{
  "schema_version": 1,
  "problem_number": "KP-2.28",
  "title": "Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 2.28 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026), recorded by T. Koberda, asks: if Γ is a finite graph that is not a join, is there an injective homomorphism from the right-angled Artin group A(Γ) to the mapping class group of a surface which sends every loxodromic element to a pseudo-Anosov mapping class? We give a partial answer; the problem remains open in general. We prove that such a homomorphism exists, with a closed surface as target, when Γ is chordal, when the complement graph of Γ is chordal, when Γ is the complement of a cycle of length at least five (this includes the pentagon), and more generally when the complement of Γ is obtained from complete graphs and cycles by gluing along cliques. The answer is also positive for disjoint unions of graphs of these kinds and of joins, with isolated vertices added, and for connected graphs whose right-angled Artin groups have finite index in the group of a connected graph with a positive answer. Further constructions treat the hexagon, graphs obtained by duplicating vertices, and complements of cycles with one more vertex. Together with a computer enumeration of small graphs, the constructions give a positive answer for every graph with at most six vertices that is not a join and for 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 we have no answer. The homomorphisms are those of Clay, Leininger and Mangahas: the generators are sent to pseudo-Anosov maps of subsurfaces, and by a theorem of these authors the property asked for is equivalent to a filling condition, which we verify. The proofs use subsurface projections (Masur–Minsky, Behrstock) and, for complements of cycles and for the hexagon, planar configurations of discs lifted to totally ramified cyclic covers of the sphere. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "right-angled Artin groups",
    "mapping class groups",
    "pseudo-Anosov mapping classes",
    "loxodromic elements",
    "chordal graphs",
    "subsurface projections",
    "curve complex",
    "branched covers",
    "partial answer",
    "K3 problem list",
    "Problem 2.28",
    "UnsolvedMath",
    "KP-2.28",
    "math.GT",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kp-2-28/",
  "pdf_url": "https://eulersolve.org/papers/kp-2-28/paper.pdf?v=2fe517faf9ed",
  "doi": "10.5281/zenodo.23248883",
  "zenodo_record_url": "https://zenodo.org/records/23248883",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A partial answer to Problem 2.28 of the K3 problem list (recorded by T. Koberda): an injective homomorphism from the right-angled Artin group A(Γ) to the mapping class group of a closed surface that sends every loxodromic element to a pseudo-Anosov class is constructed when Γ is chordal, when the complement of Γ is chordal, when Γ is the complement of a cycle of length at least five, when the complement of Γ is a clique-sum of complete graphs and cycles, for the hexagon and some further graphs, and under closure properties. By 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 by a finite-index argument). The problem remains open for general graphs; for 91 graphs with seven vertices, among them the heptagon, there is no answer. The homomorphisms are those of Clay, Leininger and Mangahas, whose theorems (2012) are used as published; disjoint unions of complete graphs or of joins were known before (Loa; Aougab, Bray, Dowdall, Hoganson, Maloni and Whitfield). For graphs that are not connected the word loxodromic is read verbatim from the definition, as the note explains. Four independent AI-assisted verification runs checked the proofs and reproduced the enumeration. Unrefereed; no priority claim is made.",
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    "source.zip": {
      "sha256": "92937f3bbc2b2a6a9937aad8a70210f9cf365f44d0df179359d6e5875dec6942"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}