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Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of characteristic 0. Its proof relies on the (relatively) advanced theory of pro-$p$-groups. It can be found with its proof in the book, "Analytic pro-$p$-groups" of Dixon, Du Sautoy, Mann and Segal, Interlude B, or in the original paper "A Group-Theoretic characterization of linear groups" in Journal of Algebra 113.

My question is:

Has this criterion being used to prove or disprove the linearity of discrete groups of independent interest?

By independent interest I mean some group interesting for mathematicians outside of the theory of pro-$p$-groups, for instance the Braid groups $B_n$, the group $Out(F_n)$, fundamental groups of interesting manifolds, etc... not some group

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  • $\begingroup$ Comment meta: If several answers are proposed, I will add the tag "long list" and will request the post to be made CW. I have no objection if people make it CW right now anyway. $\endgroup$
    – Joël
    Aug 21, 2015 at 17:45
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    $\begingroup$ According to Alex himself, this theorem is practically useless. It does not mean that it can't be applied, for instance when you have a group with assumptions that it has many quotients in some suitable sense, it can be applied. But for explicit examples of groups (e.g., given by a presentation, or as groups of automorphisms of some reasonable structure) as in the examples you provide, you don't have much info about finite quotients so you it's not practical: in examples known to be linear, the easiest way to check Lubotzky's criterion is to show that they are linear... $\endgroup$
    – YCor
    Aug 21, 2015 at 21:42
  • $\begingroup$ Thank you Yves. I consider this as an answer -- you may want to post it as such. $\endgroup$
    – Joël
    Aug 22, 2015 at 2:34

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According to Alex himself, this theorem is practically useless. It does not mean that it can't be applied, for instance when you have a group with assumptions that it has many quotients in some suitable sense, it can be applied. But for explicit examples of groups (e.g., given by a presentation, or as groups of automorphisms of some reasonable structure) as in the examples you provide, you don't have much info about finite quotients so you it's not practical: in examples known to be linear, the easiest way to check Lubotzky's criterion is to show that they are linear...

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