Editing Prime geodesic (section)

==Technical background==
This section presents some facts from [[hyperbolic geometry]] that are helpful in understanding prime geodesics.

===Hyperbolic isometries===
In the [[Poincaré half-plane model]] ''H'' of 2-dimensional [[hyperbolic geometry]], a [[Fuchsian group]] – that is, a [[discrete subgroup]] Γ of [[projective linear group|PSL(2, '''R''')]] – [[Group action (mathematics)|acts]] on ''H'' via [[linear fractional transformation]]. Each element of PSL(2, '''R''') defines an [[isometry]] of ''H'', so Γ is a group of isometries of ''H''.

There are then 3 types of transformation: hyperbolic, elliptic, and parabolic. (The loxodromic transformations are not present because only [[real number]]s are involved.) Then an element γ of Γ has 2 distinct real fixed points if and only if γ is hyperbolic. See [[Möbius transformation#Classification|Classification of isometries]] and [[Möbius transformation#Fixed points|Fixed points of isometries]] for more details.

===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.