## 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?

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If your standard for concrete is $SL(2, \mathbb Z)$ you might not like the kinds of answers you get for other mapping class groups. :) Usually people come to understand mapping class groups by their actions on things -- Teichmuller space, the space of curves in the surface (curve complexes), etc. If all you care about is elements of the mapping class group up to conjugacy, a nice way to study them is by the surface bundles over a circle you can form using the mapping as monodromy. – Ryan Budney Dec 22 2009 at 3:13
OK, so when we talk about mapping class groups, there's usually an action going on. In my case, that's probably the action on the orbits of the billiard flow or on the moduli space of translation surfaces itself. Let me think about this a bit more... – John Mangual Dec 22 2009 at 22:19

1) Let me start by dealing with punctures and higher genus mapping class groups.

Aside from a few low-genus cases, there is no easy description of the mapping class group. As you said, the mapping class group of a torus is $SL_n(\mathbb{Z})$, and adding one puncture to a torus does not change its mapping class group (adding a boundary component, however, turns it into the 3-strand braid group).

In general (ie for $(g,n)$ not equal to the degenerate cases of $(1,1)$ or $(0,k)$ with $k$ at most $3$), you can relate the mapping class group $\Gamma(g,n)$ of a genus $g$ surface with $n \geq 1$ punctures to the mapping class group of a surface with fewer punctures via the Birman exact sequence. Two forms of it are:

$$1 \longrightarrow \pi_1(S_{g,n-1}) \longrightarrow \Gamma(g,n) \longrightarrow \Gamma(g,n-1) \longrightarrow 1$$

and

$$1 \longrightarrow B(g,n) \longrightarrow \Gamma(g,n) \longrightarrow \Gamma(g,0) \longrightarrow 1$$

Here $B(g,n)$ is the $n$-strand braid group on a genus $g$ surface. The map to the cokernel comes from "forgetting" punctures, and the kernel comes from "dragging" punctures around the surface.

A good reference for this material is the book "A Primer on mapping class groups" by Farb and Margalit, which is available here.

2) As far as moduli space goes, the moduli space of complex structures on a genus $g$ surface with $n$ punctures (as long as $g$ and $n$ are not too small) is isomorphic to the quotient of Teichmuller space by the mapping class group. It is thus no accident that the mapping class group appears in the description of moduli space. Again, Farb and Margalit's book is a nice source.

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The following may or may not be seen as a "concrete description." Let pi be the fundamental group of your punctured surface. (Warning: this is written up from memory so consult an authoritative source.) Then pi is a free group endowed with a set S of n distinguished conjugacy classes, namely the conjugacy classes of loops around punctures. Now the mapping class group acts on pi -- well, to be more precise, since a diffeomorphism of the surface might move the basepoint around, you have a homomorphism

mapping class group --> Out(pi)

But these aren't just any outer automorphisms of pi; they preserve this set S (though of course they may permute the individual elements of S.) Write Out^+_S(pi) for the the group of outer automorphisms of pi which preserve S, and whose action on homology has positive determinant. Then we have a map

mapping class group --> Out^+_S(pi)

and -- here comes the "concrete" description -- this homomorphism is an isomorphism.

When (g,n) = (1,0) this is just telling you that the mapping class group is SL_2(Z), as you say.

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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

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