What is Representation of Surface Groups? I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "surface group". We define a surface group to be the group defined by the presentation $$ \Gamma _{g} = \left\langle a_{1}, b_{1}, a_{2}, \cdots, a_{g}, b_{g} \bigg| \prod _{r = 1} ^{g} [a_{r}, b_{r}] \right\rangle$$
Moreover, this is a surface group of genus $g$ surface. The representation of surface groups is the representation spaces of surface groups into semi-simple Lie groups $G$. In the subject, we define $\textbf{representation variety}$ of the fundamental group $\pi_{1}(S)$ of a closed connected surface
$S$ of genus greater than $2$, with values in a Lie group $G$. The representation variety is defined as follows: $$\text{Rep} \big(\pi_{1} (S), G\big) := \hom \big(\pi_{1}(S), G\big) / G$$ where $G$ acts on $\hom (\pi_{1}(S), G)$ by conjugation. I heard that this topic also intersects with Hyperbolic Geometry when the target group $G  = \text{PSL}_{2} (\mathbb{R})$ or $\text{PSL}_{2} (\mathbb{C})$. Also, I have heard that Classical Teichmüller theory may be viewed as a starting point of the subject.
Now my question is how to study the topic? Does the topic intersect with the Representation Theory of Semi -Simple Lie Groups? Please advise me for a roadmap of the Representation Theory of Surface Groups.
Sorry for my bad English. Please help me.
Thanking you in advance.
 A: Here is a stab:

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*"how to study the topic?"
If you want to get into this stuff, you really need an advisor, otherwise, the likeliest result is that you will try to read something only to get repeatedly lost in the unfamiliar terminology. Or, you start reading background material, spend few years on this and never get to the core subject.


*"Does the topic intersect with the Representation Theory of Semisimple Lie Groups?" Yes and no. You can get a very good feel for the material with bare minimum of the general Lie theory, just focus on representations into $SL(2, {\mathbb C})$ (or even $SL(n, {\mathbb C})$) where all you need is elementary linear algebra. But if you want to study the general case of representations to semisimple Lie groups $G$, yes, you will need some general understanding of the structure theory of semisimple Lie groups, eventually including the representation theory of Lie groups. (And the further you go, the more knowledge of algebraic groups you will need.) An advisor will guide you through this.


*"Please advise me for a roadmap of the Representation Theory of Surface Groups." I suggest you first take a look at
Sikora, Adam S., Character varieties, Trans. Am. Math. Soc. 364, No. 10, 5173-5208 (2012). ZBL1291.14022.
which covers character varieties of general finitely generated groups (not just surface groups), but also contains some basic material on character varieties of surface groups. In particular, you will see how to make sense of the quotient by the group $G$ in the definition of $Rep$. (For the record, this is a special case of a GIT quotient when you have an algebraic group acting on an algebraic variety and you are trying to form a quotient.)
Do you have enough background to read this paper? I have no idea. Try and see how it goes. If you do
not have much background in Lie theory, whenever he says "a reductive group", just think $SL(2, {\mathbb C})$.
If you manage to read through this paper, you have a choice (and again an advisor would guide you): Are you interested in analytical tools for studying character varieties of surface groups? Are you interested in algebro-geometric side? Are you interested in connections to geometric structures?...
If you like connections, curvature, etc (do you know what this means?), take a look at
F.Labourie, "Lectures on Representations of Surface Groups"
If you liked that staff, do you want to learn about gauge-theoretic side of the theory (Higgs bundles)? Then consider reading
Goldman, William M.; Xia, Eugene Z., Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces, Mem. Am. Math. Soc. 904, 69 p. (2008). ZBL1158.14001.
which deals with the "baby case" of the theory, when the target group $G$ is ${\mathbb C}^*$. After that, you can proceed to the case of representations to $SL(n, {\mathbb C})$:
Wentworth, Richard A., Higgs bundles and local systems on Riemann surfaces, Álvarez-Consul, Luis (ed.) et al., Geometry and quantization of moduli spaces. Based on 4 courses, Barcelona, Spain, March – June 2012. Basel: Birkhäuser/Springer (ISBN 978-3-319-33577-3/pbk; 978-3-319-33578-0/ebook). Advanced Courses in Mathematics – CRM Barcelona, 165-219 (2016). ZBL1388.30052.
Reading this will require some background on Riemann surfaces and complex differential geometry. Chances that you can navigate towards this without a competent advisor are very slim (and I can only repeat my suggestions in the first paragraph).
Edit. Connections to hyperbolic geometry appear for both representations to $PSL(2, {\mathbb R})$ and $PSL(2, {\mathbb C})$. The story is again quite long. You can start by reading the discussion here. I'll add more references later.
