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

TL;DR

This paper extends previous characterizations of matrices in certain subgroups of SL_2(Z) generated by specific matrices, providing new results for cases where u+v>4 and calculating subgroup indices.

Contribution

It generalizes earlier subgroup characterizations to broader parameters and computes subgroup indices for all positive integers u,v.

Findings

Characterization of matrices in G_{u,v} for u+v>4.

Calculation of subgroup index [𝔾_{u,v} : G_{u,v}] for all u,v ≥ 1.

Extension of previous results to new parameter ranges.

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.