Open Problems in Mathematical Logic — Problem 37

Definitions and mathematical statement

A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups. Here isomorphism is interpreted as topological group isomorphism; the supplied record does not explicitly specify this convention or the universe and coding of profinite groups.

For equivalence relations $E$ on a standard Borel space $X$ and $F$ on a standard Borel space $Y$, write $E\leq_B F$ if there is a Borel map $f:X\to Y$ such that \[ \forall x,x'\in X,\qquad x\,E\,x'\iff f(x)\,F\,f(x'). \] Write $E<_B F$ when $E\leq_B F$ and $F\not\leq_B E$. Let $\Delta_{{\mathbb{R}}}$ denote equality on ${\mathbb{R}}$. An equivalence relation is smooth if it is Borel reducible to $\Delta_{{\mathbb{R}}}$. Let $S_\infty$ be the group of permutations of ${\mathbb{N}}$, with the topology of pointwise convergence, and let $E_{\mathrm{gr}}$ be isomorphism of simple graphs on vertex set ${\mathbb{N}}$, coded in $2^{[{\mathbb{N}}]^2}$. This is a standard representative of the $S_\infty$-complete degree: it is an orbit equivalence relation of a Borel $S_\infty$-action to which every such orbit equivalence relation is Borel reducible.

Subject to specifying a standard Borel space $\mathcal P$ of codes for the intended profinite groups, the question is whether there exists a Borel subset $\mathcal C\subseteq\mathcal P$ such that, for \[ x\,E_{\mathcal C}\,y\iff\text{the groups coded by }x\text{ and }y\text{ are topologically isomorphic}, \] one has \[ \Delta_{{\mathbb{R}}}<_B E_{\mathcal C}<_B E_{\mathrm{gr}}. \] The source asks in particular whether such a $\mathcal C$ can consist entirely of abelian profinite groups.

The supplied record does not specify the coding space, whether the groups are required to be metrizable or countably based, or whether $\mathcal C$ must be invariant under isomorphism in the ambient coding space. These qualifications remain unresolved; the displayed formulation makes the usual Borel-reducibility interpretation of "properly between" explicit without settling them.

The source text reports that the problem was also posed during the workshop and that, by its end, Gao and Nies realized that the answer is affirmative. This is a supplied report, with no construction or proof included; it does not specify whether the affirmative answer also covers the abelian case. The registry separately labels the record open.

Short English statement

Does some Borel family of profinite groups have topological isomorphism strictly above equality of real numbers and strictly below an $S_\infty$-complete equivalence relation under Borel reducibility? Can the family consist of abelian groups? The source reports an affirmative answer by Gao and Nies, while the registry labels the problem open; the scope of that answer and the intended coding require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $801025$, source problem code catalogue entry, title Open Problems in Mathematical Logic --- Problem 37. The supplied record attributes the problem to Barmpalias, Gao et al., with proposed year $2026$, and records status open.
• Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 37. Supplied source URL: https://arxiv.org/abs/2608.26628. The provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance claims, not independently verified access or source verification.
• The supplied problem text attributes recognition of an affirmative answer by the end of the workshop to Gao and Nies. No separate reference, proof, construction, or workshop date is supplied.