TITLE: What finitely presented groups embed into GL_2? 
QUESTION [17 upvotes]: This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.) 
I think I can figure out the finite groups with this property (they can be conjugated into $\text{U}(2)$ and taking determinants reduces to the classification of finite subgroups of $\text{SU}(2)$ and an extension problem) as well as the f.g. abelian groups with this property (there can't be too much torsion). But already I don't know what finitely presented groups appear as, say, congruence subgroups of $\text{GL}_2(\mathcal{O}_K)$ for $K$ a number field. 
What can be said if you are given, say, a nice space $X$ with fundamental group $G$? I hear that in this case linear representations of $G$ are related to vector bundles on $X$ with flat connection.

REPLY [6 votes]: First, instead of subgroups of $GL(2,C)$ it suffices to consider subgroups of $SL(2, C)$. I will also restrict to subgroups which are not virtually solvable (since these should be easy to classify). Lastly, I will consider characterization of subgroups of $SL(2, C)$ "up to abstract commensuration" since the criterion is much cleaner in this setting. (Recall that two abstract groups $G_1, G_2$ are called "commensurable" if they contain isomorphic finite-index subgroups.) 
Definition. Let $p$ be a prime and $c$ an integer. A $p$-congruence structure of degree $c$ for a group $\Gamma$, is a descending chain of finite-index normal subgroups of $\Gamma$: 
$\Gamma=N_0\supset N_1 \supset ... \supset N_k ... $, such that:
(i) $\bigcap_{k=0}^\infty N_k= 1$; 
(ii)    $N_1/N_k$ is a finite $p$-group for every $k\ge 2$; 
(iii)   $d(N_i/N_j)\le c$ for all $j\ge i\ge 1$, where $d(H)$ denotes the minimal number of generators 
of a group $H$. 
Theorem. A finitely-generated (non-virtually solvable) group $\Gamma$ is commensurable to a 
subgroup of $SL(2,C)$ if and only if $\Gamma$ admits (for some prime $p$) a $p$-congruence structure of degree $c=3$. 
This theorem is an application of "A Group Theoretic Characterization of Linear Groups" by Alex Lubotzky (Journal of Algebra, 113 (1988), 207-214), combined with the classification of Lie algebras of dimension $\le 3$ over fields. Note that Lubotzky does not get a group-theoretic characterization of subgroups of $SL(n,C)$ for particular values of $n$ (even up to commensurability), however, for $n=2$ the situation is better than for the general $n$ since dimension of the Lie group is so small in this case.