Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces.

In this connection, I have read Chapter 11 of the book Primer of mapping class groups by Dan Margalit and Benson Farb.

So I have understood that the moduli space of a Riemann surface is the quotient of the Teichmuller space by the mapping class group, the action is properly discontinuous, the quotient is an orbifold, but it is not in general compact(Mumford's compactness criterion), it has "only one end", etc..

Other than these facts, does Teichmuller theory simplify the study of moduli spaces of Riemann surfaces in any way? Can we do something using Teichmuller theory which we can't do, say, using algebraic geometry? Are we able to prove theorems about moduli spaces, using Teichmuller theory methods? I would be grateful for any examples.

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What kind of things do you want to know about moduli space? –  Ryan Budney Jan 19 '10 at 21:32
I know of Moduli spaces as $\mathcal{M}_{g,n}$, the moduli of curves with n marked points,from algebraic geometry. So I am familiar with the algebraic geometry approach. I want to grasp the flavor and purpose of the Teichmuller theory approach, now that I have understood the first principles of the connection with Teichmuller space with and mapping class group. –  Anweshi Jan 19 '10 at 21:36
I misread your question at first. I've reposted my answer in a way that makes it clear I know how I'm misreading your question. :) –  Ryan Budney Jan 19 '10 at 21:41

One of the main "gains" of the Teichmuller theory approach is that you're dealing with a ball. So you're in a situation where you can readily make analytic arguments using fixed-point theory.

Thurston's homotopy-classification of elements in the mapping class group "reducible, (pseudo) anosov, or finite-order" is one example. His argument proceeds roughly along these lines (no real details included): the mapping class group acts on Teichmuller space tautologically. Thurston defined a compactification of Teichmuller space (the "projective measured lamination space") such that the action of the mapping class group extends naturally. In particular, the compactification is a compact ball. So given any element of the mapping class group, you can ask what kind of fixed points it has in this ball. Thurston's theorem is that the fixed point is in the interior if and only if the mapping is finite-order (in the mapping class group). There are exactly two fixed points on the boundary if and only if the mapping is (isotopic to) a pseudo-anosov -- moreover the action looks like a "translation" along an arc connecting the two fixed points. A necessary and sufficient condition to be reducible is that your automorphism of the projective measured lamination space has a single fixed point on the boundary.

The proof of geometrization for manifolds that fibre over the circle is of course closely related.

These techniques were used to show mapping class groups satisfy the Tits alternative (which linear groups satisfy) so it was one of the big chunks of "evidence" leading people to ask the question of whether or not mapping class groups are linear.

Another application would be the resolution of the Nielsen Realization problem: http://en.wikipedia.org/wiki/Nielsen_realization_problem

The list goes on. But these are really applications of Teichmuller space to other things -- specifically not Moduli space.

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Maybe this is not specific to moduli space. But I did not know about the things you write, and I learned something from your answer. Thanks. –  Anweshi Jan 19 '10 at 21:43
Could you expand on the first paragraph of your answer? The idea of using analytic arguments seems very interesting, but I can't imagine how one would go about it. What's an example of a result that's proved this way? –  Ilya Grigoriev Jan 20 '10 at 2:59
The first two big theorems of Thurston's I mentioned are some of the best if you're looking for analytic details. I included a bit more of a sketch, above. –  Ryan Budney Jan 20 '10 at 4:33
@Ilya. You can look into the "primer" I have linked above for getting some idea. –  Anweshi Jan 20 '10 at 12:18

I'll discuss things which are more applications of the mapping class group to moduli space rather than Teichmuller theory per se, but of course this is all tightly connected.

One of the big applications of this point of view is to the cohomology of moduli space. The moduli space of curves is not quite a classifying space for the mapping class group because the action of the mapping class group on Teichmuller space is not free, but the problem all comes from finite order elements. One can think of moduli space as a "rational classifying space" or an "orbifold classifying space" for the mapping class group. The upshot is that the group cohomology of the mapping class group is identical to the cohomology of moduli space with $\mathbb{Q}$ coefficients.

I will try to give a brief survey of this field, but it is huge and I will omit a lot of important work.

There is now a lot known about the group cohomology of the mapping class group. The most spectacular is the resolution by Madsen-Weiss of the Mumford conjecture giving the rational cohomology ring in a stable range. This is certainly not known via algebro-geometric methods.

This was proceeded by many older results. The most germane come from a series of papers by Harer in the '80's which (among other things) do the following:

1) Show that the cohomology stabilizes as the genus increases.

2) Calculate the Euler characteristic. This really is not a theorem about the mapping class group, as the proof uses a certain triangulation of moduli space rather than group theory. However, this triangulation definitely comes from Teichmuller theory rather than algebraic geometry, and it is still part of this same circle of ideas.

3) Make a number of low-dimensional calculations (up to degree 3 in published work and 4 in unpublished work).

The calculation of $H_2$ by Harer in particular is the key to calculating the Picard group of moduli space.

These low-dimensional cohomology calculations can now be (basically) done via algebraic geometry. See the paper "Calculating cohomology groups of moduli spaces of curves via algebraic geometry" by Arbarello and Cornalba. Thus the Picard group of moduli space can now be calculated via algebraic geometry.

A more recent application of this point of view comes from work of myself which calculates the Picard groups of the moduli spaces of curves with level structures (see my paper of the same title). I think it would be very interesting to try to make this same calculation using algebro-geometric methods, but I have no idea how to do so.

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I think a good example is Kerckhoff's solution to the Nielsen Realization Problem, which asks if every finite subgroup of the mapping class group is realized as a group of isometries of some hyperbolic surface. (The answer is yes.)

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Kerckhoff's solution is a major theorem, but (as you know, though maybe not everyone does) Teichmuller theory gives a simple proof of this fact for cyclic subgroups. It uses the contractibility and finite-dimensionality of Teichmuller space, so I think it's worth mentioning. Claim: if f is a self-homeomorphism of a surface so that f^n is homotopic to the identity, then f is homotopic to some g so that g^n = id and g is an automorphism for some Riemann surface structure. (There are other ways to prove this, of course.) Proof in the next comment; I'll assume n = p prime for simplicity. –  Tom Church Jan 20 '10 at 4:28
Proof: let F be the automorphism of Teichmuller space induced by f; we have F^p = id. Teichmuller space is contractible and finite-dimensional, so e.g. group cohomology tells us no finite group acts freely on it. Thus in particular F has a fixed point in Teichmuller space, which is some Riemann surface C. Tracing through the definitions, this means that F is represented by some periodic automorphism of C, as desired. –  Tom Church Jan 20 '10 at 4:29
Thanks tom for the above comments, they really show how you get a cool result by pulling on different branches of the same tree! –  Sean Tilson Jul 29 '10 at 20:22