# Verification report — KP-2.28 (K3 Problem 2.28: right-angled Artin groups in mapping class groups with every loxodromic element pseudo-Anosov)

Verification date: 2026-10-08. Paper: "Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov
Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List" (39 pages, dated October 9, 2026).

**Verdict.** The note gives a partial answer. The problem remains open in general. For a finite graph Γ that is
not a join, an injective homomorphism A(Γ) → Mod(S) that sends every loxodromic element to a pseudo-Anosov mapping
class exists, with S a closed surface, in the following cases: Γ chordal (Theorem 1.1); the complement of Γ
connected and chordal (Theorem 1.2); Γ the complement of a cycle of length at least five, which includes the
pentagon (Theorem 1.3); the complement of Γ built from complete graphs and cycles by clique-sums (Theorem 1.4);
disjoint unions of such graphs and of joins, with isolated vertices added (Theorem 1.5); the hexagon, complements
of cycles with one more vertex, and graphs obtained from these by adding twins (Theorem 1.6). With a computer
enumeration this covers every graph with at most six vertices that is not a join, and 758 of the 853 such graphs
with seven vertices; four more of these are settled by the finite-index property, and for the remaining 91 there
is no answer. Four independent verification runs, all AI-assisted, examined the proofs line by line: the first two
a working write-up of Theorems 1.1 to 1.5, the third the finished manuscript (in particular Section 8, Section 9
and Theorem 1.6), the fourth the final text as a whole. None found a mathematical error; their corrections of
citations, definitions and wording are applied. The note is unrefereed.

## Statement checked
- **Primary source.** R. İ. Baykur, R. C. Kirby, D. Ruberman (eds.), *K3: A New Problem List in Low-Dimensional
  Topology*, Math. Surveys Monogr. 295, AMS 2026, doi:10.1090/surv/295, Problem 2.28 (pp. 108–109 of the authors'
  preliminary version, which is the version that was read), recorded by T. Koberda. In paraphrase:
  - Γ is a finite graph that is not a join, and A(Γ) is its right-angled Artin group. Is there an injective
    homomorphism from A(Γ) to the mapping class group of a surface that sends every loxodromic element to a
    pseudo-Anosov mapping class?
  - Remark (4) of the problem gives a sufficient condition in terms of subsurfaces S_v, one for each vertex:
    disjoint for adjacent vertices, overlapping otherwise, and such that for every connected dominating set Y of
    the complement graph the S_y, y ∈ Y, fill the surface. No case is listed as known.
- **Wording of the source.** The source says "not a nontrivial join" and "injective map". The note writes "not a
  join" (a join in the note is a nontrivial join) and "injective homomorphism", which is the reading of the
  remarks of the problem; Section 1.1 of the note says so.
- **Definitions.** "Loxodromic" is taken from Koberda–Mangahas–Taylor (Trans. AMS 369 (2017), Sections 2.2 and
  2.4) and Kim–Koberda: g ≠ 1 and g is not conjugate into a join subgroup. They develop the notion for connected
  graphs that are not joins. The note applies the same definition to all graphs (Definition 2.1 and Section 2.2);
  for graphs that are not connected this is an extension, and it makes the powers of an isolated vertex
  loxodromic. With this definition a group A(Γ) contains loxodromic elements exactly when Γ is not a join, as in
  Remark (2) of the problem.
- **Corpus record.** ulamai/UnsolvedMath, KP-2.28 (version 1.6.0, upstream status `open`). The statement agrees
  with the source; the background field reproduces Remarks (1) to (4).

