The outside of the Teichmuller space of punctured tori in Maskit's embedding

Abstract

We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformation. We also show that the discrete groups with elliptic elements in the complement of $\overline{\Cal M}$ are boundary groups of deformation spaces of certain Kleinian groups representing a punctured torus on their invariant component.