AI 中文总结
研究有限生成自由群的波斯特对应问题，通过与有限状态换能器联系，构造相关同态，证明其不可判定，解决算法群论长期问题，还表明均衡器秩无法计算，且无算法计算虚拟自同态固定子群基及识别相关语言。
AI 中文摘要
我们证明了有限生成自由群的波斯特对应问题是不可判定的，即使两个同态之一是单射。这解决了算法群论中一个长期存在的开放问题。该结果与固定子群理论形成鲜明对比：虽然当其中一个映射是单射时，两个自由群同态的均衡器是有限生成的，但没有算法能判定这个均衡器是否平凡。证明通过与有限状态换能器的联系进行。给定一个循环标签系统\(\mathcal C\)，我们有效地构造一个有限部分确定性逆换能器\(\mathcal{T}_{\mathcal C}\)，其不动点集非平凡当且仅当\(\mathcal C\)停机。然后为任何这样的换能器关联两个同态\(g,h\colon F_Y\longrightarrow F_A\)，其中\(h\)是单射，使得它们的均衡器非平凡恰好当换能器有一个非平凡的固定循环时。因此，一般情况下这些均衡器的秩无法计算，回答了1984年斯塔尔ings提出的一个问题。进一步的结果是，我们证明没有算法能计算有限生成自由群的虚拟自同态的固定子群的基，也不能构造一个有限自动机来识别有限完全逆换能器的简化不动点语言。
英文摘要
We prove that the Post Correspondence Problem for finitely generated free groups is undecidable, even when one of the two homomorphisms is injective and has finite-index image. This resolves a longstanding open problem in algorithmic group theory. The proof proceeds through a connection with finite-state transducers. Given a cyclic tag system $\mathcal C$, we effectively construct a finite partial deterministic inverse transducer $\mathcal T_{\mathcal C}$ whose fixed-point set is nontrivial if and only if $\mathcal C$ halts. We then associate to any such transducer two homomorphisms $g,h\colon F_Y\longrightarrow F_A,$ with $h$ injective, such that their equalizer is nontrivial precisely when the transducer has a nontrivial fixed loop. As an immediate consequence, the rank of these equalizers cannot be computed in general, answering a question posed by Stallings in 1984. We further prove that there is no algorithm which decides whether the fixed subgroup of a virtual endomorphism of a finitely generated free group is trivial. Finally, we apply the main result to show that the stabilizer problem is undecidable for free subgroups of $\operatorname{SL}_4(\mathbb Z)$, and that the upper-right-corner problem is undecidable for free subgroups of $\operatorname{SL}_5(\mathbb Z)$, even when the given generators are promised to form a free basis, improving on recent results of Breuillard and Kocharyan.
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