We already know that $\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $\begin{bmatrix} 1 & 0 \\ 2 & 1\end{bmatrix}$ generate a free subgroup of $GL_2 (\mathbb{C})$. But we are then asked to conclude that every finitely generated group is a subquotient (i.e. quotient of a subgroup) of $SL_2 (\mathbb{C})$.
I don’t quite see the relation between them, so I barely know where to start. Could anyone give me some hint on doing this?
Given any positive integer $m$, the free group of rank $2$ has a subgroup which is free of rank $m$. Any group with $m$ generators is isomorphic to a quotient of this, and so to a quotient of a subgroup of your group.