TITLE: Mapping Class Groups of Punctured Surfaces (and maybe Billiards)
QUESTION [14 upvotes]: 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?

REPLY [4 votes]: Here is an arcticle of Luo, 
A Presentation of the Mapping Class Groups
http://arxiv.org/PS_cache/math/pdf/9801/9801025v1.pdf
I remember seen somewhere a neat presentation of the hyperelliptic mapping class group, 
but I can't remeber now the reference, but you can check page 12 in the article 
Hyperelliptic Szpiro inequality 
http://arxiv.org/PS_cache/math/pdf/0106/0106212v1.pdf