DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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 sequences for enumerated groups

Definition

Let $G = {g_{1}, g_{2}, \dots}$ be an enumerated countable group. A sequence $(F_{n})_{n \geq 1}$ of finite nonempty subsets of $G$ is a Folner sequence if

$\lim_{n \to \infty} \frac{∣ g_{i} F_{n} △ F_{n} ∣}{∣ F_{n} ∣} = 0 for every fixed i \geq 1.$

Equivalently, for every finite subset $S \subseteq G$ and every $ε > 0$ , all sufficiently large $F_{n}$ are $(S, ε)$ -Folner sets.

Depends on

Folner sets and the Folner condition

Used by

FALSE: every uncountable amenable group has a Folner sequence False statement
Enumerated countable amenable groups admit Folner sequences Proposition

Dependency tree · two levels

2 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)