TITLE: Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
QUESTION [31 upvotes]: Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are related to $A$-polynomials and thus to various conjectures concerning asymptotics of quantum invariants such as the AJ-conjecture.
Varieties of representations and characters of surface groups in algebraic groups have also been intensively studied, this being a survey. 
In recent years, there have been a large number of papers on varieties of representations and characters of $3$-manifold groups in $SL(n,\mathbb{C})$ for $n>2$, including the development of various tools to compute invariants associated to such varieties, e.g. work of Garoufalidis, Thurston, Zickert, and Goerner.
I have leafed through various of these papers on various occasions and heard some talks, but although understanding any sorts of representations of $3$--manifold groups is surely a worthy goal, as is generalizing results known for $n=2$ or for surface groups, I haven't really understood what motivates this direction of research or what people are aiming to achieve by studying such representations.

Question: What is the motivation for the study of representations of 3-manifold groups into $SL(n,\mathbb{C})$ for $n>2$? 

For example, is it expected that such representations should provide tools to identify geometric structures on manifolds? Are they expected to offer insight into conjectures concerning asymptotics of quantum invariants? Or maybe something else?

REPLY [14 votes]: Three manifold groups are quite special and their representation varieties have more structure than that of a random group. Casson's view point is helpful to see the point. The Heegaard decompoistion of a three manifold $Y= H_1\cup_\Sigma H_2$ means that the fundamental group of a three manifold has a quite special presentation. There is pushout diagram,
\begin{eqnarray}
\pi_1(\Sigma) & \to & \pi_1(H_1)\\
\downarrow & & \downarrow \\
\pi_1(H_2) & \to & \pi_1(Y)
\end{eqnarray} 
and hence maps in the opposite direction on representation varieties.
\begin{eqnarray}
R_G(\Sigma) & \leftarrow & R_G(H_1)\\
\uparrow & & \uparrow \\
R_G(H_2) & \leftarrow & R_G(Y)
\end{eqnarray} 
The representation variety $R_G(\Sigma)$ is a symplectic manifold when $G$ is compact Lie group (thanks to Goldman and Atiyah-Bott) and gets a Kahler structure
once metric is chosen on $\Sigma$.  If $G$ is the complexification of a compact Lie group and a metric on $\Sigma$ is choose $R_G(\Sigma)$ is Hyperkahler manifold.
(It is a slight lie (small l ;-) that these are manifolds.)
The maps $R_G(H_i) \to R_G(\Sigma)$ are injective and images (to the extent they are manifolds) are Lagrangian in case $G$ is compact and complex Lagrangian when $G$ is the complexification of a compact group.  The intersection 
$$
R_G(Y)=R_G(H_1)\cap R_G(H_2) \subset R_G(\Sigma)
$$
is then much more special, so in the compact case lies in the setting of Lagrangian Floer homology and the in the complex case in the setting is for example discussed recently by Witten and Haydys (see for example arXiv: 1010.2353 )
Related to this point of view is the fact that representation varieties of three manifolds are the set of critical points of a Chern-Simons functional on a suitable space of gauge equivalence classes of connections.
Apologies for the inaccuracies above due to haste and lazy typesetting.