Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Abstract

We start by describing how ideal triangulations on a surface degenerate under pinching of a multicurve. We use this process to construct a homomorphism from the Ptolemy groupoid of a surface to that of a pinched surface which is natural with respect to the action of the mapping class group. We then apply this construction to the study of shear coordinates and their extension to the augmented Teichm\"uller space. In particular, we give an explicit description of the action of the mapping class group on the augmented Teichm\"uller space in terms of shear coordinates.