OP-01054 Needs verification

Connes's rigidity conjecture for property (T) groups

Needs verification: A solution has been claimed in the literature; held until the record of the claim is listed here.

Submit a result

Chifan, Ioana, Osin and Sun constructed an icc group $\Gamma$ with property (T) such that $L\Gamma \cong L\Lambda$, for an arbitrary group $\Lambda$, forces $\Gamma \cong \Lambda$.

Problem. Let $\Gamma$ and $\Lambda$ be icc groups with property (T) such that $L\Gamma \cong L\Lambda$. Must $\Gamma$ and $\Lambda$ be isomorphic?

Source: A. Connes, Classification des facteurs (Kingston 1980; Proc. Sympos. Pure Math. 38, 1982); listed as O.3 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: OA — Operator Algebras
Secondary: GR
Keywords: Connes's rigidity conjecture, group von Neumann algebras, Peterson's operator algebra problems, property (T), W*-rigidity

References

@incollection{connes1982-o3,
  author    = {Connes, Alain},
  title     = {Classification des facteurs},
  booktitle = {Operator algebras and applications, Part 2 (Kingston, Ont., 1980)},
  series    = {Proc. Sympos. Pure Math.},
  volume    = {38},
  publisher = {Amer. Math. Soc.},
  pages     = {43--109},
  year      = {1982},
  doi       = {10.1090/pspum/038.2/679497},
}
@misc{peterson-o3,
  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 O.3; version of 15 May 2026},
}
@article{chifan2023-o3,
  author  = {Chifan, Ionu{\c t} and Ioana, Adrian and Osin, Denis and Sun, Bin},
  title   = {Wreath-like products of groups and their von {N}eumann algebras {I}: {$W^*$}-superrigidity},
  journal = {Ann. of Math.},
  volume  = {198},
  number  = {3},
  pages   = {1261--1303},
  year    = {2023},
  doi     = {10.4007/annals.2023.198.3.6},
}
@inproceedings{popa2007icm-o3,
  author    = {Popa, Sorin},
  title     = {Deformation and rigidity for group actions and von {N}eumann algebras},
  booktitle = {International Congress of Mathematicians, Vol. I (Madrid, 2006)},
  publisher = {Eur. Math. Soc.},
  pages     = {445--477},
  year      = {2007},
  doi       = {10.4171/022-1/18},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01054, "Connes's rigidity conjecture for property (T) groups", clopen.solutions.