Teichmüller spaces via representations
I don't have much expertise in this area but I am confused by a remark I overheard regarding Teichmüller spaces.
I was always under the impression that for a surface $S$ (say genus $\geq 2$) that the Teichmuller space of $S$ was give by $\mathcal{T}(S) = \{\text{Hyperbolic structures for S} \} / \text{homotopy}$.
I was told that this is equivalent to the space of discrete faithful representations $\phi: \pi_1(S) \rightarrow PSL_2(\mathbb{R})$ quotiented by $PGL_2(\mathbb{R})$ .
My questions are as follows:
Why is the representation mapping to $PSL_2(\mathbb{R})$? Every representation I've seen is always defined as a map $\varphi: G\rightarrow GL_n(V)$. Perhaps this is a case where the word representation is overloaded and just means a correspondence, but every source I've seen uses the word representation and this is a slight point of confusion for me.
Could you suggest a resource where the equivalence between the two definitions is proven? Everything I've read mentions that they are equivalent with very little justification.
Are there any examples where using one definition over the other vastly eases calculations/computations?
Thanks!
Here are some partial answers which outline how to approach this problem.
For the first question, if $G$ is a group and $X$ is a set with some structure (e.g. $X$ might be a group or a vector space or a metric space or whatever), a homomorphism $G \to Aut(X)$, where $Aut$ refers to the fact that we consider all bijections from $X$ to $X$ which preserve its structure, is called a representation of $G$. If $X$ is a vector space, then $Aut(X)$ means the group of linear automorphisms, i.e. $GL(X)$; it is customary to say in this case that $\rho$ is a linear representation. If the homomorphism is injective, the convention is to say that the representation is faithful.
Fix a closed oriented surface $S$ of genus $g$ with preferred basepoint $s$. Given a faithful representation $\rho: \pi_1 (S,s) \to PSL(2,\mathbb{R})$ with discrete image, we obtain a hyperbolic surface. For $PSL(2, \mathbb{R})$ can be identified with the orientation preserving group of isometries of the hyperbolic plane $\mathbb{H}^2$, and the quotient of $\mathbb{H}^2$ by the image of $\rho$ is homeomorphic to $S$. (You will want to use covering space theory to prove this; the universal cover of $S$ is homeomorphic to $\mathbb{H}^2$. Given a point in $S$, choose a point in the fiber of the universal covering projection, and map $s$ to the orbit of this point. This is a well-defined homeomorphism. Prove!)
Now, if $f:S \to X$ is a marked hyperbolic surface (this means that $f$ is a homeomorphism and that $X$ is a quotient of $\mathbb{H}^2$ by a discrete group of orientation preserving isometries), then we consider the set of pairs $(X,f)$. Teichmuller space can be defined as the set of marked hyperbolic surfaces up to equivalence, where $(X,f) \sim (Y,g)$ if $gf^{-1}$ is homotopic to an isometry. What needs to be analyzed is what is the relationship between the induced representations $f_*$ and $g_*$ (the maps on the level of fundamental groups). The claim is that $(X,f) \sim (Y,g)$ if and only if the representations are conjugate in $PGL(2,\mathbb{R})$.
If the two representations are conjugate via $A \in PGL(2, \mathbb{R})$, then the map $\Gamma_X.p \mapsto \Gamma_Y.(ApA^{-1})$, where $\Gamma_X = \rho(\pi_1(S,s)$ and $p \in \mathbb{H}^2$ and $PGL(2, \mathbb{R})$ is identified with the full isometry group, is an isometry; keep in mind that $X = \mathbb{H}^2/\Gamma_X$ is the orbit space. To show that $gf^{-1}$ is homotopic to an isometry, you again will want to appeal to covering space theory and use the fact that $S$ is a $K(\pi,1)$ space. Proposition 1B.9 in Hatcher's text should give you some ideas. But, there are many details being left to you. There is probably a much nicer way to think about all of this; but I expect it would involve using somewhat fancier notions.