Topological Isomorphism on Procountable Groups

• Topological isomorphism on procountable groups examines both algebraic and topological structures in groups formed as inverse limits of countable discrete groups.
• Recent research establishes that the descriptive complexity of these isomorphisms exceeds that of graph isomorphism, proving non-classifiability by countable invariants.
• Key methods include Borel-reduction techniques and ultrametric representations, leading to distinct complexity results across subclasses like oligomorphic and finitely generated profinite groups.

Topological isomorphism on procountable groups concerns the identification of group-theoretic and topological structure among groups presented as inverse limits of countable discrete groups, with fundamental consequences for classification theory in descriptive set theory. Recent research reveals that the complexity of topological isomorphism far exceeds that of graph isomorphism and is not Borel-reducible to any classification by countable structures. Key advances by Gao, Nies, and Paolini have sharply delineated the descriptive complexity landscape for these groups, with contrasting results in special subclasses such as oligomorphic or finitely generated profinite groups.

1. Non-Archimedean Polish and Procountable Groups

A Polish group is a topological group with a separable, completely metrizable topology. Non-archimedean Polish groups admit a neighborhood basis at the identity consisting of open subgroups. Every non-archimedean Polish group can be realized as a closed subgroup of $\mathrm{Sym}(\mathbb{N})$ (with the pointwise convergence topology).

A procountable group is a non-archimedean Polish group $G$ which can be presented as the inverse limit of a sequence of countable discrete groups:

$$G = \{ x \in \prod_n G_n : p_n\left(x(n+1)\right) = x(n) \ \forall n \}$$

where $(G_n, p_n)$ is an inverse system with surjective homomorphisms $p_n: G_{n+1} \rightarrow G_n$ and the topology is inherited from the product of discrete groups (Gao et al., 13 Dec 2025).

Profinite groups, a subclass, are compact, totally disconnected, and presented as inverse limits of finite groups. Countably based profinite groups coincide with procountable groups in the Polish setting (Nies, 2016).

2. Topological Isomorphism and Descriptive Complexity

Topological isomorphism between procountable groups $G$ and $H$ is an isomorphism $\varphi: G \rightarrow H$ that is both a group isomorphism and a homeomorphism (i.e., $G \cong_{\mathrm{pc}} H$). The primary problem is to classify procountable groups up to this equivalence.

Descriptive set theory quantifies the complexity of equivalence relations via Borel reducibility. An equivalence relation $E$ on a Polish space is Borel reducible to $F$ if there exists a Borel map $f$ such that $x E x' \iff f(x) F f(x')$ (Gao et al., 13 Dec 2025).

A central notion is classifiability by countable structures: $E$ is classifiable in this manner if $E \leq_B F$ for $F$ isomorphism on a class of countable structures (e.g., graph isomorphism GI). Friedman–Stanley’s theorem characterizes these as orbit equivalences of Borel actions by $\mathrm{Sym}(\mathbb{N})$ (Gao et al., 13 Dec 2025).

3. Sharp Complexity Results for Procountable Group Isomorphism

The work of Gao–Nies–Paolini proves that topological isomorphism on procountable groups is not classifiable by countable structures. They show that the analytic equivalence relation $\ell_\infty$ on $\mathbb{R}^\mathbb{N}$,

$$x\,\ell_\infty\,y \iff \sup_n |x_n - y_n| < \infty,$$

is Borel-reducible to topological isomorphism on procountable groups: there exists a Borel assignment $x \mapsto G_x$, such that $x\,\ell_\infty\,y \iff G_x \cong_{\mathrm{pc}} G_y$ (Gao et al., 13 Dec 2025). Since $\ell_\infty$ is known to have maximal analytic complexity (universal for analytic equivalence relations above all orbit relations), topological isomorphism sits strictly above graph isomorphism and any orbit equivalence on the Borel hierarchy.

