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.
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2$\begingroup$ Is there a 3-manifold whose fundamental group is known not to be linear? $\endgroup$– algoriCommented Oct 3, 2012 at 22:24
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1$\begingroup$ As far as I know, there is not a 3-manifold with non-linear fundamental group. $\endgroup$– lagrangiansubmanifoldCommented Oct 3, 2012 at 23:02
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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 and Przytycki-Wise.
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.
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$\begingroup$ I did not realize the question for non-compact polyhedra was even a question. Is there any published work on this? $\endgroup$ Commented Oct 3, 2012 at 22:48
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$\begingroup$ @ Igor: I don't know that it is asked anywhere, but it's a natural question (when and if the compact case is resolved), although I couldn't really tell you what linearity of groups is useful for. $\endgroup$– Ian AgolCommented Oct 4, 2012 at 3:06