TITLE: Understanding groups that are not linear
QUESTION [19 upvotes]: I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting examples of groups that aren't linear?
Are there general constructions that one can use to cook up a group or a family of groups that isn't linear?
Are there general techniques that one can use to show that a given group isn't linear?
More loosely, what is it about these groups that makes them interesting?

REPLY [4 votes]: Let me give you two examples from the theory of pro-$p$ groups:

The Nottingham group contains every finite $p$-group. Thus, it is not linear over any field.
Zubkov proved that a non-abelian free pro-$p$ groups cannot be embedded into $SL_2(R)$, where $R$ is any pro-$p$ ring. This implies that any pro-$p$ subgroup of $SL_2(R)$ satisfies a pro-$p$ identity. Lubotzky and Shalev conjectured that this is true for $SL_n(R)$ for any $n$ and any pro-$p$ ring $R$. Micheal Larsen and I proved that a non-abelian free pro-$p$ group is not linear over local fields. Our proof was a fairly easy application of Richard Pink's work which enabled us to reduce the question to showing that an open subgroup of semisimple group over local fields is not free. I think we then used some result of Lubotzky, but I am sure there are many arguments to deal with that case. Let me emphasize that our result does not imply the existence of a pro-$p$ identity (for that you need to deal with all pro-$p$ rings). However, Zelmanov claims in an unpublished work that pro-$p$ identity exists if we fix $n$ and let $p$ be big enough. Also, notice that this does not imply non-linearity in general.