For countably based profinite groups, isomorphism is $S_\infty$-complete—Borel-equivalent to countable graph isomorphism—whereas for finitely generated profinite groups, the isomorphism relation is smooth (Borel-equivalent to equality on the reals) (Nies, 2016).

4. Key Structural Constructions and Proof Methods

The principal Borel-reduction analysis proceeds in two steps (Gao et al., 13 Dec 2025):

1. Reduction from $\ell_\infty$ to Uniform Homeomorphism of Ultrametric Spaces: For sequences $x \in \mathbb{N}^\mathbb{N}$, construct pruned trees $T_x$ whose branches form ultrametric Polish spaces. Uniform homeomorphisms between these spaces correspond to bounded coordinatewise shifts, thereby encoding the $\ell_\infty$ relation in the homeomorphism problem.
2. Realizing Ultrametric Spaces as Procountable Groups: For a tree $T$, the group $G_T$ is the inverse limit of free Coxeter groups $L(T_n)$ (the free product of copies of the two-element group), with bonding determined by predecessor mapping in the tree. A uniform homeomorphism of ultrametric spaces yields a system isomorphism of the inverse systems, and hence a topological isomorphism of the corresponding procountable groups.

Together, these steps establish $\ell_\infty \leq_B$ (top-iso on procountable groups), ruling out classification by countable invariants. Further, this complexity barrier applies even to subclasses such as abelian procountable groups, though the precise situation for these or nilpotent varieties remains unresolved.

5. Coarse Groups and Stone-Type Duality

The coarse group $\mathcal{M}(G)$ is constructed from a closed subgroup $G$ of $S_\infty$ by considering the set of open cosets of open subgroups, and a ternary relation $AB \sqsubseteq C$ capturing group multiplication at the coset level (Nies et al., 2019). A filter-group $F(\mathcal{M})$ can be recovered, with a Stone-type duality showing $F(\mathcal{M}(G)) \cong G$ as topological groups when $G$ has countably many open subgroups.

For profinite groups, when $\mathcal{M}(G)$ is countable and satisfies a compactness condition (every open subgroup contains a normal open subgroup), the duality yields a Polish correspondence—enabling Borel codes for profinite groups and arithmetical control over the isomorphism problem (Nies et al., 2019).

6. Subclass Reductions: Oligomorphic and Finitely Generated Procountable Groups

Oligomorphic closed subgroups of $S_\infty$ (those with finitely many orbits on $n$-tuples) admit a distinctly lower complexity for topological isomorphism. Using coarse group machinery and model-theoretic techniques, it is shown that topological isomorphism among oligomorphic groups with weak elimination of imaginaries (WEI) is smooth—Borel-equivalent to equality on the reals (Paolini, 2024).

Similarly, for finitely generated profinite groups, the isomorphism relation is smooth (Nies, 2016). This contrast underscores a rich landscape of classification complexities, from smooth up to universal analytic, depending on structural properties of the group class.

7. Open Problems and Future Research

Significant open questions include:

• Determining whether topological isomorphism on all non-archimedean Polish groups (beyond procountable) is universal analytic.
• Classifying specific subvarieties (e.g., abelian, nilpotent procountable groups) and ascertaining whether their isomorphism relations admit Borel reduction to simpler classes (such as graph isomorphism or even smooth relations).
• Exploring weakening conditions such as partial elimination of imaginaries and analyzing impacts on the complexity of isomorphism among oligomorphic groups.
• Applying expanded-age and E-ex machinery for deeper model-theoretic correspondence and Galois-theoretic analysis in $\omega$-categorical environments.

The results of Gao–Nies–Paolini and related studies establish the non-classifiability of procountable groups by countable structure invariants, sharply delineating the boundary between tractable and intractable classification within topological group theory. This suggests a definitive stratification of isomorphism problems by analytic complexity for various subclasses of non-archimedean Polish groups (Gao et al., 13 Dec 2025, Nies, 2016, Nies et al., 2019, Paolini, 2024).