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 T1,1 of the punctured torus. The space T1,1 is embedded as a holomorphic family Gμ 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(Gμ) has a unique invariant component Ω0(Gμ) and the points in T1,1 are represented by the Riemann surface Ω(Gμ)/Gμ. 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 H3/Gμ. More precisely, they can be read off from the geometry of the punctured torus ∂C0/Gμ, where ∂C0 is the component of the convex hull boundary facing Ω0(Gμ). The surface ∂C0 has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination λ on ∂C0/Gμ. 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 Gμ. The laminations and their pleating lengths are the coordinates for M.
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