I am reading a note on Teichmuller space, and I come across a somewhat algebraic problem in the picture below, which maybe easy to experts.
I wonder how to understand this injective map:
$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\mathbb{R}))/PSL_{2}(\mathbb{R})$
explicitly?
( All I can understand is just an injective map from the fundamental group to $Aut(\mathbb{H})$ just like what the author did here: Teichmuller space as Discrete Faithful Representations up to Conjugation)
I'm also wonder how to compute the yellow pattern I highlighted in the picture, why it's $char_{2}(\pi_{1}(S))$?
And how can I compute $char_{2}(\pi_{1}(S))$?
I find something that looks just like the right hand of the mapping, which is called the representation of surface group, but I can't find a formal explanation or proof about how can it be related with Teichmuller space. Can anybody help me? Any advice or comment is welcome.