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.
Hyperbolic groups
Definition
A finitely generated group is a hyperbolic group if there exists a finite generating set such that the geometric realization of the Cayley graph , with every edge realized as a unit interval and equipped with the induced path metric, is a hyperbolic geodesic metric space. On the vertex set , this path metric restricts to the word metric associated with .
Depends on
Used by
- Elementary and non-elementary hyperbolic groups Definition
- FALSE: a hyperbolic group is just a group with a hyperbolic-plane subgroup False statement
- Finite groups and free groups are hyperbolic Proposition
- Free abelian groups of rank at least two are not hyperbolic Proposition
- Finite subgroups of a hyperbolic group have uniformly bounded order Theorem
- Hyperbolic groups admit finite Dehn presentations Theorem
- Hyperbolicity of a finitely generated group is independent of the finite generating set Theorem
- Infinite-order elements of hyperbolic groups are undistorted Theorem
- Linear isoperimetric characterisation of hyperbolic groups Theorem
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, Section 6.3 (standard reference, not scraped)