Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations

• Main Authors: Han, Sandie, Masuda, Ariane M, Singh, Satyanand, Thiel, Johann
• Format: Journal Article
• Language: English
• Published: 30-08-2019
• Subjects: Mathematics - Group Theory; Mathematics - Number Theory
• DOI: 10.48550/arxiv.1909.00108

Proc. Amer. Math. Soc. 148 (2020), 3775-3786 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
$S_{u,v}$ 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 $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in
$S_{k,k}$ and $G_{k,k}$ when $k\geq 2$ given by Esbelin and Gutan to $S_{u,v}$
when $u,v\geq 2$ and $G_{u,v}$ when $u,v\geq 3$. 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 $b/d$.