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. 



What finitely presented groups embed into $\operatorname{GL}_2$?
If the group $G$ does not have [property FA](http://en.wikipedia.org/wiki/Property_FA), then a necessary and sufficient
condition is that the group embeds in $\operatorname{GL}\_2(\mathcal{O}\_K)$, for some number
field $K$ (although there are such subgroups which do not have property FA).
This follows from [Bass-Serre theory](http://en.wikipedia.org/wiki/Bass-Serre_theory). Of course, this begs the question of classifying finitely presented subgroups of $\operatorname{GL}\_2(\mathcal{O}\_K)$ with property FA.


More generally, [Bass-Serre theory implies](http://books.google.com/books?id=MOAqeoYlBMQC&lpg=PP1&pg=PR6#v=onepage&q&f=false) that a general finitely generated
subgroup of $\operatorname{GL}\_2(K)$ will have a graph of groups decomposition
into subgroups of $\operatorname{GL}\_2(\mathcal{O}\_K)$ for some number fields $K$.


The [geometrization theorem](http://en.wikipedia.org/wiki/Geometrization_conjecture) and [ending lamination theorems](http://en.wikipedia.org/wiki/Ending_lamination_conjecture) classify
discrete subgroups of $\operatorname{PSL}\_2(\mathbb{C})$ (which up to finite-index
embed in $\operatorname{SL}\_2(\mathbb{C})$), by their topological type as a 3-orbifold
and the ending lamination data.


You ask about congruence subgroups of $\operatorname{GL}\_2(\mathcal{O}\_K)$.
If $K=\mathbb{Q}$ or $K=\mathbb{Q}(\sqrt{-D})$, for some $D>0$,
then the group is a discrete non-uniform lattice in $\operatorname{PSL}\_2(\mathbb{R})$
or $\operatorname{PSL}\_2(\mathbb{C})$, and one may classify the congruence
subgroups of $\operatorname{SL}\_2(\mathbb{Z})$ by a result of Tim Hsu (more
generally, I think there exists and algorithm to determine
if a finite-index subgroup of $\operatorname{GL}\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{-D})})$
is congruence, but I don't know if it's written down - I could describe
it for you though if you're interested). More generally, one can
determine if a discrete non-uniform arithmetic lattice in $\operatorname{PSL}\_2(\mathbb{R})$
or $\operatorname{PSL}\_2(\mathbb{C})$ is congruence.


Otherwise, [Serre essentially showed](http://www.ams.org/mathscinet-getitem?mr=272790) that a finite-index subgroup of $\operatorname{GL}\_2(\mathcal{O}\_K)$
will have the congruence subgroup property (and thus, non-uniform
lattices in a product $(\mathbb{H}^2)^k\times (\mathbb{H}^3)^l$ will
have this property if $k+l>1$).


For examples of groups which don't have property FA, there's a paper
of [Calegari and Dunfield](http://www.ams.org/journals/proc/2006-134-11/S0002-9939-06-08398-5/home.html) which constructs an ascending HNN
extension subgroup of $\operatorname{SL}\_2(\mathbb{C})$.


There are many necessary conditions which show that various groups
cannot embed in $\operatorname{GL}\_2(\mathbb{C})$, some of which you describe.
But I think a general classification is beyond reach at this point.


As you say, if a space $X$ has a $\mathbb{C}^2$ bundle with flat
connection, then you get a representation of $G=\pi\_1(X)$ into
$\operatorname{GL}\_2(\mathbb{C})$. The space of such flat bundles is computable
if $G$ is finitely presented, it amounts to computing the [character
variety](http://en.wikipedia.org/wiki/Character_variety) of $G$ into $\operatorname{GL}\_2(\mathbb{C})$. However, it is difficult
to tell if there is a faithful representation. If you can solve
the word problem in $G$, then in principle one can determine if
a representation is not faithful. Also, it seems difficult to certify that
a representation is faithful, except if it is discrete. The
difficulty is to find a nice fundamental domain for the
action on a product of symmetric spaces on which the group
acts discretely (in fact, it might not exist).


