Procountable groups are not classifiable by countable structures

TL;DR

The paper demonstrates that topological isomorphism of procountable groups cannot be classified by countable structures, showing a high level of complexity in their isomorphism relation.

Contribution

It establishes that the isomorphism relation for procountable groups is not classifiable by countable structures, advancing understanding of their complexity in descriptive set theory.

Findings

Topological isomorphism on procountable groups is not classifiable by countable structures.

The equivalence relation $ ext{ell}_ extinfty$ is Borel reducible to it.

Progress on the open problem of classifying non-archimedean Polish groups.

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 $\ell_\infty$ 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.