Deciding Non-Freeness of Rational Möbius Groups

Abstract

We explore a new computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, $\mathbb{Q}$). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, $R$) for a localization $R$ = $\mathbb{Z}$[$\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant SL(2, $R$). Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in SL(2, $R$) that contains $G$.