Let , let
and let be the group generated by a and . In this paper, we study the problem of determining when the group is not free for rational. We give a robust computational criterion, which allows us to prove that if for , then is non-free with the possible exception of . In this latter case, we prove that the set of denominators for which is non-free has natural density 1. For a general numerator , we prove that the lower density of denominators for which is non-free has a lower bound
Finally, we show that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.
"Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters." Michigan Math. J. 71 (4) 809 - 833, November 2022. https://doi.org/10.1307/mmj/20205868