Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

TL;DR

This paper explores the relationship between Ptolemy groupoids, shear coordinates, and the augmented Teichmüller space, providing a natural homomorphism and explicit descriptions of the mapping class group action.

Contribution

It introduces a homomorphism from the Ptolemy groupoid of a surface to that of a pinched surface, enhancing understanding of shear coordinates in the augmented Teichmüller space.

Findings

Constructed a natural homomorphism between Ptolemy groupoids

Provided an explicit description of the mapping class group action

Extended shear coordinates to the augmented Teichmüller 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.