发表机构
Fudan University; Westlake University; University of Glasgow; ShanghaiTech University（复旦大学; 西湖大学; 格拉斯哥大学; 上海科技大学）
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procountable群的拓扑同构是万有解析等价关系，并推广至非阿基米德Polish群，通过编码权重和树构造实现分类复杂性。
AI 中文摘要
我们证明了procountable群的拓扑同构是一个万有解析等价关系，回答了Gao、Nies和Paolini提出的一个问题。同样的结论也适用于非阿基米德Polish群。更强地，存在一个可数群$H$，使得对于$H$的满自同态序列的逆向极限，万有性已经成立。因此，所有的分类复杂性都可以由连接映射来承载。我们通过具有指定对称性的整数权重来编码无条件基本序列的置换等价。对Gao、Nies和Paolini的树构造的扩展，在允许这些对称性作用的同时，从一致连续性中恢复有界权重差。Przeździecki的几乎满函子将所得的逆向图系统转移到群上。将阶段图等同于单个图，使得从权重到连接自同态的传递是连续的。
英文摘要
We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation, answering a question of Gao, Nies, and Paolini. The same conclusion follows for non-Archimedean Polish groups. More strongly, there is one countable group $H$ for which universality already holds among inverse limits of sequences of surjective endomorphisms of $H$. Thus all the classification complexity can be carried by the bonding maps. We encode permutative equivalence of unconditional basic sequences by integer weights with prescribed symmetries. An extension of Gao, Nies, and Paolini's tree construction recovers bounded weight differences from uniform continuity while allowing these symmetries to act. Przeździecki's almost-full functor transfers the resulting inverse graph systems to groups. Identifying the stage graphs with a single graph makes the passage from weights to bonding endomorphisms continuous.
Comments16 pages
