Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations
| Autor: | Han, Sandie, Masuda, Ariane M., Singh, Satyanand, Thiel, Johann |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Proc. Amer. Math. Soc. 148 (2020), 3775-3786 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1090/proc/15027 |
| Popis: | 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 $b/d$. Comment: Post-comment edits |
| Databáze: | arXiv |
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