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

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

1996-06-12

arxiv.org/abs/math/9606216v1

Mathematics

Geometric Topology

Scientific paper

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 Teichm\"uller 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 M\"obius 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.

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