CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Under the ultrafilter lemma, groups containing a rank-two free subgroup are nonamenable

Statement

Assume the ultrafilter lemma. If a group contains a free subgroup of rank $2$ , then it is nonamenable.

Facts & Assumptions

Given: A group $G$ containing a subgroup $H ≅ F_{2}$ , and the ultrafilter lemma.

[L1]

The rank-two free group is nonamenable (The free group of rank two is nonamenable).

[L2]

Under the ultrafilter lemma, subgroups of amenable groups are amenable (Under the ultrafilter lemma, subgroups and quotients of amenable groups are amenable).

Proof

technique · direct

1.1

If $G$ were amenable, then [L2] would make its subgroup $H$ amenable.

L2givenassume-contra

2.1

But $H ≅ F_{2}$ , and [L1] says $F_{2}$ is nonamenable. This contradiction proves that $G$ is nonamenable.

L1step 1.1discharge-contradiction∎

Depends on

Used by

There exist nonamenable groups without nonabelian free subgroups Remark

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)