Procountable groups are not classifiable by countable structures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914426246922240 |
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| author | Gao, Su Nies, André Paolini, Gianluca |
| author_facet | Gao, Su Nies, André Paolini, Gianluca |
| contents | We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a bounded difference is Borel reducible to it. This marks substantial progress on an open problem of Kechris, Nies and Tent (2018): to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Procountable groups are not classifiable by countable structures Gao, Su Nies, André Paolini, Gianluca Logic Group Theory 03E15, 54H05, 22A05, 20B27 We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a bounded difference is Borel reducible to it. This marks substantial progress on an open problem of Kechris, Nies and Tent (2018): to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups. |
| title | Procountable groups are not classifiable by countable structures |
| topic | Logic Group Theory 03E15, 54H05, 22A05, 20B27 |
| url | https://arxiv.org/abs/2512.12256 |