The decorated Teichmuller space of a disk with n punctures on the boundary and one in the interior is the the space of hyperbolic metrics on such a surface with an extra marking of an horocycle at each puncture. It admits coordinates, which are called lambda-lengths, and satisfy a nice cluster algebra structure ($D_n$).
To go down to the Teichmuller space of the punctured disk, we have to remove the extra structure (the horocycles). There is a well defined procedure to forget about the extra marking of horocycles: if one scale one of these horocycles, the lambda-lengths scale in a neat way, and therefore one can define a projection of the decorated Teichmuller space to the un-decorated one.
Now, an horocycle around the interior puncture is (more or less) a circle around it. Suppose we forget all of the other horocycles, therefore projecting to a partially decorated Teichmuller space. Does this space relates to the Teichmuller space of the annulus with punctures on the boundary?
