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

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Wenchang Yang

Group Theory · Submitted 5 Oct 2026

abstract compiled · 2026-10-08 · no language-model review

The authors' main claim We prove that $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, 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.

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arXiv ID2610.08857 (version 1)
Pith ID46Y2TUM7, Pith's permanent record of this paper and its review
CitationWenchang Yang. "Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups." arXiv:2610.08857 (2026). https://pith.science/paper/46Y2TUM7
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-08: 1 request, 5,078 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.91 · compares with prior results 0.45 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We formulate a root-conjugation criterion for producing generating pairs and detecting Nielsen inequivalence by a polynomial symmetric-trace invariant. method 0.76 0.93 0.06 asserted 1.00
s1 The method simultaneously treats mapping class groups and automorphism groups of free groups. contribution 0.63 0.81 0.04 asserted 1.00
s2 We prove that $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, 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. result 1.00 0.96 0.91 asserted 1.00
s3 A sign parameter unifies the full and special automorphism-group constructions, with common generation arguments and trace polynomials. method 0.83 0.88 0.04 asserted 1.00
s4 The same homology representations give the corresponding results for $\mathrm{Sp}_{2g}(\mathbb{Z})$, $g\ge2$, and $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{GL}_n(\mathbb{Z})$, $n\ge3$. result 0.94 0.85 0.08 asserted 1.00
s5 This answers Hirose--Monden's low-genus question and Kapovich's Question 8.1. interpretation 0.83 0.76 0.08 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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  • references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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 $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, 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.

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