<![CDATA[Hyperbolic Geometry Notes #3 – Teichmuller and Moduli Space]]>

<![CDATA[Hyperbolic Geometry Notes #3 – Teichmuller and Moduli Space]]>This post introduces Teichmuller and Moduli space. The upcoming posts will talk about Fenchel-Nielsen coordinates for Teichmuller space; it’s split up because I figured this was a relatively nice break point. Hopefully, I will later add some pictures to this post.

1. Uniformization

This section starts to talk about Teichmuller space and related stuff. First, we recall the uniformization theorem:

If ${S}$ is a closed surface (Riemannian manifold), then there is a unique* metric of constant curvature in its conformal class. The asterisk * refers to the fact that the metric is unique if we require that it has curvature ${\pm 1}$ . If ${\chi(S)=0}$ , then the metric has curvature zero and it is unique up to euclidean similarities.

2. Teichmuller and Moduli Space of the Torus

Let us see what we can conclude about flat metrics on the torus. We would like to classify them in some way. Choose two straight curves ${\alpha}$ and ${\beta}$ on the torus intersecting once (a longitude and a meridian) and cut along these curves. We obtain a parallelogram which can be glued up along its edges to retrieve the original torus. This parallelogram lives/embeds in ${\mathbb{C}^2}$ , and, by composing the embedding with euclidean similarities, we may assume that the bottom left corner is at ${0}$ and the bottom right is ${1}$ . The parallelogram is therefore determined by where the upper left hand corner is: some complex number ${z}$ with ${\mathrm{Im}(z) > 0}$ . Notice that this is the upper half-plane, which we can think of as hyperbolic space. Therefore, there is a bijection:

{ Torii with two chosen loops up to euclidean similarity } ${\leftrightarrow}$ { ${z \in \mathbb{C} \, | \, \mathrm{Im}(z) > 0}$ }

This set is called the Teichmuller space of the torus. We don’t really care about the loops ${\alpha}$ and ${\beta}$ , so we’d like to find a group which takes one choice of loops to another and acts transitively. The quotient of this will be the set of flat metrics on the torus up to isometry, which is known as Moduli space.

We are interested in the mapping class group of the torus, which is defined to be

$\displaystyle \mathrm{MCG}(T^2) = \mathrm{Homeo}^+(T^2) / \mathrm{Homeo}_\circ(T^2)$

Where ${\mathrm{Homeo}_\circ(T^2)}$ denotes the connected component of the identity. That is, the mapping class group is the group of homeomorphisms (homotopy equivalences), up to isotopy (homotopy). The reason for the parentheses is that for surfaces, we may replace homeomorphism and isotopy by homotopy equivalence and homotopy, and we will get the same group (these catagories are equivalent for surfaces).

To find ${\mathrm{MCG}(T^2)}$ , think of the torus as the unit square in ${\mathbb{R}^2}$ spanned by the standard unit basis vectors. Then a homeomorphism of ${T^2}$ must send the integer lattice to itself, so the standard basis must go to a basis for this lattice, and the transformation must preserve the area of the torus. Up to isotopy, this is just linear maps of determinant ${1}$ (not ${-1}$ because we want orientation-preserving) preserving the integer lattice, which we care about up to scale, otherwise known as ${\mathrm{PSL}(2,\mathbb{Z})}$ .

Using the bijection above, the mapping class group of the torus acts on ${\{ z \in \mathbb{C} \, | \, \mathrm{Im}(z) > 0 \}}$ , and this action is

$\displaystyle \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] z = \frac{az + b}{cz+d}$

This action is probably familiar to you from complex analysis.

In summary, the Teichmuller space of the torus is (can be represented as) ${\{ z \in \mathbb{C} \, | \, \mathrm{Im}(z) > 0 \}}$ , and the mapping class group ${\mathrm{PSL}(2,\mathbb{Z})}$ acts on this space, and the quotient of this action is the set of flat metrics up to isometry, which is Moduli space. What is the quotient? A fundamental region for the action is the set

$\displaystyle \{ z\in\mathbb{C} \,\, |\,\, |\mathrm{Re}(z)| \le \frac{1}{2}, \, |z| \ge 1\}$

Which is glued to itself by a flip in the ${y}$ axis. The resulting Moduli space is an orbifold: one point is ideal and goes off to infinity, one point looks locally like ${\mathbb{R}^2}$ quotiented by a rotation of ${\frac{2\pi}{3}}$ , and the other point looks like ${\mathbb{R}^2}$ quotiented by a rotation of ${\pi}$ .

