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, 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 = Z[1/b] of 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.