M_2 and M_3 (Table 1): in each the 2^m elements have one
  trace and are pairwise non-conjugate in PSL(Λ), also up to inversion; pairwise non-conjugate in PGL(Λ) up to
  inversion except for (d, m) = (2, 1), where the two classes are conjugate in Mod^± by a reflection.

## Computations (programs and outputs in reproducibility/)
The general theorem does not depend on a computation. All programs use exact integer arithmetic.
- **The program written with the note** (`writing_stage/check_note.py`, standard library, 1 second; TOTAL
  failures: 0): the seeds from the two lists; the lattices Λ; u, v and the traces of all 24 instances (digits
  and log λ of the rows of Table 1); the words; the case (d, m) = (2, 1); the second certificate (below).
- **The programs with which the results were first obtained** (`original/`): the model against Lemma 3.1; all
  admissible pairs for n ≤ 8 and, among the pairings allowed by Lemma 3.1, for n = 10, 12 (for n = 8 only M_2
  and its mirror image; for n = 12 the surfaces of genus 2 lie in Q(2,2) and Q(1,1,1,1), in agreement with the
  minimal numbers 8, 12, 16 of Jeffreys' Table 1.1); sums up to genus 40; Λ; trace polynomials for m ≤ 3; the
  invariants in PSL(2,Z) with validation; the 24 instances; Magnus expansions up to degree 6; the reflection;
  the 550 classes of G of word length ≤ 7 (522 alone in their class of PSL(Λ); 7 classes of PSL(Λ) contain four
  classes of G each).
- **Verification run A** (`verification_run_A/`): Lemma 3.1 on 11,259 pairs (n ≤ 8) and 100,000 random pairs
  (n ≤ 20); 304 sums; chains to genus 41; the trace identity for m ≤ 2 as a polynomial identity in the entries
  of generic matrices; Lemma 4.2 on 88,592 words for each n ∈ {3, 4, 8, 12}; all 24 instances with invariants
  from reduced quadratic forms (same traces, same class counts); primitivity for d ≤ 3, m ≤ 3.
- **Verification run B** (`verification_run_B/`): the seeds with its own code; the two Dehn twists from the
  curves alone; Johnson levels exactly 2, 4, 6, 8 for the twists, c_2, c_3, c_4 on the once-punctured M_2
  (Magnus expansions up to degree 9); the handedness (τ_a = T_a^(−1), τ_b = T_b) from a test against the
  holonomy; all rows of Table 1 with its own conjugacy test; explicit conjugating matrices for m = 1; the
  second certificate; the original programs re-run with identical results.
- **Third run** (`independent_run_2/`, programs written from the statements of the note): the seeds from the
  gluing maps; Lemma 3.1 against the direct computation on all 904,284 pairs of involutions with n ≤ 10; all
  688 admissible triples with n ≤ 12 (2, 16 and 670 for n = 8, 10, 12) have only good intersection points; 400
  sums of two seeds and six chains of sums up to genus 30, by the rule of Lemma 3.2 on ribbon graphs, always
  give admissible models with good points only; the two shears as maps of the surface, with linear parts A and
  ±B; Λ of the seeds from saddle connections; all 24 instances with a new program for the cyclic words (same
  traces, same class counts); for (d, m) = (2, 1) an elementary certificate that g_0 and g_1 are not conjugate
  in PSL(Λ) (modulo 11 for M_2 and modulo 7 for M_3 every invertible X with X g_0 = g_1 X has a non-square
  determinant); Lemma 4.2 on 39,364 words for each n ∈ {3, 4, 8, 12}; the 550 classes of G of word length ≤ 7
  recounted; all elements g_ε of the 24 instances are primitive in PSL(Λ); the second certificate with a third
  cell structure.
- **The second certificate** (genus 2; does not use Lemma 4.4 nor the facts (L1), (L3)): conjugate Torelli
  classes act in conjugate ways, up to sign and a permutation of the characters, on the lattices
  H_1(S; Z_χ)/torsion of the 15 non-trivial characters χ of H_1(S; Z/2), with determinant 1 for
  orientation-preserving conjugators which fix χ. For M_2, d = 2, m ≤ 6: the classes f_ε act on the lattice of
  the holonomy character, identified with 2Λ, by ±g_ε, and their trace there occurs for no other character. So
  the class counts of Table 1 apply. Computed by run B (`vb07_covers.py`) and again, with another cell
  structure and another derivation of the twist maps, by `check_note.py`, which finds the same for d = 3
  (m ≤ 5) and d = 4 (m ≤ 4). The third run repeated the computation for d = 2, 3, 4 with a third cell
  structure (the square tiling itself) and the images of the edges under the two shears
  (`independent_run_2/r2_04_covers.py`), with the same results, and checked the argument. The argument is
  written out in `reproducibility/README.md`.
- **Re-runs.** On 2026-10-11 all programs with a recorded output were run again from an extracted copy of the
  archive, first when the note was written and a second time by the third run. Every output is identical to
  the recorded one, or identical up to the fields which record running times
  (`reproducibility/RERUN_LOG.txt`). The programs of the third run print running times on the standard error
  stream only; their outputs were reproduced byte by byte from an extracted copy of the archive (made before
  the last changes of the texts; the programs and their recorded outputs are the same files as in the final
  archive).

## Independent verification runs
The results were first obtained, with proofs and programs, in a first written version. Two independent
verification runs, both AI-assisted, followed on 2026-10-11; each wrote its own programs and did not see the
report of the other. Run A examined the statement and every proof; run B made independent computations and
examined the literature and the novelty.

| Item | Run A | Run B |
|---|---|---|
| Statement: reading of the question, definitions, indexing, the second sentence of the record | CONFIRMED_WITH_FIXES (the reading is right; the result must be presented as an observation) | CONFIRMED |
| Lemmas 2.1, 2.2 (Johnson filtration) | CONFIRMED (re-derived) | levels confirmed by computation |
| Section 3 (model, seeds, sums, all genera) | CONFIRMED (re-derived; computed) | seeds and lists confirmed by computation |
| Lemmas 4.1–4.3 (twists, ping-pong, lattice) | CONFIRMED (re-derived; computed) | twists, handedness and lattices confirmed by computation |
| Lemma 4.4 (conjugacy detected in PSL(Λ), PGL(Λ)) | CONFIRMED_WITH_FIXES (correct; rests on the theorem of Bers as stated by Leininger; part (ii) needed a reference or a short argument for one step) | not its part; for M_2, d = 2 an independent certificate |
| Lemmas 5.1, 5.2 (traces, the family) | CONFIRMED (re-derived; computed) | tested; references for the reversal identity |
| Lemma 6.1, Corollary 6.2 (trees) | CONFIRMED (re-derived) | fibres recounted |
| Theorem 1.1 | CONFIRMED_WITH_FIXES (proof correct; the fixes concern presentation and references) | not its part |
| Instances of Section 8 | all 24 recomputed: identical | CONFIRMED (all rows; coincidence for m = 1; second certificate) |
| Programs of the first version | not run | CONFIRMED (all re-run, identical results) |
| Novelty | the source's own argument extends; present as an observation | CONFIRMED_WITH_FIXES (no earlier answer found; the ingredients are known) |

Neither run found a mathematical error.

**Corrections required by the two runs**, all applied in the note:
1. (Run A) The result is presented as an observation which follows the source's own mechanism, in the title,
   the abstract, the introduction and "Scope and priority"; the ideas are those of the source and of the cited
   papers.
2. (Run A) The notion of multiplicity is stated in the first paragraph, with the reason for the reading, the
   reading by classes of H (always infinite), and the coverage of inversion and of the extended group.
3. (Run A) One sentence explains why a finitely generated subgroup of γ_d(⟨T_a, T_b⟩) is used.
4. (Run A) Lemma 4.4: the citation of Leininger's Theorem 3.2 is kept, with the reduction to the theorem of
   Bers (stated as not read); the proportionality of two quadratic differentials with the same Teichmüller
   disc is proved in three lines from Teichmüller's uniqueness theorem.
5. (Run A) No locations in books: the inclusion [γ_i, γ_j] ⊂ γ_(i+j) is proved; the tree of a free product is
   described with a short verification; standard facts on free groups and free products are named as such.
6. (Run A) Small corrections: freeness of ⟨A, B⟩ also for n = 2, the hypothesis n ≥ 3 for hyperbolicity; n ≥ 2g;
   the handedness of the two twists; "hyperbolic" in the description of the invariants; finiteness of each
   multiplicity by the theorem of Arnoux–Yoccoz and Ivanov; the remark on a discarded choice of v_0 is dropped.
7. (Run A, Run B) Farb–Leininger–Margalit, Section 2.3, is described as a construction of separating
   multicurves; Aougab–Taylor and Jeffreys are the references for single curves.
8. (Run B) The cited places of Farb–Leininger–Margalit were checked against arXiv:math/0603675v2 (Section 2.3,
   Proposition 4.8, Theorem 1.2); that paper contains no statement on multiplicities. This is said in the note.
9. (Run B) The handedness is stated once (τ_a = T_a^(−1), τ_b = T_b), and it is said that the original program
   for the Johnson levels uses the pair (τ_a, τ_b^(−1)).
10. (Run B) "Scope and priority" says that the source contains both ingredients; references for the reversal
    identity (Kapovich–Levitt–Schupp–Shpilrain; Anderson's survey) are given.
11. (Run B, optional) The second certificate is mentioned in the note and written out in the package; the
    exact Johnson levels up to degree 9 are reported.

**Changes made when the note was written** (after the two runs; examined by the third run, see the next
section).
- Proposition 3.3 is stated with part (a) (a pair of separating filling curves is a model), and the remark
  that every intersection point of an admissible model is good replaces the bookkeeping of good points in sums.
- The invariance argument of the second certificate was written out from the report of run B, and the
  computation was repeated with `check_note.py` (also for d = 3, 4).
- The text was shortened; proofs were not changed otherwise.

## Third verification run, on the final text
A third run, also AI-assisted, was made on 2026-10-11 on the final text of the note and on the package. It did
not take part in the earlier stages. It read the source again, went through every proof line by line, compared
the cited statements with the texts of the cited papers (arXiv versions), wrote its own programs from the
statements of the note (`reproducibility/independent_run_2/`), read `check_note.py` completely, and re-ran the
package from an extracted copy of the archive.

| Item | Third run |
|---|---|
| Statement: wording of Question 7.6, definitions, indexing, reading of multiplicity, the second sentence of the record, presentation as an observation | CONFIRMED |
| Lemmas 2.1, 2.2 | CONFIRMED (proofs followed line by line) |
| Section 3: Lemma 3.1, the remark on good points, the seeds, Lemma 3.2, Proposition 3.3 in its present form | CONFIRMED (proofs; all pairs with n ≤ 10, all 688 admissible triples with n ≤ 12, 400 sums, chains to genus 30) |
| Lemmas 4.1–4.3 | CONFIRMED (proofs; the shears as maps of the surface; saddle connections) |
| Lemma 4.4, with the facts (L1)–(L3) | CONFIRMED (the three facts are quoted exactly from Leininger, Section 3.2; the argument for the orientation-reversing case is complete) |
| Lemmas 5.1, 5.2; Lemma 6.1, Corollary 6.2 | CONFIRMED |
| Theorem 1.1 (Section 7), Remarks 7.1, 7.2 | CONFIRMED |
| Section 8: u, v, the traces, Table 1, all 24 instances, the case m = 1 | CONFIRMED (own programs; for m = 1 also an elementary certificate modulo a prime) |
| `check_note.py` and the second certificate for d = 2, 3, 4 | CONFIRMED (program read; recomputed with a third cell structure) |
| Programs of the package | CONFIRMED (36 comparisons with the recorded outputs, none differs) |
| Cited results and bibliography | CONFIRMED_WITH_FIXES (all statements agree with the sources; 12 DOIs resolved at Crossref; four points of credit, see below) |
| Novelty | CONFIRMED_WITH_FIXES (nothing found which states or answers the question; the ingredients are known and are now credited where they are used) |

The third run found no mathematical error. It required, and the note now contains:
1. The paragraph "Verification" describes the state after this run.
2. Locations in the source which do not depend on the version (Section 1.2 and formula (5), the proof of
   Theorem 5.10, Sections 7.1 and 7.2) instead of page numbers of the arXiv version, and a sentence at the first
   citation which says that the numbering is that of the arXiv versions.
3. Lemma 5.1(a) is called classical, with what the sources read say about it (Kapovich–Levitt–Schupp–Shpilrain:
   well known, probably going back to Klein, proof in Horowitz's paper; quoted from Horowitz in the proof of
   Masters' Lemma 4.1).
4. The theorem of Arnoux–Yoccoz and Ivanov: the note says that it uses the finiteness of the number of
   conjugacy classes below a bound, that the source quotes the theorem as the discreteness of the spectrum, and
   that the two papers were not read. Remark 7.1 now gives the reason why the values with large multiplicity
   tend to infinity without this theorem (finitely many conjugacy classes of hyperbolic elements of PSL(2,Z)
   with a given trace, and Corollary 6.2).
5. Credit in Section 4: Thurston's construction in the description of Leininger, Section 5.1; Lemma 4.2 is
   known (Leininger, Lemma 6.3 and Proposition 6.4); the first sentence of Remark 7.2 is Leininger's Lemma 6.3.
6. "The construction in the proof of" Proposition 4.8 of Farb–Leininger–Margalit (the statement of that
   proposition is a bound).
7. Small changes of wording, without change of content: the remark on good points names the four corners; the
   cyclic orders in Lemma 3.2 are those of the orientations; in Proposition 3.3(a) the two occurrences of an
   edge lie in different squares; in Lemma 4.1(ii) the class of an affine automorphism stabilises the disc; in
   Remark 7.2 the two passages through one point are different because ⟨T_a, T_b⟩ is torsion-free; u and v are
   printed in the standard basis; Aougab–Taylor use "an induction of the same kind"; the reference to Randol
   and Horowitz in Section 1.1 is not the source's.
8. Remark 7.1 states the primitivity of the elements g_ε for all 24 instances.
9. "Scope and priority" lists what the third run read (Leininger, Sections 5.1, 6.1, 6.2; the introduction of
   Farb–Leininger–Margalit; the proof of Theorem 5.1 of Aougab–Taylor), counts seven web searches, and names the
   papers of Johnson and of Bass–Lubotzky among those not read.

The abstract, the theorem, the lemmas, Table 1 and all numbers are unchanged; the note has 13 pages instead
of 12. These changes were made by the third run itself and were not examined by a further run; this holds also
for the sentences which are new in content: the attributions to Leininger's Sections 5.1 and 6, the last
sentence of Remark 7.2, and the sentence of Remark 7.1 on the values tending to infinity.

## Relation to the literature, novelty and scope
- **Searches (2026-10-11).** arXiv, OpenAlex, zbMATH Open, Crossref, Semantic Scholar (pseudo-Anosov classes
  with equal dilatation or stretch factor; multiplicities in the length spectrum of moduli space; dilatations
  in the Torelli group and the Johnson filtration; Horowitz and Randol in connection with mapping classes); the
  works which cite the source (88 citing the chapter and 49 citing the volume in OpenAlex; 124 in Semantic
  Scholar; titles and available abstracts read, texts not read); MathOverflow through its public interface;
  seven web searches on the question. The third run repeated a part of this on the same day: arXiv (9 queries,
  among them the newest papers on pseudo-Anosov classes in the Torelli group and the Johnson filtration),
  OpenAlex (the works citing the chapter and the volume since 2024; phrases in titles, abstracts and full
  texts), zbMATH Open, and one web search.
  - No text other than the source and the corpus record which states or answers Question 7.6 was found.
  - The closest items concern single pairs of non-conjugate pseudo-Anosov classes with equal dilatation; they
    do not concern the Torelli group, the Johnson filtration or unbounded multiplicity.
  - The papers on dilatations in the Johnson filtration (Farb–Leininger–Margalit; Malestein–Putman;
    Aougab–Taylor; Jeffreys) and Margalit's problem list contain no statement on multiplicities (full texts of
    the arXiv versions searched).
- **What was read** (arXiv versions): the source, Sections 1.2, 7.1–7.3 and Theorem 5.10 with its proof
  (pp. 6–7, 33–34, 44–47); Leininger (arXiv:math/0304163), Sections 2.1, 3.1, 3.2 and, by the third run,
  Sections 2.4, 5.1, 6.1, 6.2 (positive twists; Thurston's construction; Theorem 6.1, Lemma 6.3,
  Proposition 6.4);
  Farb–Leininger–Margalit (arXiv:math/0603675v2), the introduction, Section 2.3, Proposition 4.8 with its
  proof, Theorem 1.2; Aougab–Taylor (arXiv:1510.00995), Lemma 2.2 and Theorem 5.1 with their proofs; Jeffreys
  (arXiv:2210.11332), Theorem 1.2 and Table 1.1; Kapovich–Levitt–Schupp–Shpilrain (arXiv:math/0409284v2),
  introduction and Section 5; Masters (arXiv:math/9812069), Theorem 3.1 and Lemma 4.1, for the theorems of
  Randol and Horowitz. The bibliographic data were compared with Crossref, twice (all 12 DOIs resolve, and the
  data agree with the reference list); the entry of Arnoux–Yoccoz, which has no DOI, was compared with zbMATH
  Open by the third run.
- **Not read, or not accessible.** The printed volume of the source and the journal versions of the papers
  above; the original papers of Bers, Thurston, Kra and Veech (their statements are taken from Leininger's
  paper); those of Randol and Horowitz (taken from Masters' paper); those of Johnson and of Bass–Lubotzky
  (quoted through the source and through Farb–Leininger–Margalit); Anderson's survey; the papers of
  Arnoux–Yoccoz and Ivanov. The proofs of the cited theorems were not checked.
- **Caveats.** Leininger's Theorem 3.2 (unique invariant Teichmüller disc of a pseudo-Anosov class) is stated
  there without proof and attributed to Bers. The theorem of Arnoux–Yoccoz and Ivanov is used in the note (for
  the remark that each multiplicity is finite, which no proof needs) as the finiteness of the number of
  conjugacy classes below a bound, while the texts which were read quote it as the discreteness of the
  spectrum. The answer follows so directly from the source's own sketch that
  it may be regarded as known, and it cannot be excluded that the question was meant in a sense which was not
  considered. A search that finds nothing is not a proof of novelty.
- **Scope.** The note answers Question 7.6 in the negative for all g ≥ 2 and all k. It does not answer
  Question 7.7, and it does not determine the growth of the multiplicities. 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/