AI 中文总结
针对 $n\ge8$，研究自由群外自同构群中生成对的尼尔森类，构造了 $GL(n,\mathbb Z)$ 等的生成对族，推导关系模障碍，Evans例子验证其可区分稳定化后的尼尔森类。
AI 中文摘要
对于所有 $n\ge 8$，我们构造了 $GL(n,\mathbb Z)$ 和 $SL(n,\mathbb Z)$ 的生成对的显式族，这些生成对代表可数无穷多个尼尔森等价类。这些对源于 $Out(F_n)$ 及其指数为2的子群 $SOut(F_n)$ 中显式的挠生成对的同调像，在这两个外群中均产生可数无穷多个生成对的尼尔森类。在每个构造的外群族中，所有对在一次稳定化后成为尼尔森等价，该稳定化通过添加一个平凡坐标得到。我们推导了一次稳定化生成组的尼尔森等价的关系模障碍，Evans的非消去例子表明该障碍非平凡，可区分一次稳定化生成组的尼尔森类。我们还记录了一个商关系模的细化，供未来应用使用。
英文摘要
For every $n\ge 8$, we construct explicit families of generating pairs of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$ representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in $Out(F_n)$ and its index-two subgroup $SOut(F_n)$, yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.
