单关系群的 Sofic 性
Soficity of One-Relator Groups and Reducible Presentations
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中文总结 AI 辅助
本文证明了所有有限生成的单关系群均为 sofic 群，通过终端路径上的自由群单词构造上循环，并利用斜积作用实现有限置换逼近。
中文摘要 AI 辅助
我们证明了每个有限生成的单关系群都是 sofic 的。Poulin-Wrobel 的终端边替换保持端点不变；我们保留每条终端路径所携带的自由群单词，并用它来定义一个取值于 F(S) 的上循环。对于关系词 w^m，修复集的 m 个循环平移使得关系词缺陷任意小。随后，具有树可及轨道关系的斜积作用给出了所需的有限置换逼近。
英文摘要
We prove that every one-relator group is sofic, answering a question of Nate Brown. More generally, every group admitting a reducible presentation without proper powers is sofic, with one proper-power relation allowed at the final reducible step. The proof starts from the endpoint-preserving edge replacements of Poulin--Wróbel. We retain the free-group word carried by each terminal replacement and use it to define an \(F(S)\)-valued cocycle repair; in its relative form, the repair preserves the previously chosen generator coordinates while controlling the new relator defect. Coinduction transports the lower cocycle through successive one-relator-product extensions, allowing the construction to be iterated along reducible presentations. For a final relator \(w^m\), cyclic translates of the repaired set amplify a zero-defect set of measure close to \(1/m\) to one of measure close to one. A cocycle criterion then converts arbitrarily small relator defect into finite permutation approximations via a treeable skew-product orbit relation.
