OP-00975 Open

A discrete copy of the free group in the unitary group of the hyperfinite II₁ factor

Let $R$ be the hyperfinite II$_1$ factor and $\mathcal U(R)$ its unitary group.

Problem. Is there an injective homomorphism $\mathbb F_2 \to \mathcal U(R)$ whose image is discrete in the strong operator topology?

Source: Credited to N. Ozawa and listed as H.1 in J. Peterson, Open problems in operator algebras (version of 15 May 2026); stated as open by Carderi and Thom (2018).

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

Classification

Primary: OA — Operator Algebras
Secondary: GR
Keywords: discrete subgroups, free groups, hyperfinite II₁ factor, Peterson's operator algebra problems, unitary groups
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{peterson-h1,
  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 H.1; version of 15 May 2026},
}
@article{carderi2018,
  author  = {Carderi, Alessandro and Thom, Andreas},
  title   = {An exotic group as limit of finite special linear groups},
  journal = {Ann. Inst. Fourier},
  volume  = {68},
  number  = {1},
  pages   = {257--273},
  year    = {2018},
  doi     = {10.5802/aif.3160},
}
@misc{alekseev2023,
  author        = {Alekseev, Vadim and Carderi, Alessandro and Thom, Andreas and Tucker-Drob, Robin},
  title         = {About discrete subgroups of full groups of measure preserving equivalence relations},
  year          = {2023},
  eprint        = {2309.08400},
  archivePrefix = {arXiv},
}
@article{alekseev2022,
  author  = {Alekseev, Vadim and Thom, Andreas},
  title   = {Maximal discrete subgroups in unitary groups of operator algebras},
  journal = {SIGMA},
  volume  = {18},
  pages   = {052},
  year    = {2022},
  doi     = {10.3842/SIGMA.2022.052},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00975, "A discrete copy of the free group in the unitary group of the hyperfinite II₁ factor", clopen.solutions.