  We study the freeness problem for subgroups of
$\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. For $q =
r/p \in \mathbb{Q} \cap (0,4)$, where $p$ is prime and $\gcd(r,p)=1$, we
initiate the study of the algebraic structure of the group $\Delta_q$ generated
by the two matrices \[ A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \text{
and } Q_q = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}. \] We introduce the
conjecture that $\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$, the congruence
subgroup of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ consisting of all
matrices with upper right entry congruent to $0$ mod $r$ and diagonal entries
congruent to $1$ mod $r$. We prove this conjecture when $r \leq 4$ and for some
cases when $r = 5$. Furthermore, conditional on a strong form of Artin's
conjecture on primitive roots, we also prove the conjecture when $r \in \{ p-1,
p+1, (p+1)/2 \}$. In all these cases, this gives information about the
algebraic structure of $\Delta_{r/p}$: it is isomorphic to the fundamental
group of a finite graph of virtually free groups, and has finite index $J_2(r)$
in $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the
Jordan totient function.
