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

TL;DR

This paper characterizes subgroups of SL_2(Z) generated by specific matrices using continued fraction representations, providing an algorithm to determine membership in these subgroups.

Contribution

It extends previous subgroup characterizations to more general parameters and introduces an algorithm based on continued fractions for membership testing.

Findings

Provides a recursive algorithm for subgroup membership

Extends subgroup characterization to broader parameters

Uses continued fraction representations for efficient computation

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 $S_{u,v}$ 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 $b/d$.