On congruence subgroups of ${\mathrm{SL}}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements

We study the freeness problem for subgroups of ${\mathrm{SL}}_2(\mathbb{C})$ generated by two parabolic matrices. For $q=r\mathrm{/}p\in \mathbb{Q}\cap (0,4)$, where $p$ is prime and $\gcd (r,p)=1$, we initiate the study of the algebraic structure of the group ${\mathrm{\Delta }}_q$ generated by the two matrices $$A=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right),\text{ and }{Q}_q=\left(\begin{array}{cc}1& q\\ 0& 1\end{array}\right)\mathrm{.}$$ We introduce the conjecture that ${\mathrm{\Delta }}_{r\mathrm{/}p}={\bar{\mathrm{\Gamma }}}_1^{(p)}(r)$, the congruence subgroup of ${\mathrm{SL}}_2(\mathbb{Z}[\frac{1}{p}])$ 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 $r\le 4$ 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 $r\in \{p-1,p+1,(p+1)\mathrm{/}2\}$. In all these cases, this gives information about the algebraic structure of ${\mathrm{\Delta }}_{r\mathrm{/}p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in ${\mathrm{SL}}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function.