{
  "schema_version": 1,
  "problem_number": "AMR-011-0009",
  "title": "Property tau Need Not Survive Intersections of Normal Subgroups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a family of finite-index normal subgroups of a fixed finitely generated free group which has property (tau), whereas its prefix-intersection chain does not. The construction uses Kassabov's bounded-degree alternating-group expanders. Two quotient maps have identical expanding generators and send one additional generator to, respectively, the identity and a 3-cycle. Their joint image is the full product Alt(N) x Alt(N). In its action on N squared ordered pairs, the diagonal provides a mean-zero test function with lazy Rayleigh quotient 3/[2r(N-1)], where r is the fixed number of generators. Pullback transfers this bound to the regular quotient and then to the prefix chain. Thus failure already occurs for a sequence of pairwise intersections, giving a negative answer to Question 9 in Abert's list.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.GR",
    "math.CO",
    "math.DS"
  ],
  "keywords": [
    "property tau",
    "normal subgroups",
    "expander graphs",
    "alternating groups",
    "spectral gap",
    "AMR-011-0009"
  ],
  "manuscript_version_date": "2026-10-04",
  "publication_date": "2026-10-04",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-04",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-011-0009/",
  "pdf_url": "https://eulersolve.org/papers/amr-011-0009/paper.pdf?v=86db0ebf8645",
  "doi": "10.5281/zenodo.23137155",
  "zenodo_record_url": "https://zenodo.org/records/23137155",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete negative answer to the finite-index normal-subgroup prefix-intersection question in Abért's 2010 list, AMR-011-0009. Kassabov's alternating-group expansion theorem is a credited input, not a new expander construction. No claim is made about Mimura's more restrictive LEF-approximation question. Novelty and absolute priority remain undetermined. AI-assisted, self-audited and unrefereed; no independent specialist review or proof-assistant verification is claimed.",
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    "source.zip": {
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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."
}