A group without fixed price

Claims a finitely generated group with two free measure-preserving actions of different costs, so fixed price fails.

1 paper October 5, 2026 Explained by AI

Let a group shuffle points of a space while keeping total size fixed. Cost asks how many connections, on average, you need to link each point to everything the group can reach from it. A long-standing question asked whether a group's cost is the same for all its free shuffles. This paper claims a group where two such shuffles have different costs. The gap it gives is extremely small.

First step

1995

Gilbert Levitt, who introduced the cost of a measure-preserving action

Significance

Major

The fixed-price problem is a central open question in measured group theory, studied in Gaboriau's 2000 work on cost.

Trust

Claimed, not checked

Field

Group theory

1 paper

The explanation

Picture a cloud of points, each with a few arrows to other points. A group moves the points around and keeps total size fixed. The orbit of a point is everywhere the group can send it. A graphing is a family of measure-preserving maps whose arrows connect every point to its whole orbitSource 1. Its cost is the total size of the maps' domainsSource 1. The cost of the action is the smallest such total. For the integers acting freely, one arrow per point, sending each point to its shift, connects everything. So the cost is 1.

An action is essentially free when each nonidentity element fixes almost no pointsSource 2. A group has fixed price when all its essentially free probability-preserving actions have the same costSource 2. Levitt introduced cost, and Gaboriau linked it to group theorySource 3. Gaboriau showed free groups have fixed price equal to their rankSource 4. Products of two infinite groups have fixed price oneSource 5. Whether every group has fixed price stayed open.

The paper builds one group. Start with the free group A on 100 letters, a and b₁ to b₉₉. Form the word w = a b₁ a b₂ ⋯ a b₉₉ a. Count the a's: 99 + 1 = 100, so a height map χ sends w to 100Source 6. The group Γ glues A to a product with a new generator t, along the subgroup J made by the b's and wSource 6.

The paper claims two free actions with different costs. One is the Bernoulli shift: random labels at every group element, shifted by the group. Its cost is at least 1 + ηSource 7 Source 8. The other adds a height in Z/MZ, a clock with M positions. Its cost is at most 1 + 99/MSource 7 Source 9. For M = 1000 the bound is 1 + 99/1000 = 1.099. Choosing M large enough makes the second cost smaller than the firstSource 10. So the general fixed-price assertion is falseSource 10.

The gap is tiny. The paper sets K = e⁹⁶·96⁹⁹/95⁹⁵, about 1.13 × 10⁵⁰. It needs α < 1/200 and Kα³ < 1/2, with η = α/100Source 7. So α must be below about 1.64 × 10⁻¹⁷, and η below about 1.64 × 10⁻¹⁹. The obstacle was the lower bound. The paper reduces it to finite permutation modelsSource 11. Expansion then forces their ranks to exceed η times their sizeSource 12.

How sure is this? The paper has no Lean formalization. No specialist review has been published. This is a claim by the paper.

What it doesn't show. This does not have a Lean formalization, and it does not say which groups do or do not have fixed price.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. It has no Lean formalization and no published specialist review, and the cost gap is extremely small.

Words used

Measure-preserving action: A group moving points of a probability space so that every region keeps its size.
Orbit: All the points a group can send a given point to.
Graphing: A countable family of measure-preserving maps whose arrows link each point to its whole orbit.
Cost: The least total size of the domains of a graphing that links every orbit.
Essentially free: Each nonidentity group element fixes only a set of measure zero.
Fixed price: All essentially free probability-preserving actions of the group have the same cost.
Bernoulli shift: The action that shifts independent random labels attached to the group's elements.

The original paper

A group without fixed price (PDF, opens in a new tab)
October 5, 2026

All quoted lines (12)

Its cost $C(\mathcal E)$ is the sum of its domain measures; $\operatorname{Cost}(\mathcal R)$ is the infimum of these sums over generating graphings.
A group without fixed price Back to text
A group has fixed price if all its essentially free probability-measure-preserving (p.m.p.) actions have the same cost.
A group without fixed price Back to text
Gaboriau developed its connections with group theory and the fixed-price problem.
A group without fixed price Back to text
His work shows, in particular, that free groups have fixed price equal to their rank, and establishes fixed price one for many groups with commuting or amenable structure.
A group without fixed price Back to text
Recent results include the fixed-price-one theorem for products of two infinite countable groups, metric and infinite-measure action criteria, and a criterion based on small product neighborhoods of finite subsets.
A group without fixed price Back to text
The homomorphism $\chi:\Gamma\to\mathbb Z$ given by $\chi(a)=1$ and $\chi(b_i)=\chi(t)=0$ is well defined; note that $\chi(w)=100$ .
A group without fixed price Back to text
Put $K=e^{96}96^{99}/95^{95}$ , choose $0<\alpha<1/200,\qquad K\alpha^3<1/2, \qquad \eta=\alpha/100.$
A group without fixed price Back to text
Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.
A group without fixed price Back to text
The relation $w=ab_1\cdots ab_{99}a$ propagates $a$ -edges from any $99$ consecutive heights to the next height, so an initial cost $99/M$ is enough.
A group without fixed price Back to text
Thus the general fixed-price assertion is false.
A group without fixed price Back to text
Pass to finite permutation models.
A group without fixed price Back to text
The expansion property forces $\operatorname{rk}(D_V)>\eta|V|$ for large models.
A group without fixed price Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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