3. Teichmuller Space and Moduli Space for Negatively Curved Surfaces

Now we will go through a similar process for closed, boundaryless, oriented surfaces of negative Euler characteristic. It is possible to do this for surfaces with boundary, etc, but for simplicity, we will stick to multi-holed torii (this what closed, boundaryless, oriented surfaces of negative Euler characteristic are) for now.

We start with a topological surface ${S}$ . Topological meaning we do not associate with it a metric. We want to classify the hyperbolic metrics we could give to ${S}$ . Define Teichmuller space ${\mathrm{Teich}(S)}$ to be the set of equivalence classes of pairs ${(f, \Sigma)}$ where ${\sigma}$ is a hyperbolic surface and ${f: S \rightarrow \Sigma}$ is a homotopy equivalence. As mentioned earlier, anywhere “homotopy equivalence” appears here, you may replace it with “homeomorphism” as long as you replace “homotopy” with “isotopy.” The equivalence relation on pairs is the following: ${(f, \Sigma_1) \sim (g, \Sigma_2)}$ iff there exists an isometry ${i: \Sigma_1 \rightarrow \Sigma_2}$ such that ${i \circ f}$ is homotopic to ${g}$ .

Define the Moduli space ${\mathcal{M}(S)}$ of ${S}$ to be isometry classes of surfaces ${\Sigma}$ which are homotopy equivalent to ${S}$ . There is an obvious map ${\mathrm{Teich}(S) \rightarrow \mathcal{M}(S)}$ defined by mapping ${(f, \Sigma) \mapsto \Sigma}$ , and this map respects the equivalence relations, because if ${(f, \Sigma_1) \sim (g, \Sigma_2)}$ , then ${\Sigma_1}$ is isometric to ${\Sigma_2}$ (since it is isometric by an isometry commuting with ${f}$ and ${g}$ ).

As with the torus, define the mapping class group ${\mathrm{MCG}(S)}$ to be the group of homotopy equivalences of ${S}$ with itself, up to homotopy. Then ${\mathrm{MCG}(S)}$ acts on ${\mathrm{Teich}(S)}$ by ${\varphi \cdot (f,\Sigma) = (f \circ \varphi, \Sigma)}$ . The quotient of ${\mathrm{Teich}(S)}$ by this action is ${\mathcal{M}(S)}$ : clearly we never identify surfaces which are not isometric, and if ${i : \Sigma_1 \rightarrow \Sigma_2}$ is an isometry, and ${(f,\Sigma_1)}$ , ${(g,\Sigma_2)}$ are points in Teichmuller space with any ${f,g}$ , then notice ${f}$ has an inverse (up to homotopy), so if we act on ${(f,\Sigma_1)}$ by ${f^{-1}\circ g}$ , we get ${(f\circ f^{-1}\circ g, \Sigma_1)}$ , which is the same point in ${\mathrm{Teich}(S)}$ as ${(g,\Sigma_2)}$ . We are abusing notation here, because we are thinking of ${\Sigma_1}$ , ${\Sigma_2}$ and ${S}$ as the same surface (which they are, topologically). The point is that by acting by ${\mathrm{MCG}(S)}$ we can rearrange ${S}$ so that after mapping by ${f \circ i}$ we are homotopic to ${g}$ . The result of this is that

$\displaystyle \mathrm{Teich}(S) / \mathrm{MCG}(S) \cong \mathcal{M}(S)$

A priori, we are interested in hyperbolic metrics on ${S}$ up to isometry — Moduli space. The reason for defining Teichmuller space is that Moduli space is rather complicated. Teichmuller space, on the other hand, will turn out to be as nice as you could want ( ${\mathbb{R}^{6g-6}}$ for a genus ${g}$ surface). By studying the very nice Teichmuller space plus the less-nice-but-still-understandable mapping class group, we can approach Moduli space.

4. Coordinates for Teichmuller Space

Now we will take a closer look at Teichmuller space and give it coordinates.

4.1. Very Overdetermined (But Easy) Coordinates

One way to give this space coordinates is the following. Let us choose a homotopy class of loop in ${S}$ (this is a conjugacy class in ${\pi_1(S)}$ ), and we’ll represent this class by the loop ${\gamma : S^1 \rightarrow S}$ . Given a point ${(f,\Sigma) \in \mathrm{Teich}(S)}$ , there is a unique geodesic representative in the free homotopy class of the loop ${f\circ \gamma}$ . Define ${l_\gamma(f,\Sigma) = \mathrm{length}(f\circ \gamma)}$ to be the length of this representative. Let ${C}$ be the set of conjugacy classes in ${\pi_1(S)}$ . Then we have defined a map

$\displaystyle l : \mathrm{Teich}(S) \rightarrow \mathbb{R}^C$

$\displaystyle (f,\Sigma) \mapsto (l_\gamma(f,\Sigma))_\gamma$

This is nice in the sense that it’s a real vector space, but not nice in that it’s infinite dimensional. We will see that we need a finite number of dimensions.