## Readings
| Reading | Answer | Where |
|---|---|---|
| The problem as posed: every finite graph that is not a join | open; not decided by the note | Sections 1.4 and 10 |
| Γ chordal, without isolated vertices | yes, on every closed surface of genus ≥ n + 2 | Theorem 1.1 |
| complement of Γ connected and chordal, Γ without isolated vertices | yes, on every closed surface of genus ≥ 2r (r maximal cliques) | Theorem 1.2 |
| Γ the complement of a cycle C_m, m ≥ 5 (the pentagon for m = 5) | yes, genus (d − 1)(m − 2)/2 for every odd d ≥ 5 | Theorem 1.3 |
| complement of Γ built from complete graphs and cycles of length ≥ 4 by clique-sums, Γ without isolated vertices | yes | Theorem 1.4 |
| disjoint unions of such graphs and of joins; isolated vertices | yes, with "loxodromic" in the extended sense | Theorem 1.5 |
| connected graphs whose group has finite index in the group of a connected graph with a positive answer | yes | Proposition 3.8 (uses Koberda–Mangahas–Taylor, Theorem 2.2) |
| the hexagon; complements of cycles with one more vertex; twins (at most one true twin per vertex) | yes | Theorem 1.6(1)–(3) |
| every graph with at most 5 vertices that is not a join | yes | Theorem 1.5 (case analysis in the text; also enumerated) |
| every graph with 6 vertices that is not a join (112 graphs) | yes; 105 by Theorem 1.5, the other 7 by Theorem 1.6 | Theorem 1.6(4), Table 2 |
| graphs with 7 vertices that are not joins (853 graphs) | 758 satisfy the hypothesis of Proposition 9.3 (578 that of Theorem 1.5); 95 do not, all connected; 4 of these are doubles along stars and are settled by Proposition 3.8; no answer for 91 | Table 1, Remark 3.9, Sections 1.4, 9.3, 10 |
| a negative answer for some graph | none; no obstruction was found | Section 10 |
| target a closed surface | all constructions give closed surfaces | throughout |
| least genus; explicit powers of the pseudo-Anosov maps | not treated | Section 10 |
| condition (iii) of Remark (4): "adjacent to a vertex of Y" as domination or as total domination | the two readings differ only for a one-element Y, an isolated vertex of Γ | Section 3.1 |

## Results in the paper
- **Section 2.** Definitions; the quoted facts (K1) to (K4) on subsurface projections; Lemma 2.2 (elementary
  facts on subsurfaces, with proofs); Lemma 2.3 (a curve, or a multicurve, with intersection number zero with
  finitely many curves in minimal position can be made disjoint from all of them); Theorems 2.4 and 2.5
  (Clay–Leininger–Mangahas, Theorems 2.2 and 6.1 of the published version, quoted); Lemma 2.6 (the class 𝒦 is
  recognised by the atoms of a decomposition along clique separators).
- **Section 3.** Lemma 3.1 (a vertex set is contained in a join exactly when it does not induce a connected
  dominating subgraph of the complement). Proposition 3.2 (criterion): for a nice realization, the homomorphism
  of Clay–Leininger–Mangahas sends all loxodromic elements to pseudo-Anosov classes if and only if the filling
  property (F) holds. Reductions: induced subgraphs, isolated vertices, disjoint unions, finite index, doubles
  along stars (Remark 3.9, with Lemma 50 of the arXiv version of Kim–Koberda, Geom. Topol. 2013).
- **Section 4.** Theorem 1.1: one-holed tori chosen along a reversed perfect elimination ordering; Lemmas 4.1
  to 4.5. Remark 4.7: for disjoint unions of complete graphs the result is not new (Aougab et al.; Loa for two
  components).
- **Section 5.** Territory systems; Lemma 5.2 (territory lemma: nested or equal territories can be replaced by
  overlapping one-holed tori; uses the Behrstock inequality in the form of Mangahas, Lemma 2.5); Proposition 5.3
  (territories from a clique tree) and Theorem 1.2; Lemma 5.4 (nice realizations of arbitrary graphs).
- **Section 6.** Lemma 6.1 (totally ramified cyclic covers), Lemma 6.2 (units of Z/d), Lemma 6.3 (lifting
  lemma; part (4) is stated with the hypothesis on the complementary regions, part (5) gives the partition
  criterion), Proposition 6.5 and Theorem 1.3.
- **Section 7.** Lemma 7.1 (blocks: complete graphs, cycles C_m with m ≥ 5, the 4-cycle), Lemma 7.2 (gluing
  lemma for pointed territory systems), Theorem 1.4.
- **Section 8.** Proposition 8.1 (false twins), Lemma 8.2 and Proposition 8.3 (true twins, one for each of some
  pairwise distinct vertices), Lemma 8.4 and Proposition 8.5 (complements of cycles with one more vertex),
  Lemma 8.6 and Theorem 8.7 (the hexagon: six twice-punctured discs on the five-punctured sphere, lifted to a
  cyclic cover of odd degree d ≥ 5; genus 6 for d = 5).
- **Section 9.** Proof of Theorem 1.5, with a case analysis for graphs with at most five vertices. Definition
  9.1, Lemma 9.2, Proposition 9.3 and the proof of Theorem 1.6. The enumeration, Tables 1 and 2.

