Editing Prime geodesic (section)

===Closed geodesics===
The [[Quotient space (topology)|quotient surface]] ''M''=Γ\''H,'' using the upper half-plane [[Models of the hyperbolic plane|model of the hyperbolic plane]], is a hyperbolic surface – in fact, a [[Riemann surface]]. Each hyperbolic element ''h'' of Γ determines a [[closed geodesic]] of ''M'': first, the geodesic semicircle joining the fixed points of ''h'' forms the axis of ''h,'' which projects to a geodesic on ''M''.

This geodesic is closed because 2 points which are in the same orbit under the action of Γ project to the same point on the quotient, by definition.

It can be shown that this gives a [[bijection|1-1 correspondence]] between closed geodesics on Γ\''H'' and hyperbolic [[conjugacy class]]es in Γ. The prime geodesics are then those geodesics that trace out their image exactly once &mdash; algebraically, they correspond to primitive hyperbolic conjugacy classes, that is, conjugacy classes {γ} such that γ cannot be written as a nontrivial power of another element of Γ.