Solving the membership problem for certain subgroups of $S{L}_2(\mathbb{Z})$

Abstract

For positive integers $u$ and $v$, let $L_u=\left[\begin{array}{cc}1& 0\\ u& 1\end{array}\right]$ and $R_v=\left[\begin{array}{cc}1& v\\ 0& 1\end{array}\right]$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$ in $G_{u,v}$ when $u,v\ge 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v>4$. Additionally, we compute $[{\mathcal{G}}_{u,v}:G_{u,v}]$ for $u,v\ge 1$, extending a result of Chorna, Geller, and Shpilrain.

New version extends results to cases when $u=1$ and index computations/proofs are included