## Computations (scripts and outputs in reproducibility/)
All programs were run again on 2026-10-08, the last time from the extracted source archive (`rerun_all.sh`, about
three minutes). The outputs agree with the recorded ones; only lines that report running times differ.
- **Enumeration (the note depends on it for the counts and for Theorem 1.6(4)).**
  - Graphs up to isomorphism with 1, …, 7 vertices: 1, 2, 4, 11, 34, 156, 1044; not joins: 1, 1, 2, 6, 21, 112,
    853; connected and not joins: 1, 8, 68, 662 for 4, 5, 6, 7 vertices.
  - Column (i) of Table 1 (Theorems 1.1 to 1.3 with the closure properties): 93/112 and 477/853. Column (ii)
    (Theorem 1.5): 105/112 and 578/853; connected graphs: 61/68 and 394/662. Five programs that share no code
    agree: `finder/coverage*.py`, `independent_run_2/v2_coverage.py`, `lead/lead_coverage.py`,
    `independent_run_3/c_enum.py`, `independent_run_4/r4_enum.py`. They also agree on the seven graphs with six
    vertices of Table 2.
  - Column (iii) (Proposition 9.3): 112/112 and 758/853; connected graphs: 68/68 and 567/662. Computed by three
    programs that share no code (`lead/lead_coverage.py`, `c_enum.py`, `r4_enum.py`). Their lists of the 95
    graphs with seven vertices that do not satisfy the hypothesis of Proposition 9.3 agree graph by graph up to
    isomorphism; all 95 are connected and have 7 to 13 edges.
  - Four of the 95 graphs are doubles along stars of graphs with five vertices (the three programs, and
    `lead/lead_star_multiples.py`). So 91 graphs with seven vertices are without answer.
  - With Theorem 1.5 and Theorem 1.6(1)–(3) as stated, without Proposition 9.3, 710 of the 853 graphs are covered
    (`c_variants.py`, `r4_enum.py`); this is why the note attaches the number 95 to Proposition 9.3.
- **Lemma 8.6 (proved in the text; checked for the coordinates of Figure 4).**
  - `independent_run_2/v2_run_planar.py`: exact rational arithmetic; it also builds the cyclic cover for
    d = 5, 7, 9, 15 as a cell complex and confirms the conclusions of Theorem 8.7.
  - `lead/lead_hexagon_planar.py`: the complementary regions on a grid; (H1), (H2), (H3) hold, and eight control
    sets inside joins fail as they should.
  - `independent_run_3/c_hexagon.py`: exact rational arithmetic, for coordinates of its own and for the six
    polygons printed in the TikZ code of Figure 4; all 15 pairs and all 63 vertex sets; covers for
    d = 5, 7, 9, 11, 13, 15, 21, 25.
  - `independent_run_4/r4_hexagon.py`: exact rational arithmetic with a different method, for the printed
    polygons; (H1), (H2), (H3); for all 63 vertex sets the hypothesis of Lemma 6.3(4) holds exactly for the 39
    sets that are not contained in a join of the hexagon.
- **Consistency checks of statements proved in the text.** Lemma 3.1 for all graphs with at most six vertices
  (several programs; one over all 2,097,151 pairs of a labelled graph and a vertex set); the combinatorial steps
  of the proof of Theorem 1.1 and of Proposition 5.3 for all graphs with at most seven vertices to which they
  apply; the two facts in the proof of the gluing lemma (552,680 cases); Lemma 6.4 for m ≤ 12; Lemma 6.2(b) for
  odd d ≤ 151 and m ≤ 200; the cyclic covers of Proposition 6.5 for m ≤ 8 and d = 5, 7, 9, 15 as cell complexes;
  the combinatorial steps of Propositions 8.1, 8.3 and 8.5 for small graphs; Lemma 2.6 for all connected graphs
  with at most seven vertices.
- These computations do not replace any proof, with one exception: the counts of Table 1 and the statement that
  the seven graphs of Table 2 are the only graphs with six vertices outside Theorem 1.5 are results of the
  enumeration.

