Pleating coordinates for the Teichmüller space of a punctured torus

We construct new coordinates for the Teichmüller space Teich of a punctured torus into ${\text{R}} \times {{\text{R}}^ + }$. 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 C. They describe the geometry of the hyperbolic manifold ${{\text{H}}^3}{\text{/}}{G_\mu }$; 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 ${G_\mu }$.