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A group without fixed price

OpenAI · October 5, 2026

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Abstract

We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action (arXiv:1103.1063, 2011). Shows that for finitely generated groups the cost is maximal on Bernoulli actions, the general fact behind using the Bernoulli action as the high-cost action in Theorem 1.1's claimed strict cost gap; not in the paper's bibliography.
M. Abért and N. Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem (arXiv:math/0701361, 2007; J. Eur. Math. Soc. 2012). Relates rank growth of finite-index subgroups to cost and connects the fixed price problem to the rank vs. Heegaard genus conjecture, relevant to the paper's transfer from ranks of the finite-model groups $D_V$ to the Bernoulli cost (Step 2); not in the bibliography.

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 259

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-group-without-fixed-price-October-5-2026,
  author = {{OpenAI}},
  title = {{A group without fixed price}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf}{OAI:A-group-without-fixed-price-October-5-2026}},
  year = {2026}
}