# What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:

1. No-wandering-domain Theorem (Sullivan)
2. Theorems of Thurston (Classification of homeomorphism of surfaces, topological characterization of rational maps, hyperbolization theorems for special 3 manifolds)
3. Computer graphics. (Using the various metrics on the Teichmüller spaces as a substitute for Gromov-Hausdorff metric.)
4. String Theory (as elementary particles are modeled by loops, they generate a Riemann surface as they move through time.)
5. Some applications to biology.(Brain morphometry)

Note: I am very sure that this is only a small fraction of what is out there, and I plan to continue to update this list.

-
Is it possible to clarify what is understood under "Teichmuller theory" here ? I aksed a separate question about it: mathoverflow.net/questions/114787/… –  Alexander Chervov Nov 28 '12 at 18:05
There were a bunch of comments which touched on the personal politics of Teichmüller that in my judgment were off-topic, which I have now removed. –  Todd Trimble Dec 9 '13 at 19:00

Solution of the Nielsen's realization problem.

-

Although applications to Thurston's work on 3-manifolds and Bers' embedding has already been mentioned, I thought it was worth mentioning Bers' simultaneous uniformization theorem (underlying Bers' embedding), which implies that for any two points in Teichmuller space, there is a unique (marked) quasifuchsian surface, such that the two conformal structures on the domain of discontinuity realize both Riemann surfaces simultaneously. This result was generalized by Ahlfors, Bers, and Sullivan, who showed more generally that the space of geometrically finite Kleinian groups are parameterized by the Teichmuller spaces of the conformal structures on the domain of discontinuity. This is the starting point for the classification of Kleinian groups (and Thurston's work on the geometrization of Haken 3-manifolds).

Related to this is the parameterization of a complex projective structure on a surface by a conformal structure and a holomorphic quadratic differential, so that the projective structures are in bijection with the cotangent space to Teichmuller space.

A related result of Bers gave Thurston's classification of mapping classes of surfaces into pseudo-Anosov, reducible, or finite order, by minimizing the translation length of a mapping class in Teichmuller space.

Teichmuller space has also been used as an ingredient in the construction of various (projective) representations of mapping class groups, notably by Andersen based on the Hitchin connection (realizing ideas of Witten).

In another direction, Labourie and Loftin discovered that convex projective structures on a surface are parameterized by a conformal structure together with a cubic differential, so may be regarded as a bundle over Teichmuller space.

-

From the perspective of algebraic geometry, Teichmüller theory is an analytic approach to moduli spaces of curves. To keep things simple, let $g\geqq 2$ be an integer and consider the moduli space $\mathscr{M}_g$ of smooth and complete algebraic curves of genus $g$. This is a fairly complicated object: a Deligne-Mumford stack over the integers, nothing that is easy to describe in any way. The associated complex analytic "space" $\mathscr{M}_g(\mathbf{C})$ is still something with a weird structure: a complex orbifold which is not a manifold. But its universal covering space, which can be identified with the Teichmüller space $\mathscr{T}_g$, is a true complex manifold. It has several nice properties as compared with $\mathscr{M}_g$:

• Teichmüller space is biholomorphic to an open domain in $\mathbf{C}^{3g-3}$. This is Bers' embedding theorem.
• Teichmüller space is diffeomorphic (forgetting the complex structure) to $\mathbf{R}^{6g-6}$, and there is a very intuitive system of coordinates, called Fenchel-Nielsen coordinates, realising such a diffeomorphism. On the other hand, even when you forget the stack structure, $\mathscr{M}_g$ is a variety of general type for $g\geqq 23$, which means that you can only embed it in projective space $\mathbf{P}^d$ where $d$ is "much" larger than the dimension of $\mathscr{M}_g$, and you need "many" equations to cut out its image. So there is no "economical" algebraic coordinate system on $\mathscr{M}_g$ in general.
• Complex geodesics in $\mathscr{T}_g$ for a natural metric, the Teichmüller metric, give families of algebraic curves which have a nice and intuitive geometric description, called Teichmüller disks. In quite a few cases they descend to algebraically defined curves in $\mathscr{M}_g$ which are consequently called Teichmüller curves. One can say much more about their geometric and number-theoretical properties than for general curves in $\mathscr{M}_g$. They form an active area of research in these years.

Another application of Teichmüller theory to moduli spaces of curves is that it gives rise to an isomorphism between the mapping class group $\Gamma_g$ of a closed oriented surface of genus $g$ and the (orbifold) fundamental group of the moduli space $\mathscr{M}_g(\mathbf{C})$, so it provides a link between the topology of moduli spaces and the topology of surfaces.

-
In the last claim doesn't "fundamental group" need to be replaced by "orbifold fundamental group" (at least for $g = 1$)? –  Qiaochu Yuan Nov 27 '12 at 18:31
@Qiaochu: I guess you meant $g>1$? –  ThiKu Nov 28 '12 at 1:12
I've been told that $\mathcal{M}_g^{an}$ is a "more correct" notation than $\mathcal{M}_g(\mathbf{C})$, since the latter is apparently just a set. –  S. Carnahan Nov 28 '12 at 9:02
@ Qiaochu: yes. –  Robert Kucharczyk Nov 28 '12 at 16:18
In fact the "naive" fundamental group of $\mathscr{M}_g(\mathbb{C})$ is trivial. –  Robert Kucharczyk Nov 28 '12 at 16:22

The proof of the Mumford conjecture by Madsen and Weiss made essential use of Teichm\"uller theory. This becomes especially clear if you state the conjecture as being about the cohomology of the space $\mathfrak{M}_g$, Riemanns moduli space of genus $g$ complex curves. Defining this space does not require Teichmueller theory, it can be done in a purely algebro-geometric way.

The first step is that $H_{\ast} (\mathfrak{M}_g;\mathbb{Q})$ is the same as the rational homology of the mapping class group $\Gamma_g$. This uses Teichm\"ullers theorem that Teichm\"ullers space $\mathcal{T}_g$ is homeomorphic to a euclidean space (being a contractible manifold would be enough).

The second step is that $B \Gamma_g$ is homotopy equivalent to $B Diff (\Sigma_g)$, the classifying space of the diffeomorphism group. This is a result by Earle and Eells, which uses Teichm\"ullers theorem as well, albeit not so essentially, because there is a purely topological proof of this result as well.

The Madsen-Weiss theorem then computes the homology of $B Diff (\Sigma_g)$ in a range of degrees; this is differential topology/homotopy theory and not related to Teichm\"uller theory.

Older results on the homology of $\mathfrak{M}_g$ (or the Deligne-Mumford compactification) are very often based on step 1 as well. Some relevant names are Harer, Harer-Zagier, Arbarello-Cornalba- and others.

-

The dynamics of the Teichmuller geodesic flow in the moduli spaces of Abelian differentials on Riemann surfaces provides a natural "renormalization dynamics" for simpler dynamical systems of parabolic type such as interval exchange transformations and billiards in rational polygons. A very nice survey explaining this fascinating interplay between Teichmuller theory and Dynamical Systems is "Flat surfaces" by A. Zorich (see this link here http://arxiv.org/abs/math/0609392).

More concretely, this "renormalization dynamics" approach was recently successfully applied by V. Delecroix, P. Hubert and S. Lelievre (see here http://arxiv.org/abs/1107.1810) to explain a conjecture of the physicists Hardy and Weber about the abnormal "rate of diffusion" of typical trajectories in typical realizations of the so-called Ehrenfest wind-tree model of Lorenz gases. Also, the ideas coming from the study of the Teichmuller geodesic flow were a source of inspiration to A. Kappes and M. Moller in their classification of all presently known ball quotients (see here http://arxiv.org/abs/1207.5433).

-

Some important applications of Teichmuller theory are to 3-D topology (I mean the work of Thurston, and later development of this work), and to holomorphic dynamics. On this I refer on the books by Thurston and by Hubbard, and on the paper of Douady and Hubbard A proof of Thurston's topological characterization of rational functions, MR1251582, but this was already mentioned in the original question. Actually, since the work of Sullivan, Douady and Hubbard, Teichmuller theory is one of the main tools in holomorphic dynamics.

There are also applications to string theory, and I expect that some string theorist will write about them. But here is a nice example, where string theorists made a pure mathematical conjecture, and it was proved using Teichmuller theory: MR0882831 Zograf, P. G.; Takhtajan, L. A. On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0.

-