发表机构
Concordia University; Google DeepMind; Weizmann Institute of Science（康科迪亚大学; 谷歌DeepMind; 魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明随机 profinite 群几乎必然有限生成且有限呈现，并构造唯一的万有 d-生成完美 profinite 群，解决相关猜想，方法基于 crown 理论的高秩多重性洞见。
AI 中文摘要
我们证明，对于一类一般的随机 profinite 群模型，几乎必然地具有拓扑有限生成性质。由此推出，在 Liu--Wood 和 Sawin--Wood 引入的模型中，几乎必然地具有有限呈现性质，并且对呈现中的亏缺（deficiency）有几乎必然的控制。特别地，这解决了 Liu--Wood 和 Sawin--Wood 工作中提出的问题。利用相同的基本原理，我们证明存在唯一的万有 d-生成完美 profinite 群：其有限商恰好是有限 d-生成完美群。这解决了 Nikolov 的一个猜想。我们还证明该群是投射的，并且具有由 d 个生成元和 d 个关系构成的 profinite 呈现，且在 d 个生成元上关系数不能更少。主要的基本主题是引入 crown 理论的一个洞见：高秩群总是由具有大多重性的 crown 所见证。这一方法由 ChatGPT5.5 pro 和 Google DeepMind 的内部智能体 Aletheia 独立发现。
英文摘要
We prove that almost sure topological finite generation holds for a general class of random models of profinite groups. We deduce that in the models introduced by Liu--Wood and Sawin--Wood, one has that finite presentation holds almost surely, with almost sure control of the deficiency in the presentation. In particular this settles questions raised in work of Liu--Wood and Sawin--Wood. Using the same underlying principle, we show that there exists a unique universal d-generated perfect profinite group: its finite quotients are precisely the finite d-generated perfect groups. This settles a conjecture of Nikolov. We show in addition that this group is projective and admits a profinite presentation with $d$ generators and $d$ relations, and with no fewer relations on $d$ generators. The main underlying theme is to bring in an insight from crown theory: groups with high rank are always witnessed by a crown with large multiplicity. This approach was discovered independently by ChatGPT5.5 pro and Aletheia, an internal agent at Google DeepMind.
