Quasigeodesics

Quasigeodesics are curves that generalize geodesics by allowing controlled distortion in length relative to the ambient metric. In a geodesic metric space, a map γ from an interval I ⊆ R into the space is called a (λ, C)-quasi-geodesic if for all s, t in I the distance estimate (1/λ)|s − t| − C ≤ d(γ(s), γ(t)) ≤ λ|s − t| + C holds. Here λ ≥ 1 and C ≥ 0 are constants that measure multiplicative and additive distortion. A geodesic is the special case λ = 1 and C = 0, where equality holds up to reparameterization.

Quasigeodesics need not be actual geodesics, but in many spaces they are still closely related to true geodesics. In δ-hyperbolic spaces, for example, the Morse stability lemma asserts that any (λ, C)-quasi-geodesic with endpoints on a geodesic remains within a bounded Hausdorff distance of that geodesic, with the bound depending only on λ, C and δ. This “stability” is central to many arguments in geometric group theory and hyperbolic geometry.

Contexts and applications: quasigeodesics naturally arise when studying word metrics on finitely generated groups via Cayley graphs, where paths corresponding to words are quasi-geodesic, and in hyperbolic groups, where they approximate geodesics in a controlled way. They provide a robust tool for transferring geometric information along coarse, rather than exact, shortest-path structures.

Variants include piecewise or coarse quasi-geodesics with similar distortion controls. See also geodesic, quasi-isometry, and Morse lemma.