## Institute for Mathematical Sciences

## Preprint ims91-2

** L. Keen and C. Series **
* Pleating Coordinates for the Maskit Embedding of the Teichmüller Space of Punctured Tori*

Abstract: The authors introduce a new set of global parameters, which they call pleating coordinates, for the Teichm\FCller space T_{1,1} of the punctured torus. The space T_{1,1} is embedded as a holomorphic family G_{μ} of Kleinian groups, where the complex parameter μ varies in a simply connected domain M in the complex plane. This is done in such a way that the regular set Ω(G_{μ}) has a unique invariant component Ω_{0}(G_{μ}) and the points in T_{1,1} are represented by the Riemann surface Ω(G_{μ})/G_{μ}. This embedding is in fact the Maskit embedding. The new coordinates are geometric in the sense that they are related to the geometry of the hyperbolic manifold H^{3}/G_{μ}. More precisely, they can be read off from the geometry of the punctured torus ∂C_{0}/G_{μ}, where ∂C_{0} is the component of the convex hull boundary facing Ω_{0}(G_{μ}). The surface ∂C0 has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination λ on ∂C_{0}/G_{μ}. There is some specific choice of transverse measure for the pleating lamination &lamba;, which allows the authors to introduce a notion of pleating length for G_{μ}. The laminations and their pleating lengths are the coordinates for M.

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