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

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 $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_g)$, $g\ge 2$, and $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$, $\mathrm{S}\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$, $\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$ and $\mathrm{S}\mathrm{O}\mathrm{u}\mathrm{t}(F_n)$, $n\ge 3$, 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 ${\mathrm{S}\mathrm{p}}_{2g}(\mathbb{Z})$, $g\ge 2$, and ${\mathrm{S}\mathrm{L}}_n(\mathbb{Z})$ and ${\mathrm{G}\mathrm{L}}_n(\mathbb{Z})$, $n\ge 3$. This answers Hirose–Monden’s low-genus question and Kapovich’s Question 8.1.