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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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Is there a 3-manifold whose fundamental group is known not to be linear? –  algori Oct 3 '12 at 22:24
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As far as I know, there is not a 3-manifold with non-linear fundamental group. –  lagrangiansubmanifold Oct 3 '12 at 23:02

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up vote 9 down vote accepted

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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Thank you very much for this detailed answer! –  lagrangiansubmanifold Oct 3 '12 at 22:26
    
I did not realize the question for non-compact polyhedra was even a question. Is there any published work on this? –  Igor Rivin Oct 3 '12 at 22:48
    
@ 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. –  Ian Agol Oct 4 '12 at 3:06

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