DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Quasi-geodesics and quasi-geodesic metric spaces

Definition

Let $(X, d)$ be a metric space.

A map $γ : I \to X$ from an interval $I \subseteq R$ is a $(λ, c)$ -quasi-geodesic if $λ \geq 1$ , $c \geq 0$ , and

$λ^{- 1} ∣ s - t ∣ - c \leq d (γ (s), γ (t)) \leq λ ∣ s - t ∣ + c for all s, t \in I .$

A $(λ, c)$ -quasi-geodesic segment from $x$ to $y$ is such a map with domain a closed bounded interval $[a, b]$ and with $γ (a) = x$ , $γ (b) = y$ .

The metric space $X$ is quasi-geodesic if there are constants $λ \geq 1$ and $c \geq 0$ such that every two points of $X$ are the endpoints of some $(λ, c)$ -quasi-geodesic segment.

Depends on

Used by

Dependency tree · two levels

9 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. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.3 (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory, Section 8.1 (standard reference, not scraped)