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Topological isomorphism of procountable groups is universal analytic

  • Xinan Dai
  • Wenhao Deng
  • Yingdong Shi
  • Tailin Wu
  • Yuchen Yang
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026topological,
  title = {{Topological isomorphism of procountable groups is universal analytic}},
  author = {Xinan Dai and Wenhao Deng and Yingdong Shi and Tailin Wu and Yuchen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20890},
  eprint = {2609.20890},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20890},
}
RIS
TY  - UNPB
TI  - Topological isomorphism of procountable groups is universal analytic
AU  - Xinan Dai
AU  - Wenhao Deng
AU  - Yingdong Shi
AU  - Tailin Wu
AU  - Yuchen Yang
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20890
UR  - https://arxiv.org/abs/2609.20890
AB  - 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.
ER  -