quasigeodesic

A quasigeodesic is a curve in a metric space that behaves like a geodesic up to uniform distortion. Formally, a map q from an interval I in the real line into a metric space (X, d) is called a (λ, c)-quasi-geodesic if there exist constants λ ≥ 1 and c ≥ 0 such that for all s ≤ t in I,

(1/λ)(t − s) − c ≤ d(q(s), q(t)) ≤ λ(t − s) + c.

If q is parameterized by arc length, one often uses similar inequalities with distance controlling the parameter difference. The image q(I) is sometimes referred to as a quasigeodesic.

Quasigeodesics generalize geodesics, which are the case (λ, c) = (1, 0). They are central in geometric group theory and the study of spaces with non-positive or negative curvature, where one can compare discrete or continuous routes to true shortest paths.

In Gromov hyperbolic spaces, quasi-geodesics enjoy a stability property: there exist constants depending only on λ and c such that every (λ, c)-quasi-geodesic stays within a bounded distance of a true geodesic with the same endpoints. This is often called the Morse lemma or stability of quasi-geodesics. In general metric spaces, however, a quasi-geodesic need not remain close to any geodesic.

There are variations in the definition, such as local quasi-geodesics, where the inequalities are required only on subintervals of bounded length. Quasigeodesics arise in contexts including word metrics on groups, Cayley graphs, Teichmüller theory, and the geometry of hyperbolic spaces, where they provide a robust way to relate algebraic paths to geometric routes.