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!