OP-01059 Needs verification

Finitely many groups with the same group factor, for property (T) groups

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

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Problem. Let $\Gamma$ be an icc group with property (T). Are there only finitely many isomorphism classes of groups $\Lambda$ with $L\Lambda \cong L\Gamma$?

Source: S. Popa, Some open problems in W*-rigidity (2013), who asks to prove Connes's rigidity conjecture up to finite classes; listed as R.6 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: group von Neumann algebras, Peterson's operator algebra problems, property (T), W*-rigidity

References

@misc{popa2013problems-r6,
  author       = {Popa, Sorin},
  title        = {Some open problems in {W}$^*$-rigidity},
  howpublished = {Slides, Paris, June 2013},
  year         = {2013},
  url          = {https://www.math.ucla.edu/~popa/ProblemsJune2013.pdf},
}
@misc{peterson-r6,
  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 R.6; version of 15 May 2026},
}
@inproceedings{popa2007icm-r6,
  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 It will not change if the problem is solved, reclassified or withdrawn.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-01059, "Finitely many groups with the same group factor, for property (T) groups", clopen.solutions.