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The amenability problem for Thompson's group F (Geoghegan's conjecture)

Theorem 1.1: Thompson's group $F$ is not amenable. The proof finds a fixed finite set $S\subset F$ and $b>0$ with $|hA\triangle A|\ge b|A|$ for some $h\in S$ , for every nonempty finite $A\subset F$ , so no Folner sets exist. It converts approximate invariance of finite averages into an approximate fixed point of a Lipschitz self-map of a Hilbert ball, contradicting a Benyamini-Sternfeld displacement map that the paper constructs explicitly. It confirms Geoghegan's conjecture. The formal version gives no explicit expansion constant for a standard generating set, and the paper does not address other questions about $F$ such as its exact isoperimetric profile.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The amenability problem for Thompson's group F (Geoghegan's conjecture) · Amenability of Thompson's group F

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Named evidence

VibeMathed record: The amenability problem for Thompson's group F (Geoghegan's conjecture)

specialist registry · passed

No independent mathematician has checked this yet. The formalization catalogue lists comparator config ComparatorChallenges/ThompsonNonamenability.json with declaration OAI.ThompsonNonamenability.thompson_F_nonamenable_composition in OAI/GroupTheory/Thompson/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement file was read: it defines F concretely as strictly increasing homeomorphisms of [0,1] with finitely many dyadic knots, each piece affine with slope an integer power of 2; it defines an invariant mean as a positive, normalized, left-invariant real-linear functional on bounded real functions on the group; and it states that there is a group structure on F whose product is composition and that no invariant mean exists. That is the headline claim. Not rebuilt here: the Lean build and axiom check were not run. Given the problem's history of withdrawn claims, an expert reading is still wanted even with a formal proof; the Lean statement was read for fidelity here, not audited independently.

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The amenability problem for Thompson's group F (Geoghegan's conjecture)

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof: thompson_F_nonamenable_composition

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: ThompsonNonamenability

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lean-proof link supplied by the VibeMathed registry.