## Sources read
- Read in the arXiv version and in the published version, with the numberings compared: Clay–Leininger–Mangahas
  (Groups Geom. Dyn. 6 (2012)); Koberda–Mangahas–Taylor (Trans. AMS 369 (2017)).
- Read in the arXiv version (the statements that are used): Kim–Koberda (Geom. Topol. 2013, Lemma 50; Int. J.
  Algebra Comput. 2014); Masur–Minsky I and II; Mangahas (Lemma 2.5); Behrstock (Theorem 4.3 located); Loa
  (Theorem 1.1); Aougab–Bray–Dowdall–Hoganson–Maloni–Whitfield (Theorem A, Corollary 1.1, Remark 1.2);
  Katayama–Kuno (Theorem 1.3); Bering–Conant–Gaster; parts of Mangahas–Taylor (the introduction, the definition
  of admissible embeddings, Section 8) and of Runnels (version 1).
- Abstracts only: Kim–Koberda (IMRN 2014); Seo; the published version of Runnels (Geom. Dedicata 214 (2021)
  277–294), on the publisher's page. That abstract says, beyond the statements on generation and undistortedness,
  that each element is pseudo-Anosov on the largest possible subsurface. Version 1 on arXiv, the only version
  there, has no such sentence in its abstract; its main theorem concerns generation and undistortedness, and a
  search of its text finds no statement on the type of the elements. The introduction of Clay–Leininger–Mangahas
  describes their Theorem 6.1 in the same words. The note therefore treats the sentence as a statement of the
  kind of that theorem (the type of the image of each element for given supports), says that it does not address
  the question of Problem 2.28, and does not use it. The published article itself was not read.
- Not consulted, quoted in standard form: Crisp–Paris; Koberda (GAFA 2012); the books of Farb–Margalit and
  Lyndon–Schupp; Dirac, Fulkerson–Gross, Gavril, Buneman, Tarjan (chordal graphs); Thurston; Penner; Sinden;
  Farb–Mosher.
- The problem and its remarks were read in the authors' preliminary version of the K3 list.
- All 27 DOIs of the bibliography were requested from Crossref and the bibliographic data compared (fourth run);
  no Crossref record exists for the three references without DOI (Loa, Katayama–Kuno, Bering–Conant–Gaster).

## Independent verification runs
All four runs were AI-assisted. They are not peer review. The first two examined a working write-up, not the
text of the note; the final text of Sections 2 to 7 and 9.1 follows that write-up, with the corrections listed
below and with some arguments written out in more detail (Lemma 2.2(d), (e); Lemma 2.3; Lemma 2.6; the case
analysis for graphs with at most five vertices). The third run examined the finished manuscript, and the fourth
the final text.

