发表机构
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文开发两种通用框架构造具有无限围长等性质的剩余有限群，建立标记围长准则为其充要条件，探讨围长与 profinite 完备化的联系并给出线性群的同余实现结果，同时提出相关开放问题。
AI 中文摘要
我们开发了两种新的通用框架，用于构造有限生成的剩余有限群，这类群具有无限围长、不满足任何非平凡群律且不含非阿贝尔自由子群。这类群较为罕见，已知的例子包括第一个 Grigorchuk 群、Thompson 群 F 以及某些迭代 wreath 积，而我们的方法提供了这类群更明确的构造，将其实现为对称群直积的子群。我们研究了群的围长与其有限商群围长之间的关系，进而引入了“标记”围长准则，并证明该准则是剩余有限群类中无限围长的充要条件。此外，我们探讨了围长与 profinite 完备化之间的联系，给出了线性群中无限围长的同余实现结果。最后，我们提出了几个由上述结果自然引出的开放问题。
英文摘要
We develop two new general frameworks for constructing finitely generated residually finite groups that possess infinite girth, satisfy no non-trivial group law, and contain no non-abelian free subgroups. While such groups are rare--with known examples including the first Grigorchuk group, Thompson's group F, and certain iterated wreath products--our approach provides a more explicit construction of such groups realized as subgroups of direct products of symmetric groups. We investigate the relationship between the girth of a group and that of its finite quotients, leading to the introduction of a "marked" girth criterion, which we establish as a necessary and sufficient condition for infinite girth within the class of residually finite groups. Furthermore, we explore the connection between girth and profinite completions and provide a congruence realization result for infinite girth in linear groups. We conclude by posing several open questions that arise naturally from our results.
