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

Abstract

We study the freeness problem for subgroups of ${\mathrm{SL}}_2(\mathbb{C})$ 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 ${\mathrm{SL}}_2(\mathbb{Z}[\frac{1}{p}])$ 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 $r\in \{p-1,p+1,(p+1)/2\}$. 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 ${\mathrm{SL}}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function.

arXiv comment19 pages, 1 figure. Comments welcome. This version: fixed a gap in the proof of Theorem 4.2, some references added