On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements (2312.11258v2)

Abstract: 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.