I am trying to read the proof of the theorem:
On any closed hyperbolic $2$-manifold M there is a unique, closed geodesic in any non-trivial free homotopy class.
The reference I am using are these notes. I am unable to follow the argument here. He first considers the deck transformation associated to the homotopy class and then concludes that the transformation has to be a hyperbolic transformation. Then we know that a hyperbolic transformation leaves a geodesic invariant (the geodesic joining the two fixed points). He then says that the image of this geodesic is the require curve. But I am unable to see this. Thanks.