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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?

Theorem 1.1: let $A=F(a,b_1,\dots,b_{99})$ , $w=ab_1ab_2\cdots ab_{99}a$ , $J=\langle b_1,\dots,b_{99},w\rangle$ and $\Gamma=A*_J(J\times\langle t\rangle)$ . With $K=e^{96}96^{99}/95^{95}$ , $0<\alpha<1/200$ , $K\alpha^3<1/2$ and $\eta=\alpha/100$ , the free part of the Bernoulli action on $[0,1]^\Gamma$ has cost at least $1+\eta$ , while the skew action on $X\times\mathbb Z/M$ via $\chi(a)=1$ , $\chi(b_i)=\chi(t)=0$ has cost at most $1+99/M$ . Taking $M>\max\{100,99/\eta\}$ 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.

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Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

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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 $\Gamma=A*_J(J\times\langle t\rangle)$ with $A$ free of rank 100, a Bernoulli action of cost at least $1+\eta$ , and finite height extensions of cost at most $1+99/M$ ; 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.