How can I tell if a group is linear? 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 $\mathbb{C}$?
 A: Since this is not purely group theoretical and not a complete answer, this maybe should be more of a comment, but since you mentioned simplicial complexes perhaps you should check out the following paper to get you started:
Haglund, Frédéric, and Daniel T. Wise. "Special cube complexes." Geometric and Functional Analysis 17.5 (2008): 1551-1620.
Which is concerned with the fundamental groups of certain square complexes (VH complexes whose 1-cells are divided into two classes, "horizontal" and "vertical", and the attaching maps of squares alternate v-h-v-h). For instance, they prove that any fundamental group of a compact virtually clean (clean means attaching maps are embeddings, and this implies that the group splits as a clean graph of groups, as studied in [1]) VH-complex is linear.
Although this isn't purely group theoretic, it is at least mostly presentation theoretic, and the result itself isn't too hard to apply if you have for instance a finitely presented group. In this case, there is often an easy algorithm to check whether such a group has a VH-subdivision. (For instance, there is an example of such a procedure outlined in my paper with Wise [2]). After you still have to check the virtually clean condition, which may or may not be so easy depending on what you are doing.
[1] Wise, Daniel T. "The residual finiteness of negatively curved polygons of finite groups." Inventiones mathematicae 149.3 (2002): 579-617.
[2] Polák and Wise, "A Note on VH Subdivisions", To appear.
A: Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear.


*

*Schur: Suppose that $G$ is a finitely generated linear group, such that all elements have finite order. Then $G$ is finite.

*Jordan: Suppose that $G$ is a finite linear group of degree $n$ over a field of characteristic zero. Then there exists an integer-valued function $\beta(n)$ such that G contains an abelian normal subgroup of finite index at most $\beta(n)$.

*Malcev: Suppose that $G$ is a finitely-generated linear group. Then $G$ is residually finite. If $G$ is simple, then $G$ is finite.

*Platonov: Suppose that $G$ is a linear group of degree $n$ of finite Pruefer rang $r$ over a field of characteristic  $p > 0$. Then $G$ contains an abelian normal subgroup of finite index bounded in terms of $r, n$, and $p$.

*Malcev: Suppose that $G$ is a solvable linear group of degree $n$ over an algebraically closed field. Then $G$ contains a triangularizable normal subgroup of finite index bounded by a function of $n$.
A: 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).
