Coordinates on Teichmuller space

Question

We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we have the length functions of the geodesics boundaries and the twisting parameters for gluing these boundaries.

My question is the following:

We choose again a pair of pants decomposition. Then there arises the trivalent graph representing this decomposition.

We could associate a fixed a pair of pants to each of vertices, say of boundary geodesic length (1,1,1). And then glue flat cylinders of both boundary length 1 and height $h$ to form a Riemann surface(of course we will also need the twisting parameter for the sewing, but it seems that we should only need the relative twisting parameter as cylinder itself has a rotation symmetry.).

The height of these cylinders and the relative twisting parameter together form 6g-6 parameters. Will they also be a parametrization of Teichmuller space?

Answer 1

You are describing the "grafting construction". I am not an expert: however if you google "grafting a Riemann surface" there are many references available.

If you take $h$ to be non-negative you cannot reach all of Teichmuller space in the way you describe. This is because the cuffs of your pair of pants decomposition will always have annular neighborhoods of definite modulus (ie of definite width). There are Riemann surfaces where the cuffs only have very thin annular neighborhoods.