Result 259, Group theory

A group without fixed price

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.

Disproof or counterexample

The bigger picture

Why it matters

Cost measures the average number of links needed to connect points within the orbits of a group action. The manuscript reports a group whose actions have different costs, so this measure is not determined by the group alone.

What changes?

The construction uses a group generated by finitely many elements, gluing a free group of rank 100 to a direct product along a shared subgroup. It has two essentially free, probability-measure-preserving actions: nonidentity elements fix only negligible sets, and the actions preserve probabilities. Its Bernoulli action, based on independent random labels indexed by group elements, has cost bounded away from one. A sequence of finite height extensions has costs tending to one, yielding actions with different costs.

What does that help mathematicians do?

This rules out assigning a single action-independent cost to every finitely generated group. For the constructed group, researchers must distinguish the Bernoulli action from actions whose costs approach one; studying one action cannot determine the costs of all the others. The limiting statement does not itself assert an action of cost exactly one. Different costs already suffice to refute the general fixed-price claim.

Are there practical applications?

The immediate value is foundational: the example separates a group's algebraic structure from the measured complexity of its actions. It provides a counterexample against which proposed general principles about cost can be tested. The supplied sources describe no practical algorithm or deployment; the demonstrated consequence concerns the mathematical classification of group actions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A group without fixed price

October 5, 2026 27 pages

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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Cite (BibTeX) 
@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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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.