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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Ten

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Teichmüller Space

This chapter introduces another main player in our story: the Teichmüller space Teich(S) of a surface S. For g ≥ 2, the space Teich(Sg) parameterizes all hyperbolic structures on Sg up to isotopy. After defining a topology on Teich(S), we give a few heuristic arguments for computing its dimension. The length and twist parameters of Fenchel and Nielsen are then introduced in order to prove that Teich(Sg) is homeomorphic to R6g−6. At the end of the chapter, we prove the 9g − 9 theorem, which tells us that a hyperbolic ...

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