Papers › Freeness and S-arithmeticity of rational Möbius groups
Freeness and S-arithmeticity of rational Möbius groups
A. S. Detinko, D. L. Flannery, A. Hulpke
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We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius 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 M\"{o}bius 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.
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