TITLE: How can I tell if a group is linear?
QUESTION [17 upvotes]: The basic question is in the title, but I am interested in both necessary and sufficient conditions. 
I know the Tits' alternative and Malcev's result that finitely generated linear groups are residually finite, but I don't know any purely group-theoretical sufficient conditions. 
Though it's not purely group-theoretical, what if the group acts simplicially on a (finite) simplicial complex? Does this imply linearity? Does it imply linearity over $\mathbb{C}$?

REPLY [4 votes]: One more necessary condition. Let $T$ be a matrix from $SL_n(\mathbb{C})$. Then the set of all matrices $B$ such that $\lim_{n\to \infty} T^{n} BT^{-n} = 1$ is a nilpotent subgroup (an exercise, first noticed by Margulis, I think, a proof can be found here.). This implies, for example,  that the group $\langle a,b,t \mid tat^{-1}=a^2, tbt^{-1}=b^2\rangle$ is not linear (this group is residually finite).