Fix a pants decomposition 
of a closed orientable
surface 
of genus 
. Each point of the Teichmüller
space 
is represented by a marked surface 
with a hyperbolic geometry.
The free homotopy class of each 
contains a unique closed geodesic
on 
;
let 
denote its length. Cutting 
along the geodesics gives hyperbolic pairs
of pants; their boundary lengths determine the geometry of the individual pieces.
A twist parameter 
records the relative displacement when the two boundary
geodesics corresponding to 
are glued back together.
The length-twist pairs 
are called the Fenchel-Nielsen coordinates
of 
with respect to 
. The Fenchel-Nielsen coordinate map is
 |
The Fenchel-Nielsen theorem states that this map is a homeomorphism. Thus a fixed pants decomposition together with a convention for normalizing twists
supplies global coordinates on 
(Buser 1992, Ch. 3).
