On congruence subgroups generated by two parabolic rational matrices

Abstract We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$ . 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 \[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\] We introduce the conjecture that $\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$ , the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$ 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 $\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$ , where $J_2(r)$ denotes the Jordan totient function.