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Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups

  • Wenchang Yang
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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 Mod(Σg), g≥2, and Aut(Fn), SAut(Fn), Out(Fn) and SOut(Fn), n≥3, 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 Sp2g(ℤ), g≥2, and SLn(ℤ) and GLn(ℤ), n≥3. This answers Hirose--Monden's low-genus question and Kapovich's Question 8.1.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026infinitely,
  title = {{Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups}},
  author = {Wenchang Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08857},
  eprint = {2610.08857},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08857},
}
RIS
TY  - UNPB
TI  - Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups
AU  - Wenchang Yang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.08857
UR  - https://arxiv.org/abs/2610.08857
AB  - 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 Mod(Σ_g), g≥2, and Aut(F_n), SAut(F_n), Out(F_n) and SOut(F_n), n≥3, 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 Sp_2g(ℤ), g≥2, and SL_n(ℤ) and GL_n(ℤ), n≥3. This answers Hirose--Monden's low-genus question and Kapovich's Question 8.1.
ER  -