DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Folner sets and the Folner condition

Definition

Let $G$ be a group, let $S \subseteq G$ be finite, and let $ε > 0$ . A finite nonempty subset $F \subseteq G$ is an $(S, ε)$ -Folner set if

$∣ s F △ F ∣ < ε ∣ F ∣ for every s \in S .$

The group $G$ satisfies the Folner condition if for every finite $S \subseteq G$ and every $ε > 0$ there exists an $(S, ε)$ -Folner set.

Depends on

Group and abelian group

Used by

Folner sequences for enumerated groups Definition
Boxes in Zⁿ are Folner sets Example
Intervals in Z are Folner sets Example
Equivalent boundary formulations of the Folner condition Lemma
Under the ultrafilter lemma, the Folner condition is equivalent to amenability Theorem

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

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