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john mangual
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Mapping Class Groups of Punctured Surfaces (and maybe Billiards)

Where can I find a concrete description of mapping class group of surfaces? I know the mapping class group of the torus is $SL(2, \mathbb{Z})$. Perhaps, there is a simple description for the sphere with punctures or the torus with punctures. Also, I would appreciate any literature reference for an arbitrary surface of genus g with n punctures.

Mapping class groups come up in my reading about billiards and the geodesic flow on flat surfaces. I wonder: the moduli space of complex structures on the torus is $\mathbb{H}/SL(2, \mathbb{Z})$, is it a coincidence the mapping class group appears here?

john mangual
  • 22.8k
  • 4
  • 63
  • 172