Shift coordinates, stretch lines and polyhedral structures for Teichmüller space

Abstract

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space $\widetilde{{\cal T}}(S)$ that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with ${\frak b}$ boundary components and ${\frak p}$ cusps (which we call generalized pairs of pants), for all possible values of ${\frak b}$ and ${\frak p}$ satisfying ${\frak b}+{\frak p}=3$ . The parametrization of $\widetilde{{\cal T}}(S)$ by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in ${\Bbb {R}}^3$ . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with ${\frak b}$ boundary components and ${\frak p}$ cusps, for fixed ${\frak b}$ and ${\frak p}$ , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on $\widetilde{{\cal T}}(S)$ whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space $\widetilde{{\cal T}}(S)$ . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2.