LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Under the ultrafilter lemma, directed unions of amenable subgroups are amenable

Statement

Assume the ultrafilter lemma. Let $G = ⋃_{i \in I} H_{i}$ be a directed union of subgroups. If every $H_{i}$ is amenable, then $G$ is amenable.

Facts & Assumptions

Given: A directed family of amenable subgroups $(H_{i})_{i \in I}$ whose union is $G$ , and the ultrafilter lemma.

[L1]

Under the ultrafilter lemma, amenability is equivalent to the Folner condition (Under the ultrafilter lemma, the Folner condition is equivalent to amenability).

Proof

technique · direct

1.1L1givenchoose

Let $S \subseteq G$ be finite and let $ε > 0$ . Because the family is directed and $⋃_{i} H_{i} = G$ , some index $i$ satisfies $S \subseteq H_{i}$ . Since $H_{i}$ is amenable, [L1] gives a finite nonempty set $F \subseteq H_{i}$ with $∣ s F △ F ∣ < ε ∣ F ∣$ for every $s \in S$ .

2.1L1step 1.1∎

The set $F$ from step 1.1 is also an $(S, ε)$ -Folner set when viewed inside $G$ . Since $S$ and $ε$ were arbitrary, $G$ satisfies the Folner condition. Therefore [L1] makes $G$ amenable.

Depends on

Under the ultrafilter lemma, the Folner condition is equivalent to amenability

Used by

Under the ultrafilter lemma, solvable groups and locally finite groups are amenable Corollary

Dependency tree · two levels

6 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)