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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