Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

Abstract: For positive integers $u$ and $v$, let $L_u=\begin{bmatrix}1 & 0 \u&1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v \ 0 & 1\end{bmatrix}$. 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 $M=\begin{bmatrix}a & c \b&d\end{bmatrix}$ in $G_{u,v}$ when $u,v\geq 3$ 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 $[\mathscr{G}{u,v}\colon G{u,v}]$ for $u,v\geq 1$, extending a result of Chorna, Geller, and Shpilrain.