Solving the membership problem for parabolic Möbius monoids

Abstract

For $$\lambda \in {\mathbf {Q}}, \lambda > 0,$$ consider the submonoid $$S_{\lambda }$$ (resp. the subgroup $$G_{\lambda }$$ ) of $$SL_2({\mathbf {Q}})$$ generated by two parabolic matrices $$A_\lambda =\left( {\begin{matrix} 1 &{} \lambda \\ 0 &{} 1 \end{matrix}}\right) $$ and $$B_\lambda =\left( {\begin{matrix} 1 &{} 0\\ \lambda &{} 1 \end{matrix}}\right) $$ . We present new results and algorithms for the membership problem for the monoids $$S_{\lambda }$$ . For $$\lambda \in {\mathbf {N}}^*, $$ let us also consider the families of monoids and subgroups of $$SL_2({\mathbf {Z}})$$ defined by $$\begin{aligned} {\mathscr {S}}_{\lambda }= & {} \left\{ \left( {\begin{matrix} 1+\lambda ^2n_1 &{} \lambda n_2\\ \lambda n_3 &{} 1+\lambda ^2n_4 \end{matrix}}\right) \in SL_2({\mathbf {Z}}) \mid (n_1, n_2, n_3, n_4) \in {\mathbf {N}}^4 \right\} , \\ {\mathscr {G}}_{\lambda }= & {} \left\{ \left( {\begin{matrix} 1+\lambda ^2n_1 &{} \lambda n_2\\ \lambda n_3 &{} 1+\lambda ^2n_4 \end{matrix}}\right) \in SL_2({\mathbf {Z}}) \mid (n_1, n_2, n_3, n_4) \in {\mathbf {Z}}^4 \right\} . \end{aligned}$$ Using continued fractions with partial quotients in $$\lambda {\mathbf {N}},$$ we characterize the matrices of the monoid $${\mathscr {S}}_{\lambda }$$ which belong to $$S_{\lambda }.$$ Our results are analogues for monoids of the classical result for groups of I. Sanov which says that $$G_2={\mathscr {G}}_2$$ .