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!
 A: 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.
