Procountable groups are not classifiable by countable structures

TL;DR

The paper addresses the complexity of topological isomorphism among non-archimedean Polish groups by proving that, for a fixed odd prime $p$, the isomorphism relation on 2-nilpotent procountable groups of exponent $p$ is not classifiable by countable structures. The authors execute a two-step Borel reduction: first encoding the ℓ∞-relation into uniform homeomorphism of ultrametric path spaces [T] derived from pruned trees, then translating uniform homeomorphism into isomorphism of procountable 2-nilpotent groups via a Mekler-type construction $L(X)$ and an inverse-limit assembly $G_T$. This shows that the isomorphism problem for these procountable groups is complete analytic and strictly more complex than GI, thereby advancing the Kechris–Kechris–Nies program on the descriptive set-theoretic complexity of isomorphism for non-archimedean Polish groups. The work also clarifies the role of uniform homeomorphism in such reductions and raises open questions about universality and broader classifier results for related classes of groups and spaces.

Abstract

We prove that topological isomorphism on $2$-nilpotent 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 substantial progress on an open problem of [16]: to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups.

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6