OP-00993 Open

Bounded tensor length for group factors of property (T) groups

Some nontrivial finitely generated groups are isomorphic to their own square (Tyrer Jones; Meier; Baumslag and Miller). Icc groups among them give II$_1$ factors with no bound on the number of II$_1$ factors in a tensor product decomposition, but none of these groups is known to have property (T). Isono and Marrakchi showed that Meier's group is icc and not inner amenable, so its group von Neumann algebra is a full II$_1$ factor isomorphic to its tensor square.

Problem. Let $\Gamma$ be an icc group with property (T). Is there $n \in \mathbb N$ such that $L\Gamma$ is not isomorphic to a tensor product of more than $n$ II$_1$ factors?

Source: S. Popa, Some open problems in W*-rigidity (2013); listed as S.4 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), tensor product decompositions
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-s4,
  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-s4,
  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 S.4; version of 15 May 2026},
}
@article{isono2022,
  author  = {Isono, Yusuke and Marrakchi, Amine},
  title   = {Tensor product decompositions and rigidity of full factors},
  journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
  volume  = {55},
  number  = {1},
  pages   = {109--139},
  year    = {2022},
  doi     = {10.24033/asens.2492},
}
@article{tyrerjones1974,
  author  = {Tyrer Jones, J. M.},
  title   = {Direct products and the {H}opf property},
  journal = {J. Aust. Math. Soc.},
  volume  = {17},
  number  = {2},
  pages   = {174--196},
  year    = {1974},
  doi     = {10.1017/S144678870001675X},
}
@article{baumslagmiller1988,
  author  = {Baumslag, Gilbert and Miller, III, Charles F.},
  title   = {Some odd finitely presented groups},
  journal = {Bull. Lond. Math. Soc.},
  volume  = {20},
  number  = {3},
  pages   = {239--244},
  year    = {1988},
  doi     = {10.1112/blms/20.3.239},
}
@article{meier1982,
  author  = {Meier, David},
  title   = {Non-{H}opfian groups},
  journal = {J. Lond. Math. Soc.},
  volume  = {s2-26},
  number  = {2},
  pages   = {265--270},
  year    = {1982},
  doi     = {10.1112/jlms/s2-26.2.265},
}
@article{drimbe2020,
  author  = {Drimbe, Daniel},
  title   = {Prime {II}$_1$ factors arising from actions of product groups},
  journal = {J. Funct. Anal.},
  volume  = {278},
  number  = {5},
  pages   = {108366},
  year    = {2020},
  doi     = {10.1016/j.jfa.2019.108366},
}

Citing this problem

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OP-00993, "Bounded tensor length for group factors of property (T) groups", clopen.solutions.