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.
Best Answer
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.