Subgroups of $S{L}_2(\mathbb{Z})$ characterized by certain continued fraction representations

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 $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In this paper we expand on a characterization of matrices $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ in $S_{k,k}$ and $G_{k,k}$ when $k\ge 2$ given by Esbelin and Gutan to $S_{u,v}$ when $u,v\ge 2$ and $G_{u,v}$ when $u,v\ge 3$. We give a simple algorithmic way of determining if $M$ is in $G_{u,v}$ using a recursive function and the short continued fraction representation of $b/d$.

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