**1. Fenchel-Nielsen Coordinates for Teichmuller Space **

Here we discuss a very nice set of coordinates for Teichmuller space. The basic idea is that we cut the surface up into small pieces (pairs of pants); hyperbolic structures on these pieces are easy to parameterize, and we also understand the ways we can put these pieces together.

In order to define these coordinates, we first cut the surface up. A *pair of pants* is a thrice-punctured sphere.

Another way to specify it is that it is a genus surface with euler characteristic and three boundary components. We can cut any surface up into pairs of pants with simple closed curves. To see this, we can just exhibit a general cutting: slice with “vertical” simple closed curves.

This is not the only way to cut a surface into pairs of pants. For example, with the once-punctured torus any pair of coprime integers gives us a curve which cuts the surface into a pair of pants. We are going to show that a point in Teichmuller space is determined by the lengths of the curves, plus other coordinates, which record the “twisting” of each gluing curve.

Now, given a choice of disjoint simple closed surves , we associate to the family of geodesics in in the homotopy classes of the . In each class, there is a unique geodesic, but how do we know the geodesics in are pairwise disjoint?

Lemma 1Suppose is a family of pairwise disjoint simple closed curves in a hyperbolic surface , and are the (unique) geodesic representatives in the homotopy classes of the .

- The geodesics in are pairwise disjoint simple closed curves.
- As a family, the are ambient isotopic to .

*Proof:* Consider a loop and its geodesic representative . Suppose that intersects itself. Now and cobound an annulus, which lifts to the universal cover: in the universal cover we must find the lift of the intersection as an intersection between two lifts and . Because the annulus bounding and lifts to the universal cover, there are two lifts and of which are uniformly close to and . We therefore find that and intersect, which means that intersects itself, which is a contradiction. The same idea shows that the geodesic representatives are pairwise disjoint.

To see that they are ambient isotopic as a family, it is easiest to lift the picture to the universal cover. At that point, we just need to “wiggle” everything a little to match up the lifts of the and .

With the lemma, we see that to a point in Teichmuller space we get pairwise disjoint simple closed geodesics, which gives us positive coordinates, namely, the lengths of these curves. We might wonder: what triples of points can arise as the lengths of the boundary curves in hyperbolic pairs of pants? It turns out that:

Lemma 2There exists a unique hyperbolic pair of pants with cuff lengths , for any . Cuff lengths here refers to the lengths of the three boundary components.

*Proof:* We will now prove the lemma, which involves a little discussion. Suppose we are given a hyperbolic pair of pants. We can double it to obtain a genus two surface:

The curves are shown in red, and representatives of the other isotopy class fixed by the involution are in blue.

There is an involution (rotation around a skewer stuck through the surface horizontally) which fixes the (glued up) boundaries of the pairs of pants. This involution also fixes the isotopy classes of three other disjoint simple closed curves, and there is a unique geodesic in these isotopy classes. Since the are fixed by the involution, they must intersect the at right angles. If we cut along the to get (two copies of) our original pair of pants, we have found that there is a unique triple of geodesics which meet the boundaries at right angles:

Cutting along the , we get two hyperbolic hexagons:

We will prove in a moment that there is a unique hyperbolic right-angled hexagon with three alternating edge lengths specified. In particular, there is a unique hyperbolic right-angled hexagon with alternating edge lengths . Since there is a unique way to glue up the hexagons to obtain our original pair of pants, there is a unique hyperbolic pair of pants with specified edge lengths.

Lemma 3There is a unique hyperbolic right-angled hexagon with alternating edge lengths .

*Proof:* Pick some geodesic and some point on it. We will show the hexagon is now determined, and since we can map a point on a geodesic to any other point on a geodesic, the hexagon will be unique up to isometry. Draw a geodesic segment of length at right angles from . Call the other end of this segment . There is a unique geodesic passing through at right angles to the segment. Pick some point on at length from (we will be varying ). From there is a unique geodesic segment of length at right angles to ; call its endpoint . There is a unique geodesic through at right angles to this segment. Now, there is a unique geodesic segment at right angles to and . Of course, the length of this segment depends on .

If we make large, then becomes large, and there is some positive such that goes to . Therefore, there is a unique length making . We have now determined the hexagon, and, up to isometry, all of our choices were forced, so there is only one.

Since there is a unique hyperbolic pair of pants with specified cuff lengths, when we cut our surface of interest up into pairs of pants, we get a map which takes a point to the lengths of the curves cutting into pairs of pants. This map is not injective: the fiber over a point is all the ways to glue together the pairs of pants.

The issue is that when we want to glue two curves together, we have to decide whether to twist them at all before gluing. Up to isometry, there are ways to glue these curves together (all the angles). However, in (marked) Teichmuller space, there are ways to glue it up. Draw another curve (this is not the same as the before). The marking on lets us observe what happens to under , and we can see that twisting the pairs of pants around results in nontrivial movement in Teichmuller space.

The twist above results in the following new curve:

The length of determines how twisted the gluing is, since twisting requires increasing its length. That is, given the image of , there is a unique way to untwist it to get a minimum length. This tells us how twisted the original gluing was.

To understand the twisting around all the curves in , we must pick another curves; one simple way is to declare that looks like the above pictures if we are gluing two distinct pairs of pants, and like this:

if we are gluing a pair of pants to itself. This construction gives us a global homeomorphism

Here is an example of a choice of and curves. The curves get a little messy in the middle: try to fit the pictures above into the context of the one below to see that they are correct.

** 1.1. A Symplectic Form on Moduli Space **

The length and twist coordinates and are not well-defined on Moduli space, but their derivatives *are*: define the 2 form on Teichmuller space

It is a theorem of Wolpert that this 2-form is independent of the choice of coordinates, so it descends to a 2-form on Moduli space. It is very usful that Modi space is symplectic.

Hello, from Toronto, Canada

Just a quick hello from as I’m new to the board. I’ve seen some interesting comments so far.

To be honest I’m new to forums and computers in general :)

Mike

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