Math Conjectures

Soficity Conjecture

OPENLandmarkConjectureProposed 2000 · Full conjecture

Canonical statement

Every countable group is sofic. Explicitly, for every countable group GG, finite F⊆GF\subseteq G, and ε>0\varepsilon>0, there are an integer nn and a map σ:G→Sym⁡(n)\sigma:G\to\operatorname{Sym}(n) such that σ(1)=1\sigma(1)=1,
dH(σ(gh),σ(g)σ(h))<ε(g,h∈F), d_H(\sigma(gh),\sigma(g)\sigma(h))<\varepsilon\quad(g,h\in F),
and dH(σ(g),1)>1−εd_H(\sigma(g),1)>1-\varepsilon for every g∈F∖{1}g\in F\setminus\{1\}, where dHd_H is normalized Hamming distance.
View source LaTeX
Every countable group is sofic. Explicitly, for every countable group \(G\), finite \(F\subseteq G\), and \(\varepsilon>0\), there are an integer \(n\) and a map \(\sigma:G\to\operatorname{Sym}(n)\) such that \(\sigma(1)=1\),
\[
 d_H(\sigma(gh),\sigma(g)\sigma(h))<\varepsilon\quad(g,h\in F),
\]
and \(d_H(\sigma(g),1)>1-\varepsilon\) for every \(g\in F\setminus\{1\}\), where \(d_H\) is normalized Hamming distance.
ALG-GROUP-021:[Weiss2000Sofic][Pestov2008SoficHyperlinear][OpenAI2026TenAdvances]
Algebra and representation theoryDynamical systemsLogic and foundations

Notes

A group is sofic when every finite fragment can be approximated by permutations in normalized Hamming distance. Weiss introduced the terminology and asked whether all groups are sofic [Weiss2000Sofic]; Pestov surveys its links with dynamics, operator algebras, and logic [Pestov2008SoficHyperlinear]. An August 1, 2026 manuscript claims an explicit counterexample [OpenAI2026TenAdvances]. Because no independent reproduction was available at the cutoff, the stable entry remains open and the announcement is separately graded C.

Proof-claim watch (1)

Explicit non-sofic group

Grade C

Full-scope counterexample claim awaiting independent verification.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.