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?

in's theorem's even more beautiful :) $\endgroup$ – YCor Mar 28 '16 at 0:27