Geodesic length functions and Teichmüller spaces

Luo, Feng

doi:10.1090/S1079-6762-96-00008-X

Geodesic length functions and Teichmüller spaces

Author: Feng Luo
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 34-41
MSC (1991): Primary 32G15, 30F60
DOI: https://doi.org/10.1090/S1079-6762-96-00008-X
MathSciNet review: 1405967
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Abstract: Given a compact orientable surface $\Sigma$, let $\mathcal {S}(\Sigma )$ be the set of isotopy classes of essential simple closed curves in $\Sigma$. We determine a complete set of relations for a function from $\mathcal {S}(\Sigma )$ to $\mathbf {R}$ to be the geodesic length function of a hyperbolic metric with geodesic boundary on $\Sigma$. As a consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary on $\Sigma$ is reconstructed from an intrinsic combinatorial structure on $\mathcal {S}(\Sigma )$. This also gives a complete description of the image of Thurston’s embedding of the Teichmüller space.

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