Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters

Let q?C, let a=1011,bq=1q01, and Gq< SL2(C) be the group generated by a bq. In this paper, we study problem of determining when Gq is not free for |q|<4 rational. We give robust computational criterion, which allows us to prove that if q=s/r |s|?27, then non-free with possible exception s=24. latter case, set denominators r?N G24/r has natural density 1. For general numerator s>27, lower Gs/r bound 1?(1?11 s)?n=1?(1?4 s2n?1). Finally, show fixed s, there are arbitrarily long sequences consecutive r such non-free. The proofs some results computer assisted, Mathematica code been provided together suitable documentation.