Pleating coordinates for the Teichm\"{u}ller space of a punctured torus

Abstract

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