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