## BI: Teichmüller space, part I

It is a remarkable fact that the group of orientation-preserving isometries of $\mathbb{H}^2$ can be identified with the group of bi-holomorphisms of the unit disk $\mathbb{D}$ in $\mathbb{C}$. [Fun fact: Poincaré said that he discovered (basically) this while stepping on a bus.]

As a consequence, for $S$ an orientable closed surface of genus at least 2 fixed from now on, we have a natural bijection

{complex structures on $S$ } $\leftrightarrow$ {hyperbolic metrics on $S$}

The bijection is given by the fact that a complex structure on $S$ is a way of seeing $S$ as a quotient of $\mathbb{D}$ by a group of biholomophisms, and similarly for hyperbolic metrics.
There’s a more explicit way of comparing a complex structure with its corresponding  hyperbolic metric. Look at the tangent space $T_p(S)$ at some point $p$ of $S$. The hyperbolic metric (as well as any other Riemannian metric) naturally gives  a collection of circles covering $T_p(S)$. The complex structure does as well (consider multiplication by $e^{i\theta}$ for $\theta\in [0,2\pi]$). As it turns out, a complex structure and its corresponding hyperbolic metric give the same collection of circles in every tangent space.
(In other words, they induce the same conformal structure.)

Teichmüller space $Teich(S)$ is a space that parametrizes the complex structures/hyperbolic metrics on $S$. It does not parameterize them up to biholorphism/isometry, but up to biholorphisms/isometries isotopic to the identity. The formal definition is below the informal discussion in the next paragraph.

Suppose you have a homeomorphism $f$ of $S$ supported in a disk and a hyperbolic metric $g$ on $S$. Then one can take the push-forward $f_*(g)$ of $g$ through $f$ and get another hyperbolic metric on $S$. Teichmüller space does not distinguish between these metrics (if it did it would be huge!).
On the other hand, suppose you have a hyperbolic metric $g$ on $S$ with some non-trivial isometry $f$. Then $g$ and $f_*(g)$ actually give distinct points of $Teich(S)$.

And, now, if you really want to see it, here is the formal definition.

$Teich(S)=\{(X,f)| X \mathrm{\ hyperbolic\ surface}, f:S\to X \mathrm{\ homeomorphism} \} /_\sim,$

where “hyperbolic surface” just means surface endowed with a hyperbolic metric and the equivalence relation $\sim$ given by $(X_1,f_1)\sim (X_2,f_2)$ if and only if there exists an isometry $\iota: X_1\to X_2$ so that the diagram

commutes up to isotopy, i.e. $f_2^{-1}\iota\circ f_1:S\to S$ is isotopic to the identity. (One can safely substitute isotopies with homotopy equivalences.)

There is a topology on $Teich(S)$, but I will not discuss it. Here is probably the most important theorem about $Teich(S)$.

Theorem:  $Teich(S)$ if homeomorphic to $\mathbb{R}^{6g-6}$, where $g$ is the genus of $S$.

More precisely, there is a system of coordinates for $Teich(S)$, the so-called Fenchel-Nielsen coordinates. Danny Calegari has a post on this which contains way more details than the discussion below.

Digression: The very nice structure of $Teich(S)$ given by this theorem explains why it is reasonable to consider hyperbolic metrics up to isometry isotopic to the identity, rather than up to isometry, at least as a first step. Then, if one is interested in hyperbolic metrics up to isometry, one can then take the quotient of $Teich(S)$ by the natural action of the mapping class group $MCG(S)$, and hence form the so-called moduli space of $S$. This action is very nice as well, e.g. it is properly discontinuous, so that moduli space has the structure of an orbifold.

Back to the theorem. The proof is quite interesting, and is based on a pants decomposition of $S$, that is to say describing $S$ as a union of pairs of pants, i.e. the following object:

Any maximal collection of disjoint simple closed curves on $S$ gives a pants decomposition, here is an example:

One can endow a pair of paints with a hyperbolic metric so that the boundary is a geodesic in the following way. In $\mathbb{H}^2$ there exist right-angled hexagons, and one can also freely choose the lengths $a,b,c$ of three sides as in the picture.

Gluing two copies of such a hexagon along the blue sides yields a hyperbolic metric on a pair of pants. What is more, one can glue pairs of pants and still obtain a hyperbolic metric as long as corresponding boundary components have the same length. So, given a pair of pants decomposition of $S$, one can assign a length to all the curves appearing in it, consider the corresponding metrics on pairs of pants and glue them all together to get a hyperbolic metric on $S$.

Any maximal collection of disjoint simple closed curves in $S$ contains $3g-3$ curves, not $6g-6$, which means that there are other parameters to consider.
The hyperbolic metrics we constructed on pairs of pants have the feature that for any given pair of boundary components  there is a unique geodesic connecting them and orthogonal to both of them, namely the suitable blue side of one of the hexagons. Hence you see that gluing two pairs of pants in this way (the blue lines represent the geodesics we just described, in adjacent pairs of pants):

or this way:

gives different metrics. So, in order to specify a metric on $S$ we also have to assign a “twist parameter” to each simple closed curve in the decomposition. You might think that these parameters are defined modulo $2\pi$, but it is not the case: If we change one of the twist parameters by $2\pi$ we obtain a new hyperbolic metric which is indeed isometric to the first one, but there’s no isometry isotopic to the identity between them.

To sum up, we choose a maximal collection of disjoint simple closed curves in $S$, and in order to specify a hyperbolic metric on $S$ we assign to each of them a “length parameter” and a “twist parameter”. Some routine and some not-so-routine checks and the theorem is proven.

In another post I’ll talk about the tangent space of $Teich(S)$ and the metrics one can put on $Teich(S)$ using it.

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