Math Conjectures

Hyperlinear Group Conjecture

OPENMajorConjectureProposed c. 2000 · Full conjecture

Canonical statement

Every countable group GG is hyperlinear: there are integers nkn_k and an injective homomorphism
G⟶∏kU(nk)/⨁kU(nk), G\longrightarrow \prod_k U(n_k)\big/\bigoplus_k U(n_k),
where the metric ultraproduct uses normalized Hilbert--Schmidt distance and a nonprincipal ultrafilter.
View source LaTeX
Every countable group \(G\) is hyperlinear: there are integers \(n_k\) and an injective homomorphism
\[
  G\longrightarrow \prod_k U(n_k)\big/\bigoplus_k U(n_k),
\]
where the metric ultraproduct uses normalized Hilbert--Schmidt distance and a nonprincipal ultrafilter.
ANAL-FA-020:[Pestov2008SoficHyperlinear][Radulescu2008Hyperlinear][JiEtAl2021MIPRE]
Analysis and functional analysisAlgebra and representation theoryLogic and foundations

Notes

Hyperlinearity asks whether every countable group admits asymptotically multiplicative finite-dimensional unitary models. Every sofic group is hyperlinear, but the converse is unknown [Pestov2008SoficHyperlinear]. Although MIP∗=REMIP^*=RE refuted the operator-algebraic Connes embedding problem [JiEtAl2021MIPRE], it did not produce a non-hyperlinear group. This distinction is why the problem is recorded separately from both soficity and Connes rigidity.

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.