Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi\_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of the Lie group $GL(n,\mathbb{C})$. What is known about the manifold in such a case, i.e. what are the implications given the earlier question is correct? I am looking for references on any property, as asking for a specific one is too much to ask for.
Your answers are greatly appreciated.



What is known about a 3-manifold $M$ when its fundamental group is linear?
If you are considering compact 3-manifolds, then it is conjectured that the fundamental groups are always linear, so there should be no restriction on the topology.
One may as well consider 3-manifolds with indecomposable fundamental group. Then the only remaining case to consider is graph manifolds with a non-trivial JSJ decomposition and which do not admit a non-positively curved Riemannian metric. See the papers of [Yi Liu](https://arxiv.org/abs/1110.1940) and [Przytycki-Wise](https://arxiv.org/abs/1205.6742).


For noncompact 3-manifolds, the issue of linearity of the fundamental group is a wide-open problem. I don't know of any evidence against it though.


