Cost of a group and its first ℓ²-Betti number
The cost $\mathcal C(\Gamma)$ of a countable group $\Gamma$ is the infimum of the costs of the orbit equivalence relations of its free probability measure preserving actions. Gaboriau proved $\mathcal C(\Gamma) \ge \beta_1^{(2)}(\Gamma) + 1$ for every infinite countable group $\Gamma$.
Problem. Is $\mathcal C(\Gamma) = \beta_1^{(2)}(\Gamma) + 1$ for every infinite countable group $\Gamma$?
Source: Group form of a question of D. Gaboriau, Invariants ℓ² de relations d'équivalence et de groupes (2002), p. 129, who asked whether C(R) − 1 = β₁(R) − β₀(R) for every p.m.p. equivalence relation; listed as G.4 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- GR — Group Theory
- Secondary
- DS, OA
- Keywords
- cost, ℓ²-Betti numbers, measured group theory, Peterson's operator algebra problems
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{gaboriau2002,
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},
}
@misc{peterson-g4,
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 G.4; version of 15 May 2026},
}
@article{gaboriau2000,
author = {Gaboriau, Damien},
title = {Co{\^u}t des relations d'{\'e}quivalence et des groupes},
journal = {Invent. Math.},
volume = {139},
number = {1},
pages = {41--98},
year = {2000},
doi = {10.1007/s002229900019},
}
@misc{donoso2026random,
author = {Donoso-Echenique, Miguel and Silva, Eduardo},
title = {The first $\ell^2$-{B}etti number, cost and rank gradient of few-relator random groups},
year = {2026},
eprint = {2610.29502},
archivePrefix = {arXiv},
}
@misc{donoso2026burnside,
author = {Donoso-Echenique, Miguel and Silva, Eduardo},
title = {Free {B}urnside groups of large odd exponent have cost 1},
year = {2026},
eprint = {2608.20472},
archivePrefix = {arXiv},
}
@misc{poulin2026,
author = {Poulin, Antoine and Wr{\'o}bel, Konrad},
title = {Cost of one-relator groups},
year = {2026},
eprint = {2606.23506},
archivePrefix = {arXiv},
}
@misc{slutsky2026,
author = {Slutsky, Raz},
title = {A product-neighbourhood criterion for fixed price one},
year = {2026},
eprint = {2607.20273},
archivePrefix = {arXiv},
}
@misc{khezeli2025,
author = {Khezeli, Ali},
title = {Products of infinite countable groups have fixed price one},
year = {2025},
eprint = {2509.08325},
archivePrefix = {arXiv},
}
@article{hutchcroft2020,
author = {Hutchcroft, Tom and Pete, G{\'a}bor},
title = {Kazhdan groups have cost 1},
journal = {Invent. Math.},
volume = {221},
number = {3},
pages = {873--891},
year = {2020},
doi = {10.1007/s00222-020-00967-6},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00973, "Cost of a group and its first ℓ²-Betti number", clopen.solutions.