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Representation theory and topology of Teichmuller space

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. enter image description here

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.