### First run (2026-10-08): criterion, Theorem 1.1, Theorem 1.2

| Item | Verdict |
|---|---|
| Source fidelity (problem text, definitions) | checked against the source |
| Criterion and reductions (now Section 3) | confirmed, with fixes of citations and definitions |
| Theorem 1.1 (now Section 4) | confirmed; the filling property was re-derived independently |
| Theorem 1.2 with the territory lemma (now Section 5) | confirmed |

It compared Clay–Leininger–Mangahas in both versions, proved the elementary facts on subsurfaces, and gave a
direct proof of the Behrstock inequality for single curves. It found no mathematical error. Its fixes and what
was done:
1. Numbering of Clay–Leininger–Mangahas (required): the published Theorems 2.2 and 6.1 are cited, and the arXiv
   numbering is mentioned once (Section 2.6). Done.
2. Definition of loxodromic (required): g ≠ 1 is part of Definition 2.1. Done.
3. "Stable translation length" in the criterion. Done (Section 2.4, Remark 3.3).
4. The Behrstock inequality is cited in the form of Mangahas, Lemma 2.5, which is stated for multicurves. Done.
5. The index-two statement for doubles is Lemma 50 of the arXiv version of Kim–Koberda. Done (Remark 3.9).
6. Comparison with Aougab et al.: filling by two subsurfaces is weaker than distance at least three between
   boundary curves; the distance statement is now proved (Remark 4.7). Done.
7. Statement card of the write-up (a line range and file paths). Not applicable to the note.
8. A superfluous parenthesis in the proof of Theorem 1.1 was dropped; an outdated theorem name in a comment of
   `finder/verify_chordal_lambda.py` was replaced. Done.

### Second run (2026-10-08): Theorems 1.3, 1.4, 1.5, the computations, the literature

| Item | Verdict |
|---|---|
| Theorem 1.3 (now Section 6) | confirmed; lifting lemma re-derived; the pentagon worked out in full |
| Theorem 1.4 (now Section 7) | confirmed, given the territory lemma; blocks and gluing lemma re-derived |
| Theorem 1.5 and the numbers | confirmed; every number reproduced by a program that shares no code |
| Literature | no prior source for Theorems 1.2 to 1.4 or for connected chordal graphs; one sub-case of Theorem 1.1 is in the literature (Loa; Aougab et al.) |

It tried to break the lifting lemma (bigons at branch points, degrees that are not prime, exponents with
vanishing consecutive sums), the gluing lemma and the bookkeeping of Theorem 1.5, and found no gap. Its
recommended changes and what was done:
1. Cite Loa (arXiv:2103.05144) next to Aougab et al. Done (Section 1.2, Remark 4.7, Scope and priority).
2. State part (4) of the lifting lemma with the hypothesis that is used (complementary regions are discs with at
   most one branch point) and keep the partition property as a sufficient criterion. Done (Lemma 6.3(4), (5)).
3. Rewrite the description of the open cases: the seven graphs with six vertices are within reach. Done
   (Sections 1.4, 9.3 and 10; Theorem 1.6).
4. Say that "loxodromic" is used in an extended sense for graphs that are not connected, and give the numbers
   for connected graphs. Done (Section 2.2, the paragraph after Theorem 1.5, Table 1).
5. Optional: closure under twins. Done (Propositions 8.1 and 8.3).

The second run also proposed, without full proofs, the constructions that became Section 8: the picture of
Figure 4 for the hexagon, the duplication of a vertex, complement territories for the pentagon with a pendant
vertex and with a triangle on an edge, and true twins. It checked the hexagon picture by its program for
d = 5, 7, 9, 15. A further rule that it explored ("join atoms") is not used in the note.

