1 · The essentials
Can the same mathematical system have different costs depending on how it acts? Here, “cost” is a mathematical quantity, not a financial expense. The manuscript describes a system generated by finitely many elements and two ways for it to act that have different costs. Both ways preserve probabilities, and almost every point is left unchanged only by doing nothing. One particular way of acting has cost separated from one by a positive gap. A sequence of other ways has costs approaching one, so they cannot all share that first cost. The manuscript presents this as a negative answer to the general fixed-price problem, but the result has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine the same set of tools used in different arrangements, with each arrangement assigned an efficiency score. The question is whether the tools alone force the score to stay the same; the manuscript describes a mathematical setting where the arrangement matters too.
Why it matters
The fixed-price problem asks whether cost must be the same for all essentially free, probability-measure-preserving actions of a given group. The construction shown in the manuscript gives a negative answer in general. Its scope is a counterexample, not a claim that every group has actions of different costs.
2 · Going further
The precise target is a finitely generated group admitting two essentially free, probability-measure-preserving actions with unequal costs. “Essentially free” means that almost every point has no nonidentity group element fixing it, while probability preservation means that the action leaves the probability measure unchanged. The abstract identifies the question being answered as Gaboriau’s general fixed-price problem.
The group described in the abstract is an amalgam of a free group of rank with a direct product. The cost comparison is the central mechanism: its Bernoulli action has cost bounded away from , whereas a sequence of finite height extensions has costs tending to . That separation ensures that the Bernoulli action and a sufficiently far-along member of the sequence have different costs.
The supplied abstract does not specify the factors in the direct product, the subgroup used in the amalgamation, or the estimates establishing the cost bounds. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described from this source. The result is shown in the manuscript, but is not formalized in Lean and has not been formally verified.
Useful notions
- Groups and finite generating sets
- Group actions
- Probability measures and measure preservation
- Essentially free actions
- Cost and the fixed-price question
What was known before / what is new
Before
The source identifies the general fixed-price problem as a problem of Gaboriau. It concerns whether a group’s essentially free, probability-measure-preserving actions must all have the same cost. No account of earlier partial results is supplied.
New
The manuscript constructs a finitely generated group with two such actions of different costs. Its distinguishing comparison is between a Bernoulli action whose cost stays away from one and finite height extensions whose costs tend to one. The manuscript presents this construction as a negative answer to the general fixed-price problem.
Limits and assumptions
The claim concerns the constructed finitely generated group and essentially free, probability-measure-preserving actions. Costs tending to one do not by themselves assert that any action in the sequence has cost exactly one. The abstract omits the detailed construction and cost estimates, and it states no further open questions. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Finitely generated group —
- A group whose elements can all be obtained by combining finitely many chosen elements and their inverses.
- Group action —
- A way for a group’s elements to transform a space consistently with the group’s operation.
- Essentially free action —
- An action in which, outside a set of probability zero, only the identity element fixes a point.
- Probability-measure-preserving action —
- An action whose transformations leave the probability measure unchanged.
- Cost —
- The numerical quantity associated with an action that is compared in the fixed-price problem.
- Fixed price —
- The property that all essentially free, probability-measure-preserving actions of a group have the same cost.
Keywords : Fixed price · Cost of actions · Group theory
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A group without fixed price
05/10/2026We 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.