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

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 $R_v$. The membership problem for $G_{u,v}$ asks the following question: Given a 2-by-2 matrix $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$, is there a relatively straightforward method for determining if $M$ is a member of $G_{u,v}$? In the case where $u=2$ and $v=2$, Sanov was able to show that simply checking some divisibility conditions for $a$, $b$, $c$ and $d$ is enough to make this determination. We answered this question in the case where $u,v\geq 3$ by finding a characterization of matrices $M$ in $G_{u,v}$ in terms of the short continued fraction representation of $\frac{b}{d}$, extending some results of Esbelin and Gutan. By modifying our previous work, we are able to further extend our previous result to the more difficult case where $u,v\geq 2$ with $uv\neq 4$.

This is joint work with Sandie Han, Ariane M. Masuda, and Satyanand Singh.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)