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On Non-separating Simple Closed Curves in a Compact Surface
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## On non-separating simple closed curves in a compact surface

### Feng Luo

**Abstract.**
We introduce a semi-algebraic structure on the set $\Cal S$ of all
isotopy classes of non-separating simple closed curves in any
compact oriented surface and show that the structure is finitely
generated. As a consequence, we produce a natural finite dimensional
linear representation of the mapping class group of the surface.
Applications to the Teichmüller space, Thurston's measured
lamination space, the harmonic Beltrami differentials, and the first
cohomology group of the surface are discussed.

*Copyright American Mathematical Society 1995*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 18-25
- Publisher Identifier: S 1079-6762(95)01003-8
- 1991
*Mathematics Subject Classification*. 57.
*Key words and phrases*. Simple closed curve, surface
- Received by the editors April 22, 1995, and, in revised form, March 22, 1995
- Communicated by Walter Neumann
- Comments

**Feng Luo**

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

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

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