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Gaboriau's fixed price problem: do all essentially free p.m.p. actions of a countable group have the same cost?

For a countable group Γ\Gamma acting by measure-preserving Borel automorphisms on a standard probability space, the cost of the orbit equivalence relation is the infimum of the total measure of generating graphings (Levitt 1995). A group has fixed price if all its essentially free probability-measure-preserving actions have the same cost. Gaboriau showed that free groups FnF_n have fixed price nn and that many groups with commuting or amenable structure have fixed price one, and asked whether every countable group has fixed price. Does every countable group have fixed price, or are there two essentially free p.m.p. actions of one group with different costs?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Measured group theory; cost of measure-preserving actions
Posed by
Damien Gaboriau (building on the notion of cost introduced by Gilbert Levitt)
Year posed
2000
Years open
26y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: let A=F(a,b1,…,b99)A=F(a,b_1,\dots,b_{99}), w=ab1ab2⋯ab99aw=ab_1ab_2\cdots ab_{99}a, J=⟨b1,…,b99,w⟩J=\langle b_1,\dots,b_{99},w\rangle and Γ=A∗J(J×⟨t⟩)\Gamma=A*_J(J\times\langle t\rangle). With K=e969699/9595K=e^{96}96^{99}/95^{95}, 0<α<1/2000<\alpha<1/200, Kα3<1/2K\alpha^3<1/2 and η=α/100\eta=\alpha/100, the free part of the Bernoulli action on [0,1]Γ[0,1]^\Gamma has cost at least 1+η1+\eta, while the skew action on X×Z/MX\times\mathbb Z/M via χ(a)=1\chi(a)=1, χ(bi)=χ(t)=0\chi(b_i)=\chi(t)=0 has cost at most 1+99/M1+99/M. Taking M>max⁡{100,99/η}M>\max\{100,99/\eta\} gives two essentially free p.m.p. actions of one finitely generated group with different costs. It does not address the fixed-price-one question for groups with property (T), the fixed price of any specific classical group, or the rank-gradient consequences of fixed price.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (October 5, 2026). Its INPUTS.md says it was developed from an intermediate source argument and that the available source record does not establish that it was the final revision of that argument; it also says the result is distinct from the fixed-price-one question for property (T) groups.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and its proof assembly (the conclusion section) were read against the fixed-price question as Gaboriau posed it. The theorem gives the amalgam Γ=A∗J(J×⟨t⟩)\Gamma=A*_J(J\times\langle t\rangle) with AA free of rank 100, a Bernoulli action of cost at least 1+η1+\eta, and finite height extensions of cost at most 1+99/M1+99/M; both actions are essentially free, so this is a direct negative answer to the general question. The proof (deployment and compression of graphings, finite permutation models, an Arzhantseva-Ol'shanskii type rank bound) was not refereed and there is no Lean formalization. The manuscript's INPUTS.md notes that it was developed from an intermediate argument whose final revision status is not recorded.

Sources

Changelog1 change

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