Alphabeta MathDefinitionDefinition: 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.
Left-invariant means and amenable groups
Definition
A mean on is left invariant if
The group is amenable if it admits a left-invariant mean on .
Depends on
Used by
- Left and right amenability agree by inversion Lemma
- Paradoxical groups admit no invariant mean Lemma
- Finite groups are amenable Proposition
- Extensions of amenable groups are amenable Theorem
- Under the ultrafilter lemma and a matching-extension principle, a group is amenable if and only if it is not paradoxical Theorem
- Under the ultrafilter lemma, subgroups and quotients of amenable groups are amenable Theorem
- Under the ultrafilter lemma, the Folner condition is equivalent to amenability Theorem
Dependency tree · two levels
3 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)