Unique Cartan subalgebras for groups with a nonzero ℓ²-Betti number
Popa and Vaes proved that for every weakly amenable group $\Gamma$ with $\beta_1^{(2)}(\Gamma) > 0$, every free ergodic probability measure preserving action gives a II$_1$ factor with a unique Cartan subalgebra up to unitary conjugacy.
Problem. Let $\Gamma$ be a group with $\beta_n^{(2)}(\Gamma) \ne 0$ for some $n \ge 1$. Does every free ergodic probability measure preserving action $\Gamma \curvearrowright (X, \mu)$ give a II$_1$ factor $L^\infty(X) \rtimes \Gamma$ with a unique Cartan subalgebra up to unitary conjugacy?
Source: Conjecture of S. Popa and S. Vaes, Acta Math. 212 (2014); listed as C.5 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
- Cartan subalgebras, C-rigidity, ℓ²-Betti numbers, Peterson's operator algebra problems
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{popavaes2014acta,
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},
}
@misc{peterson-c5,
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 C.5; version of 15 May 2026},
}
@article{popavaes2014crelle,
author = {Popa, Sorin and Vaes, Stefaan},
title = {Unique {C}artan decomposition for {II}$_1$ factors arising from arbitrary actions of hyperbolic groups},
journal = {J. Reine Angew. Math.},
volume = {694},
pages = {215--239},
year = {2014},
doi = {10.1515/crelle-2012-0104},
}
@article{ioana2015,
author = {Ioana, Adrian},
title = {Cartan subalgebras of amalgamated free product {II}$_1$ factors},
journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
volume = {48},
number = {1},
pages = {71--130},
year = {2015},
doi = {10.24033/asens.2239},
}
@article{chifan2013,
author = {Chifan, Ionut and Peterson, Jesse},
title = {Some unique group-measure space decomposition results},
journal = {Duke Math. J.},
volume = {162},
number = {11},
pages = {1923--1966},
year = {2013},
doi = {10.1215/00127094-2331230},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00979, "Unique Cartan subalgebras for groups with a nonzero ℓ²-Betti number", clopen.solutions.