Non-freeness of certain two-parabolic groups

For each positive rational number q, we study the (non-)freeness of the group G(q) generated by 2x2 matrices a = ( (1,0), (1,1) ) and b = ( (1,q), (0,1) ) in SL(2,Q). We give a computational criterion which allows us to prove that if q=s/r for s≤27 then G(q) is non-free, with the possible exception of s=24. In this latter case, we prove that the set of positive integers r for which G(24/r) is non-free has natural density 1. In the course of the proof, it will follow that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that G_(s/r) is non-free. For the case s > 27, we describe a density estimate. (Joint work with Thomas Koberda)