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Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential simple closed curves in $\Sigma$. We determine a complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold R$ to be the geodesic length function of a hyper bolic metric with geodesic boundary on $\Sigma$. As a consequence, the Teichm\"uller space of hyperbolic metrics with geodesic boundary on $\Sigma$ is reconstructed from an intrinsic combinatorical structure on $\Cal S(\Sigma)$. This also gives a complete description of the image of Thurston's embedding of the Teichm\"uller space.

Copyright American Mathematical Society 1996

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Article Info

• ERA Amer. Math. Soc. 02 (1996), pp. 34-41
• Publisher Identifier: S 1079-6762(96)00008-6
• 1991 Mathematics Subject Classification. Primary 32G15, 30F60.
• Received by the editors April 9, 1996
• Communicated by Walter Neumann
• Comments (When Available)

Feng Luo

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903

E-mail address: fluo@math.rutgers.edu