Pleating coordinates for the Teichmüller space of a punctured torus

Abstract

We construct new coordinates for the Teichmüller space Teich of a punctured torus into $\mathbf{R}\times {\mathbf{R}}^{+}$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_{\mu }$ that depend holomorphically on a parameter $\mu$ varying in a simply connected domain in $\mathbf{C}$. They describe the geometry of the hyperbolic manifold ${\mathbf{H}}^3/G_{\mu }$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_{\mu }$; and they are directly computable from the generators of $G_{\mu }$.

6 pages