LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Equivalent boundary formulations of the Folner condition

Statement

Let $G$ be a group, $S \subseteq G$ finite, and $F \subseteq G$ finite nonempty. Then:

$∣ s F △ F ∣ < ε ∣ F ∣$ for every $s \in S$ if and only if $∣ s F ∖ F ∣ < (ε / 2) ∣ F ∣$ for every $s \in S$ .
Replacing the left translates $s F$ by right translates $F s$ gives an equivalent condition after inversion.

Facts & Assumptions

Given: A group $G$ , a finite subset $S \subseteq G$ , a finite nonempty set $F \subseteq G$ , and a real $ε > 0$ .

[L1]

An $(S, ε)$ -Folner set is defined by the symmetric-difference inequality (Folner sets and the Folner condition).

Proof

technique · direct

1.1

For each $s \in S$ , left translation by $s$ is a bijection of $G$ , so $∣ s F ∣ = ∣ F ∣$ and therefore $∣ s F △ F ∣ = ∣ s F ∖ F ∣ + ∣ F ∖ s F ∣ = 2 ∣ s F ∖ F ∣$ . Hence the symmetric-difference and one-sided boundary formulations differ only by the factor $2$ .

L1givenalgebra

2.1

Inversion is a bijection $F \mapsto F^{- 1}$ with $∣ (s F) △ F ∣ = ∣ (F^{- 1} s^{- 1}) △ F^{- 1} ∣$ . Thus the left-translate and right-translate versions are equivalent after replacing $S$ by $S^{- 1}$ .

step 1.1givenalgebra∎

Depends on

Used by

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

Dependency tree · two levels

4 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)