This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\mathbb{R}))/PSL_{2}(\mathbb{R})$$
below is the definition of Teichmüller space I used.
Suppose S is a topological surface of genus $g\geq 2$. A marked Riemann surface $(X, f)$ is a Riemann surface X together with a homemorphism $f:S\rightarrow X$. Two marked surfaces $ (X,f)\sim (Y,g)$ are equivalent if $gf^{-1}: X\rightarrow Y$ is isotopic to an isomorphism(biholomorphism). The Teichmüller space $T_g$ is given by:
$$T_g=\{(X,f)\}/\sim$$
Thanks to @Andy Putman, where he gave some hints in the comments. Let me summarize here, below S is always a hyperbolic Riemann surface of genus at least 2:
The group of biholomorphisms of D is $PSL_{2}(\mathcal{ℝ})$. This shows that $𝑆\cong 𝐷/\Gamma$, where $\Gamma$ is a discrete subgroup of $PSL_{2}(\mathcal{ℝ})$. The marking on 𝑆 lets you identify $\pi_{1}(𝑆)$ with $\Gamma$(up to conj due to basepts), so we get a point in your rep space. Reversing this process gives you the inverse map from the rep space to Teichmuller space.
About how markings of S identify $\pi_{1}(𝑆)$ with $\Gamma$(up to conj due to basepts). If 𝑆 and 𝑆′ are two surfaces of genus $𝑔\geq 1$, then they're both $𝐾(\pi,1)$'s for their fundamental group, so the set of homotopy classes of homeomorphisms from 𝑆 to 𝑆′ is the same as the set of conjugacy classes of isomorphisms between their fundamental groups. Thus a marking of a surface of genus at least 1 is the same as an identification of its fundamental group with that of your fixed reference surface (up to conjugacy)
If I don't understand wrongly, he used the property of $K(G,1)$ here, namely Proposition 1B.9 in Hatcher's book "Algebraic topology".
Let X be a connected CW-complexes, and Y be a K(G,1). Then any homomorphism $\pi_{1}(X,x_{0})\rightarrow \pi_{1}(Y,y_{0}) $ is induced by a map $(X,x_{0})\rightarrow (Y,y_{0})$, which is unique up to homotopy fixing $x_{0}$
What I want to show is the marking, which is the homotopy classes of biholomorphism of hyperbolic Riemann Surfaces, identify with $Hom(\pi_{1}({S}),PSL_{2}(\mathbb{R}))/PSL_{2}(\mathbb{R})$. So my questions are:
Are homotopy classes of homeomorphism between hyperbolic Riemann Surfaces the same thing as homotopy classes of biholomorphsim between hyperbolic Riemann Surfaces?
I want to prove if (X, f)~(Y, g), then their corresponding representations are conjugate in $PSL_2(\mathbb{R})$. The representation corresponding to (X, f) is induced from f by:
$$\pi_{1}(S)\stackrel{f_{*}}\rightarrow \pi_{1}(X)\cong \Gamma_{X}\subset Aut(D)$$
If the answer to question 1 is right, how can I show "if (X, f)~(Y, g), then their corresponding representations are conjugate in $PSL_2(\mathbb{R})$" using "the set of homotopy classes of homeomorphisms from 𝑆 to 𝑆′ is the same as the set of conjugacy classes of isomorphisms between their fundamental groups"?
In other words, we have known that:
$$\pi_{1}(S)\stackrel{f_{*}}\rightarrow \pi_{1}(X)\cong \Gamma_{X}\subset Aut(D)$$ and:
$$\pi_{1}(S)\stackrel{g_{*}}\rightarrow \pi_{1}(Y)\cong \Gamma_{Y}\subset Aut(D)$$
are two representations of $\pi_{1}(S)$ on D, then the set $Hom(\pi_{1}({S}),PSL_{2}(\mathbb{R}))/PSL_{2}(\mathbb{R})$ is bijective to the set of all the group isomorphism:
$$h:\pi_{1}(X)\rightarrow \pi_{1}(Y)$$
up to conjugacy?
P.S: Maybe the second question is easy to answer because right now I'm stuck in these definitions and maybe I ignore something obvious.
