OP-00969 Open

Cocycle superrigidity of Bernoulli actions of groups with vanishing first ℓ²-Betti number

Problem. Let $\Gamma$ be a nonamenable group with $\beta_1^{(2)}(\Gamma) = 0$. Is the Bernoulli shift action of $\Gamma$ $\mathcal U_{\mathrm{fin}}$-cocycle superrigid?

Source: S. Popa, Some open problems in W*-rigidity (2013), conjecturing that for nonamenable groups the Bernoulli action is cocycle superrigid exactly when the first ℓ²-Betti number vanishes; listed as R.8 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: DS, GR
Keywords: Bernoulli actions, cocycle superrigidity, ℓ²-Betti numbers, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-r8,
  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-r8,
  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.8; version of 15 May 2026},
}
@article{popa2007cocycle,
  author  = {Popa, Sorin},
  title   = {Cocycle and orbit equivalence superrigidity for malleable actions of {$w$}-rigid groups},
  journal = {Invent. Math.},
  volume  = {170},
  number  = {2},
  pages   = {243--295},
  year    = {2007},
  doi     = {10.1007/s00222-007-0063-0},
}
@article{popa2008,
  author  = {Popa, Sorin},
  title   = {On the superrigidity of malleable actions with spectral gap},
  journal = {J. Amer. Math. Soc.},
  volume  = {21},
  number  = {4},
  pages   = {981--1000},
  year    = {2008},
  doi     = {10.1090/S0894-0347-07-00578-4},
}
@article{peterson2012,
  author  = {Peterson, Jesse and Sinclair, Thomas},
  title   = {On cocycle superrigidity for {G}aussian actions},
  journal = {Ergodic Theory Dynam. Systems},
  volume  = {32},
  number  = {1},
  pages   = {249--272},
  year    = {2012},
  doi     = {10.1017/S0143385710000751},
}
@article{tuckerdrob2020,
  author  = {Tucker-Drob, Robin D.},
  title   = {Invariant means and the structure of inner amenable groups},
  journal = {Duke Math. J.},
  volume  = {169},
  number  = {13},
  pages   = {2571--2628},
  year    = {2020},
  doi     = {10.1215/00127094-2019-0070},
}
@article{bowen2022,
  author  = {Bowen, Lewis and Tucker-Drob, Robin D.},
  title   = {Superrigidity, measure equivalence, and weak {P}insker entropy},
  journal = {Groups Geom. Dyn.},
  volume  = {16},
  number  = {1},
  pages   = {247--286},
  year    = {2022},
  doi     = {10.4171/GGD/647},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00969, "Cocycle superrigidity of Bernoulli actions of groups with vanishing first ℓ²-Betti number", clopen.solutions.