Suppose we have a 3manifold $M$ 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.
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If you are considering compact 3manifolds, 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 3manifolds with indecomposable fundamental group. Then the only remaining case to consider is graph manifolds with a nontrivial JSJ decomposition and which do not admit a nonpositively curved Riemannian metric. See the papers of Yi Liu and PrzytyckiWise. For noncompact 3manifolds, the issue of linearity of the fundamental group is a wideopen problem. I don't know of any evidence against it though. 

