DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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.

Delta-slim triangles and hyperbolic spaces

Definition

Let $(X, d)$ be a geodesic metric space and let $δ \geq 0$ .

A geodesic triangle in $X$ is $δ$ -slim if each side lies in the closed $δ$ -neighborhood of the union of the other two sides.

The space $X$ is Gromov hyperbolic if there exists $δ \geq 0$ such that every geodesic triangle in $X$ is $δ$ -slim.

Depends on

Geodesic segments, geodesic triangles, and geodesic metric spaces

Used by

Hyperbolic groups Definition
The Gromov boundary via asymptotic sequences Definition
The hyperbolic plane is hyperbolic Example
Hyperbolic spaces have thin geodesic quadrilaterals Lemma
Cayley trees are 0-hyperbolic Proposition
Slim triangles, the Gromov product, and the four-point condition are equivalent up to constants Theorem

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

Clara Löh, Geometric Group Theory, Section 6.2.1 (standard reference, not scraped)