OP-00970 Open

Trivial fundamental group for Bernoulli crossed products

Problem. Let $\Gamma$ be a nonamenable group with $\beta_1^{(2)}(\Gamma) < \infty$, and $(X_0, \mu_0)$ a nontrivial standard probability space. Does the crossed product $L^\infty(X_0^\Gamma) \rtimes \Gamma$ of the Bernoulli shift action $\Gamma \curvearrowright (X_0, \mu_0)^\Gamma$ have trivial fundamental group?

Source: Popa and Vaes, in this form in S. Popa, Some open problems in W*-rigidity (2013); listed as R.9 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, fundamental group of II₁ factors, ℓ²-Betti numbers, Peterson's operator algebra problems
Literature checked: Oct. 26, 2026 by clemens-admin

References

@misc{popa2013problems-r9,
  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},
}
@incollection{popavaes2011,
  author        = {Popa, Sorin and Vaes, Stefaan},
  title         = {On the fundamental group of {II}$_1$ factors and equivalence relations arising from group actions},
  booktitle     = {Quanta of maths},
  series        = {Clay Math. Proc.},
  volume        = {11},
  publisher     = {Amer. Math. Soc.},
  pages         = {519--541},
  year          = {2011},
  eprint        = {0810.0706},
  archivePrefix = {arXiv},
}
@misc{peterson-r9,
  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.9; version of 15 May 2026},
}
@article{popa2006,
  author  = {Popa, Sorin},
  title   = {Strong rigidity of {II}$_1$ factors arising from malleable actions of {$w$}-rigid groups. {II}},
  journal = {Invent. Math.},
  volume  = {165},
  number  = {2},
  pages   = {409--451},
  year    = {2006},
  doi     = {10.1007/s00222-006-0502-3},
}
@article{popavaes2014acta-r9,
  author  = {Popa, Sorin and Vaes, Stefaan},
  title   = {Unique {C}artan decomposition for {II}$_1$ factors arising from arbitrary actions of free groups},
  journal = {Acta Math.},
  volume  = {212},
  number  = {1},
  pages   = {141--198},
  year    = {2014},
  doi     = {10.1007/s11511-014-0110-9},
}
@article{gaboriau2002-r9,
  author  = {Gaboriau, Damien},
  title   = {Invariants $\ell^2$ de relations d'{\'e}quivalence et de groupes},
  journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.},
  volume  = {95},
  pages   = {93--150},
  year    = {2002},
  doi     = {10.1007/s102400200002},
}

Citing this problem

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 It will not change if the problem is solved, reclassified or withdrawn.

OP-00970, "Trivial fundamental group for Bernoulli crossed products", clopen.solutions.