Work

Topological isomorphism of procountable groups is universal analytic

Year: 2026

Type: preprint

Abstract: We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation. This answers a question of Gao, Nies, and Paolini and determines the complexity of topological isomorphism of non-Archimedean Polish groups. We encode permutative equivalence of unconditional basic sequences by integer weights on a countable structure. A tree construction turns bounded changes in these weights into uniform homeomorphisms, while a finite-level counting argument recovers the weights up to bounded error. A graph-to-group functor whose nontrivial homomorphisms, modulo inner automorphisms of the target, come from unique graph maps transfers this equivalence to inverse systems of countable groups. We then obtain a Borel realization of their limits as closed permutation groups.

Source: Zenodo (CERN European Organization for Nuclear Research)

DOI: 10.5281/zenodo.22705277

Authors: Dai, Xinan, Deng, Wenhao, Shi, Yingdong, Wu, Tailin, Yang, Yuchen

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7212316420

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