Teichmuller theory and moduli of Riemann surfaces

Question

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.

Answer 1

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/disc. 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). You can think of this part as an elaboration of the theorem that isometry groups of hyperbolic manifolds are finite. There are exactly two fixed points on the boundary (and the automorphism acts as a translation along a line connecting the two points) if and only if the mapping is (isotopic to) a pseudo-anosov. A necessary and sufficient condition to be reducible is that your automorphism of the projective measured lamination space is not of the other two types, i.e. it could have one fixed point on the boundary or any number, so long as it is not precisely two acting as a translation from one to the other.

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.

Answer 2

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.

Answer 3

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.)