{
  "abstract": "We formulate a root-conjugation criterion for producing generating pairs and detecting Nielsen inequivalence by a polynomial symmetric-trace invariant. The method simultaneously treats mapping class groups and automorphism groups of free groups. We prove that $\\mathrm{Mod}(Σ_g)$, $g\\ge2$, and $\\mathrm{Aut}(F_n)$, $\\mathrm{SAut}(F_n)$, $\\mathrm{Out}(F_n)$ and $\\mathrm{SOut}(F_n)$, $n\\ge3$, have countably infinitely many Nielsen classes of generating pairs. A sign parameter unifies the full and special automorphism-group constructions, with common generation arguments and trace polynomials. The same homology representations give the corresponding results for $\\mathrm{Sp}_{2g}(\\mathbb{Z})$, $g\\ge2$, and $\\mathrm{SL}_n(\\mathbb{Z})$ and $\\mathrm{GL}_n(\\mathbb{Z})$, $n\\ge3$. This answers Hirose--Monden's low-genus question and Kapovich's Question 8.1.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08857",
  "author_total": 1,
  "authors": [
    {
      "name": "Wenchang Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9cqewcjhrdag.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08857",
  "fulltext_url": null,
  "id": "d9cqewcjhrdag",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08857",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220951942",
  "pdf_url": "https://arxiv.org/pdf/2610.08857",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9cqewcjhrdag.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.08857",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08857"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups",
  "title_text": "Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/d9cqewcjhrdag",
  "venue": "arXiv",
  "versions": []
}