4.2. Dimension Counting

Method 1

Let’s try to count the dimension of ${\mathrm{Teich}(S)}$ . Suppose that ${S}$ has genus ${g}$ . We can obtain ${S}$ by gluing the edges of a ${4g}$ -gon in pairs (going counterclockwise, the labels read ${a_1}$ , ${b_1}$ , ${a_1^{-1}}$ , ${b_1^{-1}}$ , ${a_2}$ …, ${a_g}$ , ${b_g}$ , ${a_g^{-1}}$ , ${b_g^{-1}}$ ). Since we will be given ${S}$ a hyperbolic metric, let us look at what this tells us about this polygon. We have a hyperbolic polygon; in order to glue it up, we must have

The paired sides must have equal length.
The corner angles must add to ${2\pi}$ .

For a triangle in hyperbolic space, the edges lengths are enough to specify the triangle up to isometry. Similarly, for a hyperbolic 4-gon (square), we need all the exterior edge lengths, plus 1 angle (the angle gives the length of a diagonal). By induction, a ${n}$ -gon needs ${n}$ side lengths and ${n-3}$ angles. For our ${4g}$ -gon, then, we need to specify ${4g}$ side lengths and ${4g-3}$ angles. This is ${8g-3}$ dimensions. However, we have ${2g}$ pairs, each of which gives a constraint, plus our single constraint about the angle sum. This reduces our dimension to ${6g-4}$ . Finally, we made an arbitrary choice about where the vertex of this polygon was in our surface. This is an extra two dimensions that we don’t care about (we disregard those coordinates), so we have ${6g-6}$ dimensions.

Method 2

A marked hyperbolic structure on ${S}$ gives a ${\pi_1(S)}$ -equivariant isometry ${\widetilde{\Sigma} \rightarrow \mathbb{H}^2}$ . That is, an element of ${\mathrm{Teich}(S)}$ is ${(f,\Sigma)}$ , which tells us how to map ${\pi_1(S)}$ isomorphically onto ${\pi_1(\Sigma)}$ , which is the same as the deck group of the universal cover ${\widetilde{\Sigma}}$ , which is ${\mathbb{H}^2}$ . Therefore, to an element of ${\mathrm{Teich}(S)}$ is associated a discrete faithful representation of ${\pi_1(S)}$ to ${\mathrm{PSL}(2,\mathbb{R})}$ , the group of isometries of ${\mathbb{H}^2}$ , and this representation is unique up to conjugacy (if we conjugate the image of the representation, then the quotient manifold is the same). The dimension of ${\mathrm{Teich}(S)}$ is therefore the dimension of the space of representations of ${\pi_1(S)}$ in ${\mathrm{PSL}(2,\mathbb{R})}$ up to conjugacy.

The fundamental group of ${S}$ has a nice presentation in terms of the polygon we can glue up to make it; the interior of the polygon gives us a single relation:

$\displaystyle \pi_1(S) = \langle a_1, b_1, \cdots, a_g, b_g \,| \, \prod_i [a_i,b_i]\rangle$

So ${\mathrm{Hom}(\pi_1(S), \mathrm{PSL}(2,\mathbb{R}))}$ is the subset of ${\mathrm{Hom}(F_{2g}, \mathrm{PSL}(2,\mathbb{R}))}$ such that ${\prod_i [a_i,b_i] = 1}$ (here ${F_{2g}}$ is the free group on 2 generators, which is what we get if we forget the single relation). Now a representation in ${\mathrm{Hom}(F_{2g}, \mathrm{PSL}(2,\mathbb{R}))}$ is completely free: we can send the generators anywhere we want, so

$\displaystyle \mathrm{Hom}(F_{2g}, \mathrm{PSL}(2,\mathbb{R})) \cong \left( \mathrm{PSL}(2,\mathbb{R}) \right)^{2g}$

Since ${\mathrm{PSL}(2,\mathbb{R})}$ is 3-dimensional, the right hand side is a real manifold of dimension ${6g}$ . Insisting that ${\prod_i [a_i,b_i]}$ map to ${1}$ is a 3-dimensional constraint (it gives 4 equations, when you think of it as a matrix equation, but there is an implied equation already taken into account). Therefore we expect that ${\mathrm{Hom}(\pi_1(S), \mathrm{PSL}(2,\mathbb{R}))}$ will be ${6g-3}$ dimensional. However, we are interested in representations up to conjugacy, so this removes another 3 dimensions, giving us the same dimension estimate for ${\mathrm{Teich}(S)}$ as ${6g-6}$ dimensional.

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