Is there a group that is not sofic?
Problem. Is there a group that is not sofic?
Source: Sofic groups were introduced by M. Gromov (1999) and named by B. Weiss (2000), the question being whether every group is sofic; listed as G.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- GR — Group Theory
- Secondary
- DS, OA
- Keywords
- group approximation, Peterson's operator algebra problems, sofic groups
References
@article{gromov1999,
author = {Gromov, Misha},
title = {Endomorphisms of symbolic algebraic varieties},
journal = {J. Eur. Math. Soc.},
volume = {1},
number = {2},
pages = {109--197},
year = {1999},
doi = {10.1007/PL00011162},
}
@article{weiss2000,
author = {Weiss, Benjamin},
title = {Sofic groups and dynamical systems},
journal = {Sankhy{\=a} Ser. A},
volume = {62},
year = {2000},
}
@misc{peterson-g1,
author = {Peterson, Jesse},
title = {Open problems in operator algebras},
howpublished = {Web page},
year = {2026},
url = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
note = {Problem G.1; version of 15 May 2026},
}
Resolution claims
We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field $K$ of characteristic two, a finitely presented group $G$, and finite sums $a,b\in K[G]$ with $ab=1$ but $ba\ne1$. The group $G$ is nonsofic. The same elements define a cellular automaton on $K^G$ that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: math @ 7ef5c0c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01064, "Is there a group that is not sofic?", clopen.solutions.