### Third run (2026-10-08): the finished manuscript — Sections 8 and 9, Theorem 1.6, the lemmas added in Section 2

| Item | Verdict |
|---|---|
| Lemma 2.2(d), (e); Lemma 2.3; Lemma 2.6 | confirmed |
| Case analysis of Theorem 1.5 (Section 9.1) | confirmed |
| Proposition 8.1 (false twins) | confirmed |
| Lemma 8.2 and Proposition 8.3 (true twins) | confirmed, with a fix of wording (fix 3 below) |
| Lemma 8.4 and Proposition 8.5 (complements of cycles with one more vertex) | confirmed; the case m = 5 worked out in full |
| Lemma 8.6 and Theorem 8.7 (the hexagon) | confirmed, also by an exact program of its own and by cell complexes of the covers of eight degrees |
| Definition 9.1, Lemma 9.2, Proposition 9.3; Theorem 1.6 | confirmed, with fix 3 |
| Tables 1 (all columns) and 2; the numbers 95, 4, 91 | reproduced by a program written from the text; its list of the 95 graphs equals the list of `lead/lead_coverage.py` up to isomorphism |

It re-derived every proof in its scope, and the two steps of the territory lemma and parts (1) to (4) of the
lifting lemma on which Section 8 rests. It tried to break the territory lemma with partners (curves that cut a
territory and miss both tori in it), the hypotheses of Proposition 8.5 (for 2|I| = m the territory of the new
vertex can be disconnected: an example with three components), and the reading of every rule of the enumeration
(`c_variants.py`: each hypothesis that the proofs need is visible in the count). It found no mathematical error.
Its three required corrections of wording and what was done:
1. The number 95, and "all of them are connected", belong to the hypothesis of Proposition 9.3, not to
   "Theorems 1.5, 1.6": with the two theorems as stated only 710 of the 853 graphs are covered. Done (Section 1.4,
   Remark 3.9; also the caption of Table 1 and Section 1.3).
2. "The smallest graph without clique separator that is neither complete nor a cycle is the triangular prism" is
   false: K_{2,3} has no clique separator either. Done (Section 10; the new text also says that no pointed
   territory system was constructed for the complement of K_{2,3}).
3. "True twins of pairwise distinct vertices" must exclude a twin of a twin: one true twin for each of some
   pairwise distinct vertices of the graph to which twins are added. Done (Proposition 8.3, Theorem 1.6(3),
   Proposition 9.3(d)).

Its optional precisions were applied: the normalisation in the proof of Lemma 2.2(d); the multicurve version of
Lemma 2.3, which is used in the proof of Lemma 2.2(e); "general position" in the proof of Lemma 2.3; in Lemma 8.2,
the later sets B_l contain both boundary curves of a pair. Its programs and outputs are in
`reproducibility/independent_run_3/`.

### Fourth run (2026-10-08): the final text as a whole

| Item | Verdict |
|---|---|
| Statement fidelity (problem, remarks, conventions) | confirmed; one clarification (fix 4 below) |
| Proofs of Sections 2 to 9, read completely and re-derived | confirmed; no gap |
| Cited statements against the sources (Clay–Leininger–Mangahas, Masur–Minsky, Mangahas, Koberda–Mangahas–Taylor, Kim–Koberda, Loa, Aougab et al., Katayama–Kuno, Bering–Conant–Gaster) | as quoted |
| Abstract and introduction against the theorems; remarks; Section 10 | confirmed, with fixes 1 to 3 |
| Figures 1 to 5 and Tables 1, 2 against the text and the outputs | confirmed; Figure 4: the printed polygons have (H1) to (H3); Figure 5: the drawings are the graphs of Table 2 |
| Computations: own programs for Table 1 (all columns), Table 2, the 95 graphs, the doubles, Lemma 8.6, Lemma 6.2 | all reproduced |
| Reproducibility package, run from the extracted source archive (before and after the last revision) | all outputs reproduced; only lines that report running times differ |
| Literature (arXiv API, OpenAlex, zbMATH Open, Crossref, one web search; the cited articles read again for credit) | no source answering the problem or giving the filling property for these classes of graphs; one more earlier construction to credit (fix 5 below) |

It found no mathematical error. Its required changes, of wording and of credit, and what was done:
1. State the result for seven vertices exactly and in the same way everywhere: 853 graphs are not joins; 758
   satisfy the hypothesis of Proposition 9.3; 95 do not, all connected; 4 of these are doubles along stars; 91
   are without answer. Done (abstract, Sections 1.3, 1.4, 9.3, 10, caption of Table 1).
2. The sentences on the article of Runnels: say what was read and what kind of statement its published abstract
   makes. Done (Section 10, Scope and priority).
3. The Verification paragraph and Section 11 describe the final state (third and fourth runs, the programs that
   now reproduce column (iii)). Done.
4. Say that the source words the problem with "injective map" and "not a nontrivial join". Done (Section 1.1).
5. Credit: Mangahas and Taylor (Section 8 of the arXiv version) realize the complement of the cycle C_{2n} nicely
   on a surface of genus n + 1 and find convex cocompact free subgroups of the resulting group; they do not
   consider the filling property. This is prior work next to Theorem 1.3 and was not mentioned. Done
   (Section 1.2, Scope and priority).
6. Fixes 1 to 3 of the third run, each checked again (710 of 853 for the two theorems as stated; three graphs
   with five vertices without clique separator that are neither complete nor cycles). Done, see above.

Two optional precisions were applied: Remark (2) of the problem holds for all graphs with the definition of the
note (Section 2.2), and Corollary 1.1 of Aougab et al. is cited together with their Theorem A (Section 1.2). Its
programs and outputs are in `reproducibility/independent_run_4/`.

### What no run checked
- The proofs of the published theorems that are used as stated: Theorems 2.2 and 6.1 of Clay–Leininger–Mangahas,
  the results (K1) and (K2) of Masur and Minsky, the Behrstock inequality, Theorem 2.2 of
  Koberda–Mangahas–Taylor, Lemma 50 of Kim–Koberda.
- The published article of Runnels (only its abstract was read); the sources listed above as not consulted.
- MathSciNet and Google Scholar.

## Relation to the literature, novelty and scope
- **Searches (2026-10-08).** arXiv listings and arXiv API (keyword searches on right-angled Artin groups,
  mapping class groups, pseudo-Anosov, loxodromic); OpenAlex (works citing Koberda–Mangahas–Taylor and
  Clay–Leininger–Mangahas); Crossref; zbMATH Open; eight web searches in all stages together. No source was found
  that answers Problem 2.28, and none that gives a positive answer for connected chordal graphs, graphs with
  chordal complement, complements of cycles, clique-sums, or the hexagon. All requests were anonymous; they are listed in
  `reproducibility/literature/`. The arXiv API answered some requests of the writing stage with 429 or 503; they
  were repeated later at a slower rate.
- **Known before.** The criterion is the theorem of Clay, Leininger and Mangahas, as Remark (4) of the problem
  says. The positive answer for disjoint unions of complete graphs or of joins is due to Loa (two components)
  and to Aougab, Bray, Dowdall, Hoganson, Maloni and Whitfield; Theorem 1.1 overlaps with these results for
  disjoint unions of complete graphs. The use of a branched cover for the pentagon goes back to Section 2.4 of
  Clay–Leininger–Mangahas, and Mangahas–Taylor (Section 8 of the arXiv version) realize the complements of even
  cycles nicely; the filling property is not considered in either place.
- **Not searched.** MathSciNet and Google Scholar. This negative search is not a proof of priority, and none is
  claimed.
- **Scope.** A partial answer: explicit classes of graphs. The homomorphisms are those of Clay, Leininger and
  Mangahas, into mapping class groups of closed surfaces. The proofs rely on published theorems that were not
  re-derived: Theorems 2.2 and 6.1 of Clay–Leininger–Mangahas, the results (K1) and (K2) of Masur and Minsky, the
  Behrstock inequality, and Theorem 2.2 of Koberda–Mangahas–Taylor (for the finite-index statement). Not treated:
  the general problem, a negative answer for any graph, the least genus, explicit powers. Finite-index subgroups
  other than doubles along stars were not examined.

## 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/