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.
Closed surface groups are hyperbolic
Example
The fundamental group of a closed hyperbolic surface is a hyperbolic group.
Facts & Assumptions
Given: A closed hyperbolic surface and its fundamental group .
The hyperbolic plane is hyperbolic (The hyperbolic plane is hyperbolic).
The Švarc-Milnor lemma transfers geometric actions on proper geodesic spaces to quasi-isometries with finitely generated groups (The Svarc-Milnor lemma).
Hyperbolicity is invariant under quasi-isometry (Hyperbolicity is a quasi-isometry invariant of geodesic spaces).
Verification
The group acts properly discontinuously and cocompactly by deck transformations on the universal cover of .
By [L2], is quasi-isometric to , and [L1] shows that is hyperbolic. Therefore [L3] makes hyperbolic.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- Clara Löh, Geometric Group Theory, Sections 4.4 and 6.3 (standard reference, not scraped)