On the membership problem for some subgroups of $$SL_2(\mathbf {Z})$$ S L 2 ( Z )

Abstract

For \(\lambda \in \mathbf {Z}, \lambda \ge 2,\) consider the group \(G_{\lambda }\) generated by two matrices \(A_\lambda =\left( \begin{array}{cc} 1 &{} \lambda \\ 0 &{} 1 \\ \end{array}\right) \) and \(B_\lambda =\left( \begin{array}{cc} 1 &{} 0\\ \lambda &{} 1 \\ \end{array}\right) \). We present new algorithms for the membership problem for the groups \(G_{\lambda }\). Let us consider another family of subgroups of \(SL_2(\mathbf {Z})\) defined by $$\begin{aligned} {\mathcal {G}}_{\lambda } = \left\{ \left( \begin{array}{cc} 1+\lambda ^2n_1 &{} \lambda n_2\\ \lambda n_3 &{} 1+\lambda ^2n_4\\ \end{array}\right) \in SL_2(\mathbf {Z}) \mid (n_1, n_2, n_3, n_4) \in \mathbf {Z}^4 \right\} . \end{aligned}$$ Using continued fractions with partial quotients in \(\lambda \mathbf {Z},\) we characterize the matrices of the group \({\mathcal {G}}_{\lambda }\) which belong to \(G_{\lambda }.\) Our results are extensions of the classical result of I. Sanov for \(G_2={\mathcal {G}}_2\).