Editing Geodesic (section)

=== Topology and geometric group theory ===

* In a surface with negative [[Euler characteristic]], any (free) homotopy class determines a unique (closed) geodesic for a [[Hyperbolic surface|hyperbolic]] metric. These geodesics contribute significantly to the geometric understanding of the action of [[Mapping class group of a surface|mapping classes]].
* [[Geodesic metric space|Geodesic metric spaces]] and [[Length space|length spaces]] behave particularly well with isometric [[Group action|group actions]] ([[Švarc–Milnor lemma|Švarc-Milnor lemma]], [[Hopf–Rinow theorem#Variations and generalizations|Hopf-Rinow theorem]], [[Quasi-isometry#Quasigeodesics and the Morse lemma|Morse lemma]]...). They are often an adequate framework for generalizing results from Riemannian geometry to constructions that reflect the geometry of a group. For instance, [[Hyperbolic metric space|Gromov-hyperbolicity]] can be understood in terms of geodesic triangle thinness, and [[CAT(k) space|CAT(0)]] can be stated in terms of angles between geodesics.