A group without fixed price

1 paper

Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.

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A group without fixed price

OpenAI · October 5, 2026

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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@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}
}

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