Let $A \subset \mathsf{SL}(2; \mathbb{C})$ be a finite set of matrices. Consider the set $$S = \overline{\langle A \rangle},$$ where we take the closure in $\mathsf{SL}(2; \mathbb{C})$, and where $\langle A \rangle$ denotes the free group of $A$.
Clearly $S$ is a group, and by construction $S$ is closed in $\mathsf{SL}(2; \mathbb{C})$. Therefore, by Cartan's closed subgroup theorem, it must be the case that $S$ is a Lie subgroup of $\mathsf{SL}(2; \mathbb{C})$. I am interested in understanding what are the necessary and sufficient conditions on $A$ for it to be the case that, in fact, $S = \mathsf{SL}(2; \mathbb{C})$?
It is easy to come up with a few concrete conditions for insufficiency: for instance, if all the elements of $A$ are mutually commuting, or if $A \subset \mathsf{SL}(2; \mathbb{R})$, or if $A \subset \mathsf{SU}(2)$. But I am interested in more systematically characterizing when $S$ equals $\mathsf{SL}(2; \mathbb{C})$.
Any thoughts or references are welcome. Thank you!
First of all, the list of closed connected subgroups of $SL(2, {\mathbb C})$ is not very long, see my answer here. From that, you conclude that a subgroup $\Gamma< SL(2, {\mathbb C})$ is either virtually solvable (i.e. contains a solvable subgroup of finite index), or is discrete, or is dense in $SL(2, {\mathbb C})$ or its closure equals a conjugate of $SU(2)$ or $SL(2, {\mathbb R})$ or contains $SL(2, {\mathbb R})$ as an index 2 subgroup.
Thus, effectively, you are asking for necessary and sufficient conditions for discreteness of a finitely generated subgroup of $SL(2, {\mathbb C})$. There is one standard necessary condition for discreteness of 2-generated subgroups $\Gamma$ which are not virtually solvable, called the Jorgensen Inequality: $$ {\displaystyle \left|\operatorname {Tr} (A)^{2}-4\right|+\left|\operatorname {Tr} \left(ABA^{-1}B^{-1}\right)-2\right|\geq 1.\,} $$ Here are $A, B$ are generators of $\Gamma$. From this, one deduces the following:
(I am replacing your condition that $\Gamma$ is free by the weaker condition that $\Gamma$ is not virtually solvable.) There are other conditions which are either necessary or sufficient for discreteness, but the one above is the most concise. See also the discussion in this Mathoverflow question.
For some recent work on discreteness conditions see
Klimenko, Elena; Kopteva, Natalia, A two-dimensional slice through the parameter space of two-generator Kleinian groups, Int. J. Algebra Comput. 28, No. 8, 1535-1564 (2018). ZBL1412.30132.
and references therein.
If you want to read more about discrete subgroups of $SL(2, {\mathbb C})$, there are many books to choose from, for instance,
Beardon, Alan F., The geometry of discrete groups, Graduate Texts in Mathematics, 91. New York - Heidelberg - Berlin: Springer-Verlag. XII, 337 p. (1983). ZBL0528.30001.
Matsuzaki, Katsuhiko; Taniguchi, Masahiko, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 253 p. (1998). ZBL0892.30035.