Is every group sofic, or hyperlinear?
Green asks whether every group is well approximated by finite groups, which is two questions: by symmetric groups with the normalised Hamming distance (sofic groups), and by unitary groups (hyperlinear groups). Every sofic group is hyperlinear, and the converse is not known. The Higman group was not known to be either.
Problem. Is every group sofic? Is every group hyperlinear?
Source: Sofic groups: M. Gromov (1999) and B. Weiss (2000); hyperlinear groups: from Connes' embedding problem (F. Rădulescu); see V. G. Pestov, Hyperlinear and sofic groups: a brief guide (2008). Listed as Problem 100 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- GR — Group Theory
- Secondary
- OA
- Keywords
- approximate homomorphisms, Higman group, hyperlinear groups, sofic groups
References
@article{gromov1999endomorphisms,
author = {Gromov, Misha},
title = {Endomorphisms of symbolic algebraic varieties},
journal = {Journal of the European Mathematical Society},
volume = {1},
number = {2},
pages = {109--197},
year = {1999},
doi = {10.1007/PL00011162}
}
@article{pestov2008guide,
author = {Pestov, Vladimir G.},
title = {Hyperlinear and sofic groups: a brief guide},
journal = {Bulletin of Symbolic Logic},
volume = {14},
number = {4},
pages = {449--480},
year = {2008},
doi = {10.2178/bsl/1231081461}
}
@article{helfgott2019higman,
author = {Helfgott, Harald A. and Juschenko, Kate},
title = {Soficity, short cycles, and the {H}igman group},
journal = {Transactions of the American Mathematical Society},
volume = {371},
number = {4},
pages = {2771--2795},
year = {2019},
doi = {10.1090/tran/7534}
}
@misc{kun2026nonsofic,
author = {Kun, Gabor and Thom, Andreas},
title = {Nonsofic wreath products of residually finite groups},
year = {2026},
eprint = {2608.06222},
archivePrefix = {arXiv}
}
@misc{fournierfacio2026torsionfree,
author = {Fournier-Facio, Francesco},
title = {A torsion-free non-sofic group},
year = {2026},
eprint = {2608.02025},
archivePrefix = {arXiv}
}
@misc{alekseev2026hyperlinear,
author = {Alekseev, Vadim and Liu, Jihao and Thom, Andreas},
title = {Spectral gap, internality of relative commutants, and non-hyperlinear groups},
year = {2026},
eprint = {2609.37761},
archivePrefix = {arXiv}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 100; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
We prove that the relative commutant of a finite unitary tuple in a tracial matrix ultraproduct is internal whenever its conjugation operator has a spectral gap off the commutant. This implies the existence of a non-hyperlinear group.
This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $Γ<G$ be such that $\{g\in G:gΓg^{-1}\leqΓ\}$ generates $G$ as a group, and suppose that both $Γ$ and $G$ have property $(T)$. If $Γ$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/Γ}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Open → Solution claimed | The registry's curators | Spectral gap, internality of relative commutants, and non-hyperlinear groups v1 recorded from arXiv:2609.37761v2. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01047, "Is every group sofic, or hyperlinear?", clopen.solutions.