Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

Abstract

For positive integers u and v, let Lu = 1 0 u 1 and Rv = 1 v 0 1 . Let Gu,v be the group generated by Lu and Rv. In a previous paper, the authors determined a characterization of matrices M = a b c d in Gu,v when u, v ≥ 3 in terms of the short continued fraction representation of b/d. We extend this result to the case where u, v ≥ 2 with uv = 4.