Procountable groups are not classifiable by countable structures

Abstract: We prove that the relation of topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ that expresses that two sequences of reals have bounded difference is Borel reducible to it. This marks progress on an open problem of [15], to determine the exact complexity of isomorphism among all non-archimedean Polish groups.