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Geodesic length functions and Teichm\"uller spaces
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## Geodesic length functions

and Teichm\"uller spaces

### Feng Luo

**Abstract.**
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`

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