Procountable groups are not classifiable by countable structures
Logic
2026-03-30 v4 Group Theory
Abstract
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 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.
Cite
@article{arxiv.2512.12256,
title = {Procountable groups are not classifiable by countable structures},
author = {Su Gao and André Nies and Gianluca Paolini},
journal= {arXiv preprint arXiv:2512.12256},
year = {2026}
}