I am new to mapping class groups for surfaces and representation theory.

I would like to know why people care about representations of mapping class groups. I think in general representation theory reduces a problem into a problem in linear algebra so that we can attack the problem.

But I don't know how this works. I want to know:

When we talk about a mapping class group representation (of any kind), what do we expect to get? What are people trying to do with it? Is there any kind of representation that people still look for? What are the good results on mapping class groups obtained using a representation?


1 Answer 1


Well, think about what the mapping class group is: we can view it as a group of diffeomorphisms of a surface where we identify isotopic ones. But two isotopic diffeomorphisms induce the same action on the fundamental group of the surface (I will completely ignore basepoints here; all my surfaces are closed, connected and oriented and everything I write preserves orientation).

Therefore, one of the fundamental ways to understand the mapping class group is to understand what it does to isotopy classes of curves on your surface. In fact, the essential closed curves inside surfaces determine many geometric properties of the surface. And the mapping class group is the thing that moves isotopy classes of these curves around.

Now look: $\pi_1(S)$ is generally a non-abelian group, but it has an abelianization isomorphic to $H_1(S,\mathbb{Z})$. And since the MCG acts on $\pi_1$, it acts on homology; by tensoring with $\mathbb{R}$, we get a linear action of the mapping class group on a real vector space $V=H_1(S,\mathbb{R})$ preserving a lattice. That is, we get a representation of the MCG on a $V$ with has additional structure preserved by the action of the MCG, and this representation has a lot to say about the surface.

For example, via this representation one can prove that the mapping class group has a torsion free subgroup of finite index and already gain information about the group and the surface. This is a highly non-trivial result and the symplectic representation on homology plays a fundamental role.

It turns out however that a big portion of the MCG is not seen by its representation on homology. The missing part, called the Torelli subgroup, is the subgroup that acts trivially on homology, and it is complicated. Think of the linear action on comology as rearranging cells, and the remaining non-linear part as involved in the knotting and twisting of curves in the surface.

There is a vast amount of things to be said about the linear and non-linear actions of the mapping class group. After a reasonable modification, it becomes isomorphic to the outer automorphism group of the fundamental group (this is a huge result). This provides an algebraic group structure on MCG, and the representation theory of algebraic groups is rich and elegant.

I can go on forever, but I prefer to give you an excellent reference: A primer on mapping class groups, by Farb and Margalit. It develops in an accessible way everything I wrote and goes far beyond. If you are familiar with the basics of the MCG, you can in fact jump directly to chapter 6 which is the most relevant part to your question. I did not talk at all about the external properties of representations of the MCG or how they inform the Teichmuller theory of the surface, but I hope Farb and Margalit's book will provide enough incentive for you to look into these things.


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