# Verification report — AMR-109-0231 (Mosher, Problem 3.3 of Farb's problem book: an algorithm for the exponent M)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete affirmative answer** to the question, as a promise problem, **modulo the
published theorems listed in its Section 3**; every other step is proved in the note.
- **Settled.** There is an algorithm which, given g ≥ 2, n ≥ 1 and words for mapping classes φ1, …, φn of a closed
  oriented surface S of genus g, promised to be pairwise independent pseudo-Anosov, halts and outputs an integer M
  such that for all integers α1, …, αn ≥ M, chosen independently, the classes φi^αi freely generate a free group H
  of rank n whose orbit map into the curve graph C(S) is a quasi-isometric embedding; hence H is convex cocompact
  (Schottky) and Γ_H is word hyperbolic (Theorem A = Theorems 5.1 and 6.1). Outside the promise the search may not
  halt, but any output is correct (Remark 5.3).
- **Also proved.** For n ≥ 3 no M depending only on S exists (Remark 2.1).
- **Not new mathematics in its ingredients.** Every ingredient was in print. The closest prior work is H. Kim,
  J. Algebra 554 (2020) 106–138 (arXiv 1908.04460), whose Theorem A(i) is a semi-algorithm certifying convex
  cocompactness (stability) of one given subgroup by the same local-to-global method in C(S). What the note adds is
  the part uniform in the exponents (Lemma 4.3, Step 2 and the bookkeeping of Theorem 5.1) and a written, complete
  proof. Experts may regard the answer as folklore. No priority is claimed.
- **Limits.** The constants are crude (δ = 340; M is huge and far from optimal); the algorithm has not been
  implemented and no curve-graph computation was run. The decidability of the promise is not proved.

The note is unrefereed.

## Statement checked
- **Primary source.** L. Mosher, "Problems in the geometry of surface group extensions", Chapter 16 of B. Farb (ed.),
  Problems on Mapping Class Groups and Related Topics, Proc. Sympos. Pure Math. 74, AMS 2006,
  doi:10.1090/pspum/074/2264544.
  - Read in the open copy of the book, https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf (2,724,624 bytes,
    sha256 `f37c6a1dbc875105c2b294196de1a03a88595b75e8705e6077e9d48f066e402a`), PDF pages 268–270 (printed
    pages 261–263; Sections 1–3 in full); the chapter occupies printed pages 261–273 (PDF pages 268–280) in this
    copy, while Crossref lists pages 245–256. The note cites the printed pages of the copy read.
  - Conventions: S closed, oriented, genus g ≥ 2; MCG(S) = Homeo(S)/Homeo0(S) (orientation-reversing classes
    included); Γ_G is the preimage of G in MCG(S, p) ≅ Aut(π1(S, p)). Independence: disjoint fixed pairs on the
    Thurston boundary, "equivalently" different pairs.
  - Theorem 3.1 [Mosher, Proc. AMS 125 (1997)]: for pairwise independent pseudo-Anosov φ1, …, φn there is M > 0
    such that for integers α1, …, αn ≥ M the powers freely generate H with Γ_H word hyperbolic. Theorem 3.2: H is
    Schottky (via Theorem 2.1 = Farb–Mosher).
  - Problem 3.3, quoted verbatim in the note: "Does there exist an algorithm which produces the integer M in
    Theorem 3.1, given φ1,...,φn?" The sentence before it asks to make the theorem "more effective"; after it the
    source names Hamidi-Tehrani's algorithm, which gives freeness only.
- **Corpus record.** ulamai/UnsolvedMath, AMR-109-0231 (snapshot d371ee9; status `open`, research classification
  OPEN-TRIAGE). Its statement is the source's question verbatim, with the source item "Problem 3.3, PDF page 270".
  Nothing is truncated.

## Readings
| Reading | Answer | Where |
|---|---|---|
| An algorithm that, on inputs promised to be pairwise independent pseudo-Anosov, outputs some M valid for all tuples αi ≥ M (independent exponents) | yes | Theorem A (Theorems 5.1, 6.1) |
| The same, with the promise removed (a total algorithm) | not proved; a sketch is given and labelled as not written out and not checked | Remark 5.3 |
| An M depending only on S | does not exist for n ≥ 3; exists for n = 2 by Fujiwara (not used) | Remark 2.1 |
| The least M; an efficient algorithm | not asked, not given | Section 1.2 |
| Freeness only | weaker; Hamidi-Tehrani's algorithm (named in the source) | Section 1.1 |
| n = 1 | trivial (M = 1) | Section 1.2 |

## Results in the paper
- **Lemma 4.1.** If every triangle of geodesic edge-paths in a connected graph is k-centred (the definition of
  Hensel–Przytycki–Webb), the vertex set satisfies ⟨x,z⟩_w ≥ min(⟨x,y⟩_w, ⟨y,z⟩_w) − 20k; with k = 17 this gives
  δ = 340 for C(S). Five steps: (1) d(w, s_xy) ≤ ⟨x,y⟩_w + 4k; insize 8k; thinness 16k; slimness 16k; 4k + 16k.
- **Lemma 4.2** (local-to-global for broken sequences): consecutive distances ≥ L, middle Gromov products ≤ C,
  L > 2C + 2δ ⇒ ⟨y0, y_{k+1}⟩_{y_k} ≤ C + δ and d(y0, ym) ≥ m(L − 2C − 2δ) + 2(C + δ).
- **Lemma 4.3** (periodic sequences): z_k = f^{kP}x, l = d(x, f^P x), c = ⟨f^{−P}x, f^P x⟩_x, l > 2c + 3δ ⇒
  d(z_{k1}, z_{k3}) ≥ (k3 − k1)λ with λ = l − 2c − 2δ > δ, and ⟨z_{k1}, z_{k3}⟩_{z_{k2}} ≤ c + 2δ.
- **Algorithm** (Section 5): Step 1 periods P_i (first P with l > 2c + 3δ) and R_i; Step 2, for every pair of
  ends, the first N with d(x,u) > G + c_i + 4δ and d(x,v) > G + c_j + 4δ; Step 3 the constants C, L, K_i and
  M = max P_i K_i. Every quantity is an exact curve-graph distance between explicit curves.
- **Theorem 5.1** (correctness for all tuples αi ≥ M; parts (i)–(v), including the passage to H⁺ = H ∩ Mod(S)).
- **Theorem 6.1** (termination): Masur–Minsky translation length for φi² and the Morse lemma (Step 1); Mosher's
  Theorem 3.1 with an even exponent, Farb–Mosher, Kent–Leininger and the Morse lemma (Step 2). The non-explicit
  constants enter only the termination proof.
- **Remark 2.1** (no uniform M for n ≥ 3); **Remark 5.3** (promise).

Published results used as black boxes (Section 3 of the note): Hensel–Przytycki–Webb Theorem 1.1;
Masur–Minsky (hyperbolicity; Proposition 3.6 of the arXiv version, numbered 4.6 in the journal version according
to Kent–Leininger, not compared); Bridson–Haefliger III.H.1.7; Kent–Leininger Theorems 1.3/7.1 (and Hamenstädt);
Farb–Mosher via the chapter's Theorem 2.1; Mosher's Theorem 3.1; computability of the action on curves (Penner
1984) and of curve-graph distance (Leasure, Shackleton, Webb, Birman–Margalit–Menasco; Bell–Webb).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation; the algorithm has not been implemented; no curve-graph computation was run.
- `check_note.py` (written with the note; standard library; about 2 s; 9 checks, 0 failures): the finite and exact
  parts (constants, identities (I1)–(I3), algebra of Lemma 4.2, identity of Lemma 4.3, bookkeeping and case split of
  Theorem 5.1, the relation of Remark 2.1 in the free group, arithmetic of Theorem 6.1) and Lemma 4.1 by brute force
  on 14 small graphs.
- `original/toy_tree_check.py` (about 1.4 s): Steps 1–3 for F2 acting on its Cayley tree, δ = 0 and the deliberately
  too large δ = 1, 3; 27 runs, recorded output of 28 lines (a header and 27 runs); zero violations; for the Remark
  2.1 triples M > K and rank 2 at exponent K. **In a tree δ = 0, so this test checks only the bookkeeping of Steps
  1–3 and of Theorem 5.1 and cannot detect an error in any δ-coefficient; it is not part of the proof.**
- Programs of verification runs A and B (tests of the δ-terms, evidence only): `lemma42_finite_test2.py` (run A;
  Lemma 4.2 in eight finite graphs with exact four-point constant d4 ∈ {1, 2, 3}; 200,524 admissible sequences,
  zero violations; control δ = d4/2: 41,400 violations); `vB_lemma41_graphs.py` (run B; Lemma 4.1 on 53 graphs,
  worst ratio 2; the false bound k*/4 and Lemma 4.2 with δ := 0 are refuted, while its control dropping δ from the
  conclusion of Lemma 4.2 alone found no violation); `vB_h2_model.py` (run B; Steps 1–3 for PSL(2,Z) on the hyperbolic plane, δ = log 2 and 1;
  44 runs, zero violations; a wrong M is refuted; dependent pairs never halt within the cap).
- Programs of the third verification run (`independent_run_2/`, evidence only): `r2_constants.py` (11 exact or
  symbolic checks of the finite claims, 0 failures); `r2_lemma41_42_graphs.py` (every intermediate bound of the proof
  of Lemma 4.1 on 37 graphs with the exact centring constant, largest ratios 2–4 against the proved 3–20, false
  smaller bounds refuted; Lemma 4.2 exhaustively, 409,246 admissible sequences, 0 violations; controls δ' = d4 − 1/2
  and d4/2: 10,076 and 10,676 violations); `r2_freeprod_model.py` and `r2_h3_model.py` (Steps 1–3 and checks (i)–(iv)
  for free products of cyclic groups on their Cayley graphs, 50 runs, and for PSL(2, Z[i]) on hyperbolic 3-space,
  20 runs; 0 violations; wrong M refuted; dependent pairs reach the cap). **These two models hardly exercise the
  δ-terms** (`r2_freeprod_excess.py`: excess 0 over c in the free products; at most 0.03δ in hyperbolic 3-space) and
  smaller δ-coefficients are not refuted there: they check the bookkeeping with δ > 0, not the coefficients of δ.
  Of all the tests, only those of Lemma 4.2 in finite graphs detect a too small δ.
- All recorded outputs were reproduced byte for byte on 2026-10-20, those of the first five programs again by the
  third verification run from a fresh copy of the released source archive.

## Independent verification runs
The result was first obtained with a written proof and the toy test. Three independent verification runs, all
AI-assisted, followed on 2026-10-20, each with its own programs: runs A and B on the first write-up, then a third
run on the text of the note (its own section below).

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, reading against the source | CONFIRMED (source read) | CONFIRMED (source read) |
| Lemma 4.1 (20k; δ = 340) | CONFIRMED (re-derived; tested on 29 graphs) | CONFIRMED (re-derived; tested on 53 graphs) |
| Lemmas 4.2, 4.3 | CONFIRMED (re-derived; Lemma 4.2 tested in non-tree graphs with a δ/2 control) | CONFIRMED (re-derived; tested on graphs and in H²) |
| Theorem 5.1 (i)–(v), passage to H⁺ | CONFIRMED | CONFIRMED |
| Remark 2.1 | CONFIRMED | CONFIRMED |
| Theorem 6.1 | CONFIRMED (Morse lemma not re-read) | CONFIRMED (Morse lemma not re-read) |
| Black boxes HPW, KL, MM, source theorems | CONFIRMED (read in the arXiv versions) | CONFIRMED (read in the arXiv versions) |
| Distance algorithms | bibliographic data checked; scope not checked | abstracts read (Shackleton: all surfaces) |
| Literature and novelty | one web search; agrees: not found in print | found Kim (2020), missing from the first write-up |
| Overall | CONFIRMED (four minor fixes) | CONFIRMED_WITH_FIXES (Kim; one line count) |

Corrections required and made (see the note): the toy-test caveat (δ = 0) wherever the test is described, with
computational checks of the δ-terms added to the package; the journal numbering for Masur–Minsky; the scope of the
distance algorithms (Birman–Margalit–Menasco is stated for closed surfaces; Shackleton "applies to all surfaces")
and a reference for the action on curves (Penner 1984); the sentence that Kent–Leininger's convex cocompactness is
Farb–Mosher's notion and that Farb–Mosher is applied to H⁺ < Mod(S) for closed S; the promise (Remark 5.3, with the
unchecked sketch of its removal); Kim (2020) cited as closest prior work, with the difference stated, in the
abstract, the introduction and the scope paragraph; the line count of the toy output (28 lines). Optional points
done: Bell–Webb, Bestvina–Bromberg–Kent–Leininger, the remark on Fujiwara's bound for n = 2, Runnels' Dehn-twist
example (p. 3 of the arXiv version) and Fujiwara's Example 18.

## Third independent verification run (on the text of the note)
AI-assisted, 2026-10-20; it examined the written note line by line, as if nothing had been checked, and wrote its
own programs before re-running the package.

| Item | Third run |
|---|---|
| Statement and conventions against the source (chapter re-downloaded, PDF pp. 266–272 read) and the corpus record | CONFIRMED (faithful; the record does not restate Theorem 3.1) |
| Kim (2020), Theorem A(i) and §2.1 (arXiv v1 read): one subgroup, not one M for all tuples | CONFIRMED as described in the note |
| Lemma 4.1: Step 1 (3k, (1) with 4k), Step 2 (3k, 8k), Step 3 (16k), Step 4, Step 5 (20k, δ = 340), degenerate triangles, HPW's definition | CONFIRMED (re-derived; tested on 37 graphs) |
| Lemma 4.2 (case split impossible exactly when L > 2C + 2δ; distance bound) | CONFIRMED (re-derived; 409,246 sequences) |
| Lemma 4.3 (l > 2c + 3δ needed for (b); λ > δ; c = l − d(x, f^{2P}x)/2) | CONFIRMED (re-derived) |
| Algorithm and Theorem 5.1 (i)–(v): stopping rule of Step 2, C, L, K_i, M, all four sign patterns, reduced-word case split, H⁺, KL in Mod(S), FM for closed S | CONFIRMED (re-derived; tested in two non-tree models) |
| Theorem 6.1 (MM for φ_i², quasi-geodesics in the sense of BH, Morse bound on c(P); Theorem 3.1 with even a*, FM, KL, the path h_r); no non-explicit constant in the output | CONFIRMED (Morse lemma not re-read) |
| Remark 2.1, Remark 5.2, Remark 5.3 (promise not used for correctness; its removal only mentioned) | CONFIRMED |
| Cited statements: HPW, KL, MM, Fujiwara, Runnels (arXiv texts), BMM (introduction), Shackleton, Webb, Bell–Webb (abstracts) | CONFIRMED |
| Bibliography: all 20 DOIs in Crossref | CONFIRMED |
| Re-run of the package from the released archive | all outputs identical |
| Literature (arXiv API, zbMATH, Semantic Scholar, Crossref, one web search; OpenAlex citing list refused) | no printed answer found |
| Overall | CONFIRMED with minor fixes; no mathematical error |

Corrections required and made: a precise account of the controls of `vB_lemma41_graphs.py` (one of its three
controls found no violation); the abstract's description of the tests (the tree test is no longer called the main
one; it still says that it checks only the bookkeeping because δ = 0); the statement of Fujiwara's Theorem 16 in
Section 2 and in the related work; two further sources not seen (Hamidi-Tehrani's preprint and Mosher's
train-track paper, both named in the chapter); the record of this run in the Verification paragraph, Section 7,
this report and `reproducibility/independent_run_2/`, and its searches in the scope paragraph.

## Relation to the literature, novelty and scope
- **Searches (20 October 2026).** arXiv API; OpenAlex works citing the chapter (5, none about Problem 3.3);
  zbMATH (one citing document, not relevant); Semantic Scholar citers of Mosher 1997 (52), Kent–Leininger 2008 (134)
  and Fujiwara II (20), screened by title; Crossref (bibliographic data, all DOIs of the note checked again when the
  note was written); five web searches in all. The third verification run repeated the search on the same day
  (arXiv API, zbMATH, Semantic Scholar citers of Mosher 1997, Kent–Leininger and Kim, Crossref for all 20 DOIs, one
  web search) and confirmed these counts; OpenAlex confirmed the count of 5 citing works but refused to return the
  list (rate limit), which that run therefore did not re-check. No paper was found which states an algorithm for
  M or answers Problem 3.3. A search that finds nothing is not a proof of novelty.
- **Closest prior work.** Kim (2020): Theorem A(i) is a partial algorithm that halts iff one given finitely generated
  subgroup of the mapping class group is stable (= convex cocompact), with an orbit in C(S), a local-to-global
  principle with computable constants and the distance algorithms (Kim, Section 2.1; arXiv version 1 read). Problem
  3.3 needs one M valid for infinitely many tuples; Kim's procedure certifies one subgroup and, run tuple by tuple,
  never produces such an M. Kim does not mention Mosher's problem.
- **Other related work.** Kent–Leininger, Theorem 1.4 and its proof (non-effective template, common exponent);
  Fujiwara (Trans. AMS 367 (2015)): Theorem 16, a uniform M(S) for n = 2 (no algorithm, nothing for n ≥ 3), Theorem 14
  (acylindrical actions), Example 18; Runnels (Geom. Dedicata 214 (2021)), Theorem 4.1 (explicit common exponent,
  RAAGs, undistorted after enlarging); Bestvina–Bromberg–Kent–Leininger (Crelle 2020); Hamidi-Tehrani (freeness only;
  thesis not seen); Anderson–Aramayona–Shackleton (2007; freeness only; abstract read); Mangahas (AJM 2013; read only
  in part at the finding stage).
- **What was read.** The chapter (Sections 1–3); arXiv versions of Hensel–Przytycki–Webb, Kent–Leininger,
  Masur–Minsky, Fujiwara, Runnels in the parts cited; Kim arXiv v1 (introduction, Section 2); Birman–Margalit–Menasco
  arXiv v2 (introduction and the scope of Theorem 1.1); abstracts of Shackleton, Bell–Webb, Hamenstädt, BBKL,
  Anderson–Aramayona–Shackleton; the zbMATH review of Penner (1984). Not seen: Leasure's thesis, Hamidi-Tehrani's
  thesis and preprint, Mosher's train-track paper suggested in the chapter as an approach. Not re-read: Bridson–Haefliger. Farb–Mosher quoted through the chapter's Theorem 2.1; Durham–Taylor through
  Kim's Theorem 2.2. Journal versions not compared; proofs of the cited theorems not checked.
- **Scope.** A complete affirmative answer modulo the cited published theorems; modest novelty; crude constants;
  not implemented. 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/