<![CDATA[Hyperbolic Geometry Notes #4 – Fenchel-Nielsen Coordinates]]>

<![CDATA[Hyperbolic Geometry Notes #4 – Fenchel-Nielsen Coordinates]]>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 ${0}$ surface with euler characteristic ${-1}$ 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 ${3g-3}$ “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 ${3g-3}$ curves, plus ${3g-3}$ other coordinates, which record the “twisting” of each gluing curve.

Now, given a choice of ${3g-3}$ disjoint simple closed surves ${\{\alpha_i\}}$ , we associate to ${(f, \Sigma) \in \mathrm{Teich}(S)}$ the family of geodesics in ${\Sigma}$ in the homotopy classes of the ${f(\alpha_i)}$ . In each class, there is a unique geodesic, but how do we know the geodesics in ${\{f(\alpha_i)\}}$ are pairwise disjoint?

Lemma 1 Suppose ${\{\alpha_i\}}$ is a family of pairwise disjoint simple closed curves in a hyperbolic surface ${\Sigma}$ , and ${\{\gamma_i\}}$ are the (unique) geodesic representatives in the homotopy classes of the ${\alpha_i}$ .

The geodesics in ${\{\gamma_i\}}$ are pairwise disjoint simple closed curves.

As a family, the ${\{\gamma_i\}}$ are ambient isotopic to ${\{\alpha_i\}}$ .

Proof: Consider a loop ${\alpha}$ and its geodesic representative ${\gamma}$ . Suppose that ${\gamma}$ intersects itself. Now ${\alpha}$ and ${\gamma}$ 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 ${\tilde{\gamma}}$ and ${\tilde{\gamma}'}$ . Because the annulus bounding ${\alpha}$ and ${\gamma}$ lifts to the universal cover, there are two lifts ${\tilde{\alpha}}$ and ${\tilde{\alpha}'}$ of ${\alpha}$ which are uniformly close to ${\tilde{\gamma}}$ and ${\tilde{\gamma}'}$ . We therefore find that ${\tilde{\alpha}}$ and ${\tilde{\alpha}'}$ intersect, which means that ${\alpha}$ intersects itself, which is a contradiction. The same idea shows that the geodesic representatives ${\gamma_i}$ 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 ${\alpha_i}$ and ${\gamma_i}$ . $\Box$

With the lemma, we see that to a point in Teichmuller space we get ${3g-3}$ pairwise disjoint simple closed geodesics, which gives us ${3g-3}$ 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 2 There exists a unique hyperbolic pair of pants with cuff lengths ${(l_1, l_2, l_3)}$ , for any ${l_1, l_2, l_3 > 0}$ . 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 ${\alpha}$ 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 ${\beta_i}$ in these isotopy classes. Since the ${\beta_i}$ are fixed by the involution, they must intersect the ${\alpha_i}$ at right angles. If we cut along the ${\alpha_i}$ to get (two copies of) our original pair of pants, we have found that there is a unique triple of geodesics ${\beta_i}$ which meet the boundaries at right angles:

Cutting along the ${\beta_i}$ , 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 ${(l_1/2, l_2/2, l_3/2)}$ . Since there is a unique way to glue up the hexagons to obtain our original ${(l_1, l_2, l_3)}$ pair of pants, there is a unique hyperbolic pair of pants with specified edge lengths. $\Box$

Lemma 3 There is a unique hyperbolic right-angled hexagon with alternating edge lengths ${(l_1, l_2, l_3)}$ .

Proof: Pick some geodesic ${g_1}$ and some point ${x_1}$ 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 ${l_1}$ at right angles from ${x_1}$ . Call the other end of this segment ${x_2}$ . There is a unique geodesic ${g_2}$ passing through ${x_2}$ at right angles to the segment. Pick some point ${x_3}$ on ${g_2}$ at length ${y}$ from ${x_2}$ (we will be varying ${y}$ ). From ${x_3}$ there is a unique geodesic segment of length ${l_2}$ at right angles to ${g_2}$ ; call its endpoint ${x_4}$ . There is a unique geodesic ${g_3}$ through ${x_4}$ at right angles to this segment. Now, there is a unique geodesic segment at right angles to ${g_1}$ and ${g_3}$ . Of course, the length ${z}$ of this segment depends on ${y}$ .

If we make ${y}$ large, then ${z}$ becomes large, and there is some positive ${y}$ such that ${z}$ goes to ${0}$ . Therefore, there is a unique length ${y}$ making ${z = l_3}$ . We have now determined the hexagon, and, up to isometry, all of our choices were forced, so there is only one. $\Box$

Since there is a unique hyperbolic pair of pants with specified cuff lengths, when we cut our surface of interest ${S}$ up into pairs of pants, we get a map ${\mathrm{Teich}(S) \rightarrow (\mathbb{R}^+)^{3g-3}}$ which takes a point ${(f, \Sigma)}$ to the ${3g-3}$ lengths of the curves cutting ${S}$ 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 ${\alpha}$ curves together, we have to decide whether to twist them at all before gluing. Up to isometry, there are ${\mathbb{R}/\mathbb{Z}}$ ways to glue these curves together (all the angles). However, in (marked) Teichmuller space, there are ${\mathbb{R}}$ ways to glue it up. Draw another curve ${\beta}$ (this ${\beta}$ is not the same as the ${\beta_i}$ before). The marking on ${S}$ lets us observe what happens to ${\beta}$ under ${f}$ , and we can see that twisting the pairs of pants around ${\alpha}$ results in nontrivial movement in Teichmuller space.

The twist above results in the following new ${\beta}$ curve:

The length of ${\beta}$ determines how twisted the gluing is, since twisting requires increasing its length. That is, given the image of ${\beta}$ , 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 ${3g-3}$ curves in ${S}$ , we must pick another ${3g-3}$ curves; one simple way is to declare that ${\beta}$ 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

$\displaystyle \mathrm{Teich}(S) \rightarrow (\mathbb{R}^+)^{3g-3} \times \mathbb{R}^{3g-3} \cong \mathbb{R}^{6g-6}$

Here is an example of a choice of ${\alpha}$ and ${\beta}$ curves. The ${\beta}$ 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 ${l_i}$ and ${t_i}$ are not well-defined on Moduli space, but their derivatives are: define the 2 form on Teichmuller space

$\displaystyle \omega = \sum_i dl_i \wedge dt_i$

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.

]]>