Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Julien Rogers (Rutgers)

Abtsract: Given a punctured surface S, its Ptolemy groupoid is a natural object associated to ideal triangulations on the surface. The action of the mapping class group on ideal triangulations extends to a natural homomorphism to this groupoid. Using hyperbolic geometry, in our context shear coordinates on Teichmuller space, this can be used to construct representations of the mapping class group in terms of rational functions. This was described first by R. Penner using the closely related lambda-length coordinates. In this talk we will describe how this construction behaves when pinching simple closed curves on S. This has combinatorial implications, with the construction of ideal triangulations on pinched surfaces and the effect on the Ptolemy groupoid, and geometrical, with a natural extension of shear coordinates to the augmented Teichmuller space. In both cases we explain how this applies to the action of the mapping class group. If time permits we will describe some possible applications to the study of quantum Teichmuller theory.