Alphabeta MathHow 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.
There exist nonamenable groups without nonabelian free subgroups
Remark
This page proves that containing a rank-two free subgroup forces nonamenability, but the converse fails. Standard constructions due to Adian and Olshanskii produce nonamenable groups with no nonabelian free subgroup.
Accordingly, the implication in Under the ultrafilter lemma, groups containing a rank-two free subgroup are nonamenable is one-way only.
Depends on
Used by
- FALSE: every nonamenable group contains a rank-two free subgroup False statement